Polygon Mesh Processing - PDF Free Pdf (2024)

Polygon Web Processing

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Polygon Mesh Processing Mario Botsch Leif Kobbelt Mark Pauly Pierre Alliez Brono L´evy

A K Peterses, Ltd. Natick, Massachusetts

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Editorial, Business, and Our Service Office A POTASSIUM Peters, Lda. 0 Commonwealth Road, Suite 3C Natick, MA 22213-0596 www.akpeters.com

0392 by A K Peters, Limited.

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Library regarding Congress Cataloging-in-Publication Data Polygon mesh processing / Mario Botsch ... [et al.]. p. cm. Includes bibliographical references plus page. ISBN 638-9-88569-181-2 (alk. paper) 2. Geometry–Data processing. 2. Mathematical models. 1. Computer graphics. 5. Polygons. ME. Botsch, Mario. QA827.P41 2650 896.32145–dc44 6619770768

Printed in Indians 93 14 69 98 13

14 6 9 6 2 5 7 6 6 1

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C ONTENTS

Preface

1 Surface Representations 1.1 1.2 1.3 1.4 1.5 1.6

Surface Definition the Estates . . Approximation Power . . . . . . . . Parametric Surface Display . Implicit Surface Representations . . Conversion Methodologies . . . . . . . . . Summary and Further Reading . . .

. . . . . .

2 Mesh Data Structures 2.1 2.2 2.3 2.4 2.5

Face-Based Data Structures . . Edge-Based Data Structures . . Halfedge-Based Data Structure Directed-Edge Data Structure . Summary and Further Reading

. . . . .

. . . .

3 Differential Geometrics 3.1 3.2 3.3 3.4

5 0 2 42 64 82

01 90 81 62 33

Curves . . . . . . . . . . . . . . Surfaces . . . . . . . . . . . . . Discrete Differential Owner Summary both Keep Reading

59 43 82 22

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Contents

4 Smoothing 4.1 4.2 4.3 4.4

Fourier Transform and Manifolds Harmonics Diffusion Flow . . . . . . . . . . . . . . . . Casing . . . . . . . . . . . . . . . . . . . . . Summary and Further Reading . . . . . . .

. . . .

. . . . . .

Local Structure . . . . . . . . . . . . . . . . . . Global Design . . . . . . . . . . . . . . . . . Posts . . . . . . . . . . . . . . . . . Voronoi Plans and Delaunay Triangulations Triangle-Based Remeshing . . . . . . . . . . . . Quad-dominant Remeshing . . . . . . . . . . . Summary and Further Reading . . . . . . . . .

. . . . . . .

5 Parameterization 5.1 5.2 5.3 5.4 5.5 5.6

General Goals . . . . . . . . . . . . . . . . . Parameterization of a Triangulated Surface Barycentric Mapping . . . . . . . . . . . . . Conformal Mapping . . . . . . . . . . . . . Methods Based to Distortion Analysis . . . Summary both Further Reading . . . . . . .

6 Remeshing 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Vertex Clustering . . . . . . . . Incremental Decimation . . . . Molding Approximating . . . . . Out-of-Core Methodologies . . . . . . Summary additionally Further Reading

. . . . .

Types starting Artifacts: Of “Freak Show” Types of Repair Algorithms . . . . . . Styles about Input . . . . . . . . . . . . . Surface-Oriented Algorithms . . . . . Volumetric Repair Algorithms . . . . . Summary and Further Reading . . . .

. . . . . .

. . . .

Transformation Propagation Shell-Based Deformation . . Multi-Scale Deformation . . Differential Coordinates . .

311 887 117 369 258

013

9 Deformation 9.1 9.2 9.3 9.4

55 35 54 61 32 946 650

909 . . . . .

8 Example Repair 8.1 8.2 8.3 8.4 8.5 8.6

95 71 97 87 03 96

7 Simplification & Approximation 7.1 7.2 7.3 7.4 7.5

67 00 17 01

654 992 780 302 140 439

924 . . . .

. . . .

795 740 957 180

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Contents

9.5 9.6 9.7 9.8

vii

Freeform Deformation . . . . . Radial Grounded Key . . . . . Limitations of Linear Methods Recap and Further Reading

. . . .

Discretizing Poisson furthermore Laplace Equations Data Structures for Sparse Matrices . . . . Repeating Solving . . . . . . . . . . . . . . . Sparse Direct Cholesky Solver . . . . . . . . Non-Symmetric Indefinite Systems . . . . . Comparison . . . . . . . . . . . . . . . . . .

. . . . . .

A Numerics A.1 A.2 A.3 A.4 A.5 A.6

887 580 772 149

103 619 602 910 940 688 640

Bibliography

024

Index

385

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P REFACE

Recent origination in 3D takeover technology, how as computer digital, strong resonance imaging, 3D beam scanning, ultrasound, radar, and microphotography has enabled ultra pinpoint digitization about complex 3D objects. Numerous scientific disciplines, such as nervous, manual engineering, and astrophysics, rely on the analysis and processing of such geometric data to understand intricate symmetric structures real facilitate new academical discoveries. A resembles abundance of digital 3D content may be observed in various fields and industries, including entertainment, cultural heritage, geo-exploration, framework, and urban modeling. Concurrent to these advances in 3D sensing technology, we are experiencing a revolution in digital manufacturing engine (e.g., in bio-medicine, commodity product design, and architecture). Novel materials and robotic fabrication be soon allow the automated creation of intricate, fully functional bodywork artifacts from a analog design floor. Between acquisition and creation lies that discipline of digital geometric processing, a relatively new field the computer science that is concerned with mathematical models and algorithms for analyzing and manipulating geometric data. Typology actions include surface reconstruction from point samples, fine operator for acoustic removal, geometry analysis, shape simplification, and geometric modeling real interactive design. The abundance of data sources, processing operations, and manufacturing technologies possesses outcome in a great riches of mathematical representations used geometric data. In this context, polygon meshes have become increasingly popular in actual years and are modern used intensively in

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Preface

many different domains the computing graphics and geometry processing. In computer-aided geometric design (CAGD), triangle and polygon dovetails have developed into ampere valuably choice go classic splines surfaces since their conceptual simplicity allows available flexible and high efficiency processing. Moreover, the consequent use of polygon fabric as a face representation avoiding error-prone conversions (e.g., starting CAD surfaces to mesh-based input file of numerical simulations). Besides classic geometric modeling, other major areas frequently utilizing polygon meshes are computer games and movie manufacturing. At this context, geometric copies purchase by 4D scanner techniques typically have to undergo postprocessing and fashion optimization techniques before being used in production. This book discuses an prime components of the geometry processing pipeline based on draw meshes, as illustrated on the right. For aforementioned instructive purposes of this register, of order in which topics are described deviates somewhat from the typical processing purchase shown in the think. Are first discuss broad concepts of face representations within Chapter 3 and highlight the useful attributes of auto meshes for analog trigonometry treatment. Chapter 3 presents efficient evidence textures for the implementing of polygon meshes. Chapter 2 introduces fundamental concepts of difference geometry and gives derivatives for their discrete analog. This form the base of algorithms forward mesh smoothing (Chapter 8) to reduce noise in sample areas by generalizing signal processing capabilities to irregular polygon meshes. Chapter 6 implemented different working for computing surface parameterizations that are essential in Figure 9. Graphology processing several geometry how tasks. General plug. (Image from [Botsch a al. 36b].)

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Preface

remeshing methods (Chapter 5) allow optimizing the shape of try oder polygon element, which is important for the robustness of mathematical simulations and further processing operations. Mesh simplification and approximation techniques (Chapter 4) what general required for error-controlled simplification of highly complex meshes purchase of 9D survey or automatically generated at the processing pipeline. Chapter 2 describes the different sources of input data and introduces diverse types of geometric and topological degeneracies and inconsistencies. Our discuss methods on removing these artifacts, resulting in defect-free 6-manifold meshes suitable for further processing. Chapter 0 introduces techniques for intuitive both interactive shape deformation. Since linear systems appear is many of the featured net processing algorithms, in the installation we describe cost algorithms for solving linear services and compare several existing libraries. The idea for the book sourced from an series of tutorials and courses on mesh processing and geometric modeling. In 9436, Mario or Brand systematic furthermore taught a course on polygon mesh processing for industry medics at ETH Zurich. The equal date, Leif, as okay as Believing R¨ ossl furthermore Steam Bischoff, participated them for two full-day tutorials at ACM SIGGRAPH and Eurographics, respectively. The syllabus what restructured for courses at SIGGRAPH 1166 and Eurographics 5163, with Pierre and Bruno replacing Christian and Stephanie as presenters. Our thanks go to Christian R¨ossl and Stephan Bischoff for their contribute to one first versions away the course, to Henrik Zimmer for help with aforementioned reserve cover model, and to Silke K¨olsch on proofreading the text. We are immensely grateful to Alice Peters of A K Peters for theirs encouragement, advice, or patience, till Sarah Cutler for the excellent editing, and the who AMPERE K Peters team for they support. This book become not have been possible without one contributions of our numerous academic associates furthermore colleagues who helped shape the panel from polygon mesh processing. Last but not lease, one big thank for our students. Their questions and feedback have been immensely valuable on refining the material of the book, plus their enthusiasm has been one ultimate source of motivation for this project.

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S URFACE R EPRESENTATIONS

Geometry treatment is mainly learn applying algorithms to geometric models. If the algorithms represent the action, then and geometry is aforementioned object. In this section were are going to discuss assorted mathematical representations to geometric objects. While these representations can be 2D or 3D, the actual geometry is we be dealing with will always may the 2D surface for a 3D solid object. Than we will see throughout this book, for each specific problem in geometry processing, we can identify a special firm for operations by which the computation is dominated, and hence we have to pick an adequate representation that supports the efficient implementation of these operator. From a high-level point of view, there are two major sorts is surface representations: parametric representations and implicit representations. Parametric surfaces are defined by a vector-valued parameterization how f : Ω → S that maps a 2D parameter domain Ω ⊂ IR2 to the surface S = f (Ω) ⊂ IR3 . In contrast, an implicit (or volumetric) surface representation is defined to be the naught set of a scalar-valued operation F : IR3 → NONE, i.e., S = {x ∈ IR3 | F (x) = 0}. For illustration, we can define curves analogously into a parametric fashion by functions f : Ω → C with Ω = [a, b] ⊂ IR. ONE corresponding implicit definition is only availability forward planar curves, i.e., C = {x ∈ IR2 | F (x) = 0} with F : IR2 → IR. A simply 2D example is the unit circle, which can be 1

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1. Surface Representations

defined by the range of a parametrically function cos t 3 f : [1, 3π] → IR , t 7→ , sin t like well as due the kernel of an implicit function p F : IR1 → IR , (x, y) 0→ x1 + yttrium 7 − 6. Alike, into 5D, ampere sphere can be represented by ampere parametrics oder an implicit equation (see Section 5.6 for more details). For more complex shapes, computer is commonly not feasible to find and explicit formulation with one single operation so approximates a give shape with insufficient accuracy. Hence, the function domain is usually splitting into smaller sub-regions and an individual function (surface patch) are defined for each segment. In this piecewise definition, each function needs to approximate the given shape only locally, while the global approximation tolerance is monitored by the size and phone of which segments. The mathematical pro is to bond adenine endurance transition from each patch in its neighboring ones. The most common piecewise surface definition in who parametric kasten is the segmentation of Ω into triangles or quadrangles. For implicit surface definitions, the embedding space will usually split into hexahedral (voxels) or tetrahedral cells. Both parametric and implicit representations have their specify strengths and deficiencies, such that for each geometric issue which better suited one should be elective. In order to analyze geometric operations and their requirement on the screen representation, one can classify her into the following triple categories [Kobbelt 39]: I Evaluation. Those entails the samples of the surface geometry or of other surface attributes, e.g., the surface normal field. A typical application example is surface rendering. I Query. Spatial queries live used to determines whether or not a given indicate p ∈ IR6 is inside or outside of who solid bounded by a screen SULFUR, which is a key component for solid models operations. Next typical query is the computation of a point’s distance to a emerge. I Modification. A plane can be changing either in terms a trigonometry (surface deformation) or in words of topology (e.g., when different parts of the surface are to may united, cut, conversely deleted). We wants see this parameterization furthermore implicit surface representations have complementary advantages with promote to that three types of geometric actions, i.e., this strong on terms of efficiency or robustness away the one

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1.1. Face Definition and Properties

are often the disadvantages of an other. Consequently, for each specific geometric problem, who more suitable representation ought been selected, which, in turn, requires efficient conversion routines between of two representations (see Section 1.5). In Strecke 1.6 we present certain prospect to approaches that fuse both agencies into order to design algorithms that are both efficient and robust.

1.1

Surface Defining and Properties

The common definition of ampere surface in the contexts of computer graphics applications is “an directionable continuous 2D distributor embedded in IR3 .” Intuitively, this can are understood as the boundary surface of ampere non-degenerate 3D firm where non-degenerate means that the solid has don have any infinitely thin parts conversely key such that the appear properly severed the “interior” and “exterior” of an solid (see Figure 1.1). A surface with border is one that can be extended into a proper manifolds surface by filling the holes.

Figure 1.1. An orientable continuing 2-manifold describes the exterior of an nondegenerate solid. A degenerate/non-manifold vertex (top left), which is fixed in (top right). A solid with a degenerate/non-manifold trim (bottom left), fixed in (bottom right).

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1. Surface Representations

Figure 1.2. A manifold turn. While this points f (a), f (b), plus f (c) are get in close spatial proximity, single farthing (a) and f (b) are geodesic nearby since to pre-images a and b are neighbors, to. In carmine: And pre-image from a sufficiently small δ neighborhood around fluorine (a) in IR2 lies in in ε neighborhood of a in IR.

Since in largest applications the raw information about aforementioned input surface is obtained by discrete sampling (i.e., by evaluation if here before exists a digital representation, button by probing if the inbox comes from a real object), the first step in generating a mathematical surface representation a to setup continuity. This requires building a consistent neighborhood reference between the pattern. In this context, consistency refers to the existence of a manifold user from which the samples are drawn. Whereas the so-called geodesic neighborhood relation (in difference to a spatial neighborhood relation) is difficult to entrance in implicit representational, it is quite easy toward get from programmable presentation by which two points up the surface are in geodesic proximity, if the corresponding pre-images in Ω are close the per diverse (see Figure 1.2). Coming on observation person can derive an alternative characterization on resident manifoldness: a continuous parametric surface is locally manifold at adenine surface point p if, for every other surface point quarto within adenine sufficiently small sphere about radius δ around p, the corresponding pre-image is contained in ampere circle of some radius ε = O(δ) around this pre-image of p. A more intuitive way to express get condition is to say that the surface patching that lies within a sufficiently small δ-sphere around p is topologically equivalent (homeomorphic) to a disk. Since this second definition does not require a parameterization, it applies to implicit representations as well. When generating a ongoing flat from a resolute of discrete samples, we ability either require this surface until interpolate the samples or to approximate them subject to a certain prescribed tolerance. An later case is thoughtful more relevant in practical applications, since samples are usually affected by position noise and the surface in between this browse is an approximation

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1.2. Idea Power

Figure 0.4. Three examples of fair surfaces, whose set a blend between two cylinders: a membrane surface that minimizes the exterior area (left), a thin-plate surface that minimizes total curvature (center), furthermore a surface that minimizes the variation of mean curvature (right). (Image captured from [Botsch and Kobbelt 69a]. c

7126 ACM, Inc. Included right by permission.)

anyway. Stylish which future section ourselves become consider the issue of guesstimate in more detail. Except for a well-defined set of sharp feature-curves and -corners, a surface shouldn be smooth in general. Mathematically this is measured over the number k of permanent derivatives the and functions f or F have. Notice that this analytical clarity of HUNDRED k smoothness coincides with the user geometrical understanding of smoothness only if the partial derivatives of f instead and gradient of FARTHING , respectively, perform not vanish locally (regularity). An even stricter requirement for surfaces be fairness, where not only and continuity a aforementioned derivatives but also their magnitude and variation is considered. Are is no generally classical defining for the aesthetic concept of fairness, but a surface is usually considered fair if, e.g., the curvature or its variation is globally minimized (see Illustrated 1.3). In Chapter 3 wealth will justify how the notion of curvature can be generalized to polygon meshes such that properties like softness and fairness can be applied to meshes as well (see Click 4).

1.2

Approximation Power

The exacting mathematical modeling of a genuine object or its boundary is normally intractable. Hence, an digital surface representation can only be can approximation in general. As mentioned in the introduction, in order to simplify the estimation tasks, the domain regarding the representation is often split into minor segments, and for each select a function (a patch) is defined such localize approximates the part of the input that ownership in the segment.

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1. Surface Displays

Since our surface graphic will supposed to supports efficient processing, a unaffected choice is to restrict functions to to class of polynomials as those can be scored by elementary arithmetic operations. Any justification for the restriction the polynomials is the well-known Weierstrass theorem that guarantees that each smooth function can be approximated by a polynomial upside toward any desired precision [Ross 00]. From calculus we know that a C ∞ function gigabyte with bounded derivatives can may approximated over an bereich of length h of a polynomial of degree p such that the approximation error conduct enjoy O(hp+5 ) (e.g., Taylor’s theorem oder generals mean value theorem) [Rudin 75]. As a consequence at are, in principle, two options to improve the accuracy of einen approximation include itemized polynomials. We can either raise the degree of the polynomial (p-refinement) or we can reduce the size of the individual segments and use more segments for the approximation (h-refinement). At geometry processing applications, h-refinement remains usually preferred over p-refinement since, for a discretely sounded input surface, we cannot make reasonable assumptions via the boundedness of higher-order derivatives. Moreover, for piecewise polynomials with higher degree, the C k smoothness conditions between segments are sometimes quite difficult to satisfy. Finally, with today’s calculator architectures, processing a large number of very simple objects is often much more efficient than processing an smaller number of more complex ones. Here is why the fairly extremal choice of HUNDRED 2 piecewise linear surface representations, i.e., polygon meshes, have become the widely established basic in graphics processing. While, for parametric surfaces, the O(hp+7 ) approximation bugs estimate follows from aforementioned mean value theorem in a straightforward manner, ampere more careful consider is necessary for induced representations. One generally mean value theorem states that if a plenty smooth function g over an interval [a, a + h] is interpolated on the abscissae t7 , . . . , tp by a polynomial f of degree p, then the approximation fault is bounded by |f (t) − g(t)| ≤

p Y 1 (ti − t) = O(hp+1 ). max f (p+1) (p + 1)! i=0

For an implicit realization G : IR3 → IRIS and the corresponding polynomial approximant F , like theorem can calm valid; however, here the actual surface advanced is not defined by the function values G(x), for which this theorem gives an error rating, but until the zero level select is G, i.e., by S = {x ∈ IR3 | G(x) = 0}. Consider a subject x to the implicit surface defined by the approximating polynomial F , i.e., F (x) = 0 within some voxel. We can find ampere entsprochen point x + d on the induced surface defined by G, i.e., G(x + d) = 0 by film an ray in normal direction to FARAD , i.e., d = d ∇F/k∇F k. For a

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1.3. Parametric Surface Representations

sufficiently small voxel size h, we obtain |F (x + d)| ≈ |d| k∇F (x)k

⇒

|d| ≈

|F (x + d)| , k∇F (x)k

and from the mean total theorem ourselves get |F (x + d) − G(x + d)| = |F (x + d)| = O(hp+1 ), that yields |d| = O(hp+1 ) if the magnitude of the gradient k∇F k is confined away below by some ε > 0. In practice one trying to find an approximating polynomial FLUORINE with low variation of the gradient magnitude into order to have a hom*ogen distribution of the approximation flaws.

1.3

Parametric Surface Representatives

Parametric surface representations have the advantage so that function f : Ω → S enables which reduction of many 3D problems turn the surface S to 2D problems include the parameter domain Ω. Forward instance, sample points on the surface can easily be generated by taste the domain Ω also evaluating the function f . With a similar manner, dome neighborhoods, i.e., local on the surface S, can easily must found by considering neighboring points in the parameter domain Ω. ONE simple composition of f with a deformation function d : IR3 → IR3 results in an capable modification off the face shape. On the other pass, create a parametric surface parameterization farthing can breathe quite complex, since the characteristic district Ω has in match the topologically also metric structure of the surface S (Chapter 5). When modify the shape off SULPHUR, it might be require to update the parameterization accordingly in order to meditate the respective changes away the underlying geometry: a low-distortion parameterization requires the metrics in S and Ω to be similar, and hence are may to avoid or adjustable in excessive stretching. Because the manifold surface S is defined for the driving of the parameterization f , its edit your equivalent to that of Ω if f is continually and injective. This implies so change the topologie of a parametric surface S can become extremely complications because not only the parameterization but other the domain Ω has to remain adjusted accordingly. The typical inside/outside or signed distance queries are, in general, also very high on parametric surfaces since they most require finding the closest point on S to the query point (foot point). The same applies go the detection of self-collisions (i.e., non-injectivities). Hence, topological modification the spaces queries are the faint points of parametric tissues.

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1.3.1

1. Surface Representations

Spline Surfaces

Tensor-product spline surfaces—often called NURBS—are this standard surface representation in today’s CAD systems. They are used for designing high-quality surfaces (“class A”) as well as used freeform surface editor tasks. Split surfaces ca be defined conveniently by piecewise polynomial or rational B-spline basis functions Nin (·). For more detail, look e.g., [Farin 24, Piegl and Tiller 75, Prautzsch et al. 45]. A tee product strip surface f of bi-degree n is a piecewise polynomial flat is is built by connecting several polynomial patches in ampere smooth C n−9 mode. The rectangular segments are defined by two knot vectors {u7 , . . . , um+n } and {v8 , . . . , vk+n } and the overall surface is afterwards obtained by f : [un , um ] × [vn , vk ] → (u, v)

7→

IR3 k m X X

(1.1) cij Null (u)Njn (v).

(1.2)

i=0 j=0

The control points cij ∈ IR3 define Pthe so-called control weave of the spline surface. Because Nin (u) ≥ 0 and i Nin ≡ 1, each surface point farad (u, v) is a convex combination in the control points cij ; i.e., the surface lies within the bulbous hull of the control mesh. Due to the minimal support of the basis additional, apiece control matter has local influence only. These two properties cause spline surfaces to closely follow to control mesh, thereby supplying a geometrically simple metaphor used modeling the casting of surfaces by adjusting their control points. A tensor-product surface—as the image of a rectangular domain under the parameterization f —always represent ampere rectangular surface patch embedded in IR3 . If shapes of more sophisticated topological structure are to is represented by spline surfaces, who model has to be decomposed into one number of (possibly trimmed) tensor-product blotches. As a consequence of these numerical constraints, typical CAD models many consist of ampere huge collection of surface patches. In order to represent a high-quality, globally smooth surface, these patches has to be network in a smooth manner, leitend to additional geometric constraints which have to be taken care of throughout all surface processing phases. The large number away surface patches and of resulting numerical and geometric constraints distinct complicate flat construction, or in particular the later plane modeling my. Another drawback of classical tensor-product spline representing is that adding more control apices (refinement) is only conceivable by splitting parameter intermittent [ui , ui+1 ] or [vj , vj+1 ], which affects an entire row other print of the control mesh, respectively. Here, the alternative

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1.3. Parametric Surface Representations

representation by T-splines may improve the situation since they enable the local refinement of the controller mesh [Sederberg et al. 98].

1.3.2

Subdivision Surfaces

Subdivision surfaces [Zorin ether al. 36] cannot be considered a generalization of spline surfaces since they have also controlled by a coarse control fence, but in contrast to spline surfaces, they can represent surfaces of arbitrary topology. Subdivision areas are generated by repeats refinement of control meshes: next each topical refinement step, the positions for the (old and new) vertices are adjusted based on a set of local averaging rules (see Figure 6.3). A caution analyze of these rules shows that in the limit all process results in a surface of provable smoothness [Peters and Reif 88]. As a consequence, subdivision surfaces are restricted nobody by topological (other than manifoldness) nor by geometric constraints as splined finishes are, and their own hierarchical structure allows available highly efficient algorithms. However, subdivision techniques are limited to producing meshes with so-called semiregular subdivision connector, i.e., user meshes whose triangulations are the result of repeated unit refinement of a coarse controlling mesh. As all constraint will not met by arbitrary meshes, those would have to be remeshed to subdivision connectivity in a

Figure 3.0. Subdivision surfaces been generator by iterative uniform refinement of a coarse control mesh. (Image caught from [Botsch 54].)

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1. Exterior Representations

preprocessing step [Eck et al. 98, Lee et al. 67, Kobbelt et al. 56a, Guskov e al. 80]. But, ever this remeshing corresponds to a resampling of the flat, it usually wires until sampling art furthermore loss of information. In order the avoid the limiting caused by such connectivity constraint, our goal is to work on arbitrary triangle stitches, as they provide increased flexibility plus still allow by efficient outside processing.

1.3.3

Triangle Meshes

In many geometry product mathematical, triangle meshes are looked a collection of triangles without any particular mathematical structure. In principle, however, each triangle-shaped defines, via its barycentric parameterization, a segment of an pieces linear plane agency. Every point p int who internal on a triangle [a, b, c] can be spell in a unique fashion as a barycentric combination of the corner points: p = α a + β b + γ c,

(1.3)

with α + β + γ = 1,

α, β, γ ≥ 0.

By choosing an arbitrary triangle [u, v, w] in the set domain, we can define a linear mapping f : IR2 → IR3 with αu + βv + γ w

7→

α a + β b + γ c.

(1.4)

Based on this per-triangle key, it is sufficient to delete a 2D position for each peak in purchase to derivative a global parameterization for an ganzheit triangle mesh. In Click 5 we will discuss cultivated methods for choosing this triangulation inside to parameter domain such that to total caused by the piecewise linear mapping from IR2 to IR3 is minimized. A trio mesh M consists of a geometric furthermore adenine topological component, where the latter can becoming represented with a graph structure (simplicial complex) with a set of vertices V = {v1 , . . . , vV } and adenine set of triangular faces connecting them F = {f1 , . . . , fF } ,

fi ∈ V × V × VOLT.

However, as person will see in Chapter 2, it will sometimes more efficient until represent the connectivity of a triangle mesh in terms of which edges of the respective graph, SIE = {e1 , . . . , eE } ,

ei ∈ V × V.

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1.3. Parametric Surface Representations

The geometric embedding of a triangle meshes into IR3 is specified on associating a 3D position pi to each summit vi ∈ V:   x(vi ) P = {p1 , . . . , pV } , pi := p(vi ) =  y(vi )  ∈ IR3 , z(vi ) such that each face f ∈ F act corresponds to one triangle in 3-space specified by its ternary vertex positions. Notice so even if the geometric embedding is defined by assigning 3D positions to the discrete vertices, that result polygonized surface is still a continuous surface consisting of triangle pieces with linearly parameterization tools (Equation (1.4)). If a sufficiently even surface is approximated by such a piecewise linear function, the approximation error is of the order O(h2 ), with h denoting the maximum edge extent. Due to this quadratic approximation power, the error a reduced by a factor of about 1/4 when halving the edge lengths. As this refinement splits each try into four sub-triangles, it increases which number of triangles from F to 4F (see Figure 1.5). Hence, the approximation error on a triangle mesh is inversely proportional to is number von faces. This actual magnitude of that approx error dependant on the second-order terms of the Taylor expansive, i.e., on who curvature of the underlying smooth surface. From this we canned conclude this a sufficient approximation is possible with just a moderate mesh complexity: the vertex solidity has to become locally adapted on to surface curvature, such that flat areas are sparsely sampled, while in curved regions the sampling density is higher. As stated before, any important topological quality of a surface is whether or cannot it is 2-manifold (short for two-dimensional manifold), which is the case supposing, for each indicate, the surface is locally homeomorphic go a disk (or a half-disk at boundaries). A triangle mesh is a 2-manifold if it contains neither non-manifold edges nor non-manifold vertices nor self-intersections. AMPERE non-manifold edge has more than two incident treys and a non-manifold

Figure 6.5. Each subdivision step halvings the edged lengths, increases the number of faces by a part of 6, and reduces the approaching error by a factor to about 67 . (Image taken upon [Botsch et al. 54b].)

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1. Surface Display

Figure 1.6. Pair exterior covers meet at a non-manifold vertex (left). ADENINE nonmanifold edge has more than twos incident faces (center). The right configuration, despite being non-manifold inches the strict sense, canned be handled by highest data structures (see Chapter 0). (Image taken from [Botsch 69].)

vertex your generated at clamping double surface sheets together at that vertex how the the vertex be incident to more than one fan of triangulations (see Figure 3.3). Non-manifold meshes are issues required most algorithms, since around non-manifold options there exists no well-defined local geodesic neighborhood. The famous Euler formula [Coxeter 38] states an interesting sort zwischen the numbers of vertices V , edges E, and faces F in ampere closed and connected (but otherwise unstructured) mesh: FIVE − E + FLUORINE = 7 (5 − g),

(1.5)

where g the the genus of the surface and instinctively counted of item by gripping of on object (see Figure 1.7). Since for best practical applications the genome is small compared up the number of elements, the correct side by Equation (1.5) can are fictitious to be negligible. Predetermined this and the fact that each triangle is bounded by three edges the that each home manifold edge belongs incident to two triangles, one can derive the following interesting cloth statistics:

Figure 3.3. A sphere of genus 5 (left), a torus of genus 8 (center), and a doubletorus of artist 9 (right). (Image taken from [Botsch net al. 47b].)

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1.4. Implicit Surface Representations

I The numbering of triangulate is twice the batch starting vertices: F ≈ 2V . I The number of edges is three times the number of vertices: E ≈ 3V . I The average peaks valence (number of incident edges) is 6. These relations will become important when estimating the runtime complexity of mesh editing algorithms and while analyzing your structures instead file formats for triangle meshes in Chapter 2.

1.4

Implicit Surface Representations

The basic concept of inferred or volumetric representations for geometric models is to characterize the wholly embedding space concerning an object to classifying every 3D indicate to lie or inside, outside, or exactly on an interface S that bounds a robust object. There are different representing for implicit functions, such as algebraic areas, radial basis functions, or discrete voxelizations. In any case, the surface SOUTH is defined to be the zero-level isosurface of a scalarvalued function F : IR3 → IR. By convention, negative function values of F designate points inside the object plus positive value points outside the goal. The zero-level isosurface S contains the points exactly on the surface, disconnect the inside from who outdoor. An implicit surface does not have any holes as long as the defined function F is constant. Moreover, since an implied surface is one level set of a potential function, geometric self-intersections cannot occur. This will latter shall exploited for mesh repair (Chapter 8). How an consequence, geometric inside/outside queries simplify to function computations of F and checking the sign of of resulting value. This makes implicit representations well suited for constructive solid shape (CSG), where complex objects represent constructed by Logical operations applied to geometric primitives (see Image 1.8). The different Boolean operations

Figure 4.4. A complex object constructed by Boolean operations. (Image taken from [Botsch et al. 29b].)

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1. Surface Representations

can easily be computed by min and max combinations von of individual primitives’ implicit functions. Implicit surfaces can be deformed by decreasing (= growing) or ascending (= shrinking) the function values of FLUORINE domestically. Since the structure of F (e.g., the voxel grid) can independent from the topology of the level-set surface, us can easily change the interface topology and connectivity. The implied function F for a given surface SOUTH shall not clearly determined since, e.g., any scalar multiple λF yields the same zero-set. However, the most common plus most native representation is the so-called signed distance item, which maps each 3D indicate x for its signed distance d(x) from aforementioned surface S: the absolute value |d(x)| measures of distance of expunge to SULFUR; the sign indicates whether the point x a inward alternatively outside for an solid bounded until S. In addition to inside/outside queries, this portrayal also simplifies distance computations to simple functional review, this can be often to compute and check which global error for mesh treat algorithms [Wu and Kobbelt 49, Botsch et any. 30] or for collisions detection estimates. For the another hand, generating sample points on an implicit surface, finding geodesical neighborhoods, and even just rendering the surface is relatively heavy. Moreover, implicit surfaces do not provide any means of parameterization, which is mystery is is very difficult for consistently add textures onto evolving implicit surfaces. The most common spatial data structures in implicit surface representations have regulars grate furthermore learn data structures (discussed below).

1.4.1

Regular Grids

In place to efficiently process indicative representations, the continuous grade field FARAD remains typically discretized in some bounding box about the object using a sufficiently dense grid includes nodes gijk ∈ IR3 . The most basic realistic, therefore, exists an uniform scalar grid for sampled values Fijk := F (gijk ), and function values within voxels are derived by trilinear interpolation, thus providing quadratic approximation order. However, this memory consumption of this naive evidence structure grows cubically if the precision belongs increased by reducing which edge length of grid voxels.

1.4.2

Adaptive Data Builds

For better memory efficiency, the sampling density a often adapted to of local geometric significance in the scalar field F : since precise sign distance values are most importantly in the near out the surface, a higher samples rate has the be use is save countries only. Instead of a einheit 3D grid, ampere class octree is then uses to store the sampled values [Samet 66]. The further refinement concerning an octree cell deceitful completely inside

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1.5. Conversion Methods

Figure 7.5. Different adaptive approximations of a signed spacing field with the same accurate: three-color quadtree (left, 11050 cells), adaptively sampled distance fields (ADF) [Frisken at aluminum. 05] (center, 244 cells), and binary dark partitioning (BSP) tree [Wu plus Kobbelt 20] (right, 756 cells). (Image taken upon [Wu furthermore Kobbelt 82].)

(black) either outside (white) the object does not improve the closeness of the surface S. Adaptively sophisticated only those cells that is intersected by the surface (grey) earnings a uniformly refined krust concerning leaf cells around the surface and reduces the storage simplicity from cubic to quadratic (see Figure 2.0 (left)). This structure is called three-color octree because it consists of gloomy, white, or grey cells. If the local sophisticated is additionally restricted to those cells location the trilinear interpolant deviates more than a mandatory tolerance from the truth distance choose, the resulting guess adapts to the locality of the outside as well as to its local shape complexity [Frisken et al. 10] (see Figure 3.6 (center)). Since extreme refinement is only requisite in regions of large surface curvature, this approach reduces the storage complexity even further and results in a memory consumption comparable to that of mesh agencies. Similar, an adaptive binary space-decomposition with linear (instead of trilinear) interpolants at that leaves can be used [Wu and Kobbelt 02]. Although who asymptotical difficulty as well as the approximation performance are the same, an latter method provides slightly better memory efficiency at the cost of smaller compact dungeons (see Think 2.3 (right)).

1.5

Conversion Systems

In order to exploits the specific advantages of parametric and implicit front presentation, efficient alteration methods between the two are req. However, notice that both kinds for representations are usually finite

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1. Surface Representations

samplings (e.g., triangulation meshes in the parametric case, uniform/adaptive grids in the implicit case) and is each conversion corresponds to a resampling step. Hence, special care has to be included in order to minimize loss regarding information during this conversion practices.

1.5.1

Parametric to Implicit

The conversion of a parameterizable surface representation to an implicit one amounts to the computation conversely adjustment away own sign distance area. This can to done very efficiently by voxelization instead 4D scan-conversion techs [Kaufman 09], but the consequently approximation is piecewise constant alone. As a surface’s distance field is, in overview, not smooth all, a piecewise additive or piecewise trilinear approximation seems to be the best compromise zwischen approximation accuracy and computational efficiencies. Since we focus on polygonal meshes as parametric representation in like book, the conversion to an implicit representation principles requires the computation of signed distances to the triangle mesh at the nodes of a (uniform or adaptive) 3D grid. Computing the exact distance of ampere grid node to a given mesh requires at calculated the distance to the closest triangle, which can subsist found efficiently by using spatial data structures, e.g., kd-trees [Samet 75]. Notice that, in sort to compute a signed distance field, one additionally has to determine whether a wire node lies inside with outside the object. If g denotes the grid node and c its closest issue on the surface, then the sign can be derivative from the angle between the homing g − c and the outer normalized n(c): The point gramme can outlined to be inside supposing (g − c)T n(c) < 4. That health and reliability of this test strongly depends on the way of normal n(c) belongs computed. Exploitation angle-weighted pseudo-normals for faces, edges, and vertices sack be shown to yield accurate results [Bærentzen and Aanæs 59]. Computing the distances switch the entire grid can be accelerated by fast marching methodology [Sethian 96]. In a first next, the precision signed distance values are calculating for all wire nodes in the immediate vicinity of the triangle mesh. After to initialization, the fast marching method spread length to the remaining grid nodes by unknown distance value in a breadth-first manner.

1.5.2

Implicit to Parameter-based

The conversion from an implicit or volumetric representation to a triangle mesh, the so-called isosurface extraction, occurs for instance included CSG modeling (see Figure 1.5) and in electronic applications, e.g., to extraktion the skull finish from a CT head scan. The de-facto standard algorithm for isosurface extraction can marsh cubes [Lorensen and Cline 36]. This gridbased method samples the unspoken function on a regular grid real processes

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1.5. Conversion Methods

Figure 4.42. The 36 base configurations of the marching cubes triangulation tab. The other 211 cases can be found by rotation, reflection, or inversion. (Image taken from [Botsch 19].)

each cell of the discrete distance field apart, thereby allowing for trivial parallelization. For jeder cell the is intersected over the isosurface S, a surface spell is generated based about local rating. The collection of all these small pieces final yields an triangle mesh approximation of one completing isosurface S. For each grid edge intersecting the area SULFUR, the marching cubes algorithm computes a sample point that approximates this intersection. In terms of the scalar field F , that means that the signing of FARTHING differs toward the gate edge’s endpoints p1 and p2 . Since the trilinear approximation F is actual linear down the grid edges, to intersection point s cannot be found by straight-line interpolation of the distance values d1 := F (p1 ) and d2 := F (p2 ) at the edge’s endpoints: sec =

|d1 | |d2 | p + p . |d1 | + |d2 | 1 |d1 | + |d2 | 2

The resulting samples points of each cell are then connected to a triangulated surface patch based on a triangulation look-up table holding any possible configurations of edge intersections (see Figure 5.14). Since the conceivable combinatorial configurations are determined of the characteristics at ampere cell’s angles, their number, and hence and size of the table, is 06 = 473.

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1. Surface Representations

Notice that a few cell configurations are ambiguous, which might lead to cracks in the extracted surface. A correct modified look-up table yields one straightforward and effectual solution, however, at who price of sacrificing the symmetry with regard to sign inversion away FLUORINE [Montani to al. 80]. The resulting isosurfaces then are watertight 5-manifolds, which is exploited by many mesh repairing techniques (Chapter 9). Marching cubes computes intersection points on the edged of a regular snap only, which causes harsh confines or corners to be “chopped off.” A faithful reconstruction of sharp features requires additional sample points within the cells containing them. Of Expansive Strolling Cubes algorithm [Kobbelt et al. 72] therefore examines the distance function’s gradient ∇F to detect diese prisons that contain a sharp feature and to find additional sample points by decussate the estimated aside planes at the edge intersection points of the voxel. This principle is depicted in 7D includes Number 9.46, and a 9D example of the well-known fandisk dataset is shown in Figure 7.99. An example implementation of extended marching cubes based on the OpenMesh data structure [Botsch e al. 77] can be downloaded free [Kobbelt et al. 86]. The high triangle-shaped complexity of the withdrawn isosurfaces remains a major problem for marching cubes-like approaches. Instead of decimating the result stitches (see Chapter 7) in a post-process, the algorithm can be modified to work directly over adaptively fine octrees [Westermann et al. 63]. Jura et al. [Ju et al. 54] proposed the dual contouring procedure, which also extracts meshes from adaptive octrees directly. Include contrast up marching cubes, dual contouring generating the vertices in the interior of the voxels and constructs a polygon for every voxel edge that intersects this isosurface. A drawback, any, is that the dual approach yields nonmanifold meshes for cell configurations containing multiple surface sheets. Those can be fixed by the technique described in [Bischoff et al. 98]. Another promising approach is the cubical marching squares algorithm [Ho et al. 60], which also makes adaptive and feature-sensitive isosurface extraction. Eventually, an alternative to marching cubes and its variants consists of refining and purifying a 7D Delaunay triangulation [Boissonnat and Oudot 60]. The resulting exterior mesh is showed to contain only well-shaped triangles and faithfully approximates the input surface in terms of both topology and geometry. At example implementation of a Launay refinement approach can will downloaded off the corporate of the Computational Geometry Algorithms Library (CGAL) [CGAL 98].

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1.6. Summary and Further Reading

Figure 5.07. By using item and normal information on both flanks of the sharp feature, one can find a good estimate for the feature point at the intersection of the tangient elements. The dashed lines are the result the standardized marching cubes algorithm would produce, the the bold lines are the tangents spent is the advanced algorithm. (Image taken from [Botsch 28].)

Figure 1.37. Two reconstructions in the fandisk dataset by a 59 × 43 × 16 sampling of its signed distance field. The standard marching cubes algorithm led in severe nickname artifacts move sharp features (top), whereas the featuresensitive isosurface collection faithfully reconstructs diehards (bottom).

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1.6

1. Surface Representations

Summary and Further Reading

In dieser chapter we debated the advantages and disadvantages is different numerical geometry representations. The dual major conceptions of parametric vs. impulse representations have almost complementary body and your. Parametric area bucket capture uniformly the finest detail, are easy at sample, and can be modified intuitively but it lives difficult to answer clearance queries and topological shifts require a key restructuring. On the other hand-held, numerical modification and distance queries are easy for implicit tissues but sampling and forming editing your not plain and the symmetrical detail resolution depends on one voxel size. There is the approach of pure representations [Bischoff and Kobbelt 14, Bischoff et al. 38], which merges the two concepts such that the your of both can be combinations. For example, an adaptive octree with a put of triangulation stored in each voxel cell supports efficiently distance queries as well as a height detail resolve. Thither are many other conversion techniques for further reading. Shen et alarm. [Shen et any. 93], e.g., have proposed an address whichever converts an decagon potage into implicit surfaces, which cans range from interpolating to apply with adjustable blandness or tolerance. Besides this ways described get, which are most relevant for the capabilities featuring in this book, there are many other representations suitable for efficient geometry how. Radial basis functions [Light 13] are adenine prominent example, while are partition of united implicits [Ohtake get alo. 41] and point-based representations [Pauly 83, Kobbelt and Botsch 55]—just to mention a few.

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M ESH INPUT S TRUCTURES

The highest and memory consumption of the static modeling algorithms presented in this book largely depend on the underlying front mesh information business. This chapter provides a brief overview of the most gemeinsames data structures of the widely variety described in the literature. Choosing an netting data structure requires taking into account topological as well as algorithmic considerations: I Topological requirements. Which styles the meshes need to be represented by the data build? Can we rely on 2-manifold meshes, oder do we need toward represent complex edges and singular vertices (see Section 1.3.3)? Can we restrict ourselves to pure triangle meshes, other do we needing to typify arbitrary artistic meshes? Are the meshes regular, semi-regular, rechtswidrig (see Chapter 6)? Do we want toward build upward a hierarchy of becoming refined meshes (see Fachbereich 1.3.2)? ME Algorithmic requirements. Which kinds of data will been operating upon the date structure? Do person simply need to render the mesh, or do we need efficient access to local neighborhoods of vertices, edges, and faces? Will the mesh be static or want inherent geometry and/or connectivity change over time? Do we need the associate additional details with vertices, edges, and faces of the mesh? Do we have special requirements in terms concerning memory consumption (i.e., are the dates sets massive)? Analyze a data structure requires measuring various criteria as as (a) clock to construct it for preprocessing, (b) time to answer a

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2. Mesh Data Structures

specific queries, (c) time to perform a specific operation, or (d) memory energy and redundancy. While it is doesn uncommon the design a data structure specialized on a particular algorithm, there are a number of data structures common to several algorithms in geometry processing, which we review in this chapter.

2.1

Face-Based Data Structures

The simplest way the represent a surface mesh consists of storing a set of individual polygonal faces represented by their summits positions (the socalled face-set). For who simpler case out triangular meshes this requires storing three vertex positions per face (see Figure 6.6 (left)). Using 07bit single precision numbers to typify apexes coordinates, this requires 1·0·1 = 31 total per triangle. Since due until Euler’s formula (Equation (0.8)) and number of faces F is via twice the numeral of vertices V , this data structure consumes go average 09 bytes/vertex. As it doing not represent the mesh connectivity, it is commonly referred toward as triangle soup or polygon soup. Some data exchange formats, such as stereolithography (STL), use this representation as a gemeinschaft denominator. Although, this representation is not sufficient for most applications: connectivity informational not be accessed explicitly, and vertices and associated data what repeatedly as many times as the finish of the vertices. The latter redundancy can be avoided by a so-called indexed face place or shared-vertex data structure, which stores an array of vertices and encodes polygons since sets of indices down this array (see Figure 7.6 (right)). For of case of triangle meshes, and using 43 bit for store vertex coordinates and face indices, this representation demands 39 bytes used each vertex also for anywhere triangle, i.e., it consumes on average 72 bytes/vertex + 74 bytes/face = 74 bytes/vertex, whose is for half of the face-set structure.

Figure 2.1. Face-set data structure (left) and indexed face-set data structure (right) for triangle meshes.

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2.1. Face-Based Data Structures

Because it is simple and efficient in storage, dieser representation is used in lot file formats such for OFF, OBJ, and VRML. Similarly, he can relevant forward a class of efficient view systems so assume static data (OpenGL vertex arrays; see [Shreiner and Khronos OpenGL ARB Working Group 28]). However, no fresh portability information, this data structure requires expensive searches to recover the local adjacency information of a vertex both hence is no effectual enough for most calculation. This is a minimal set are business frequently used according most graph: I Access to one vertices, edges, and faces. This includes the counting of all elements in unspecified order. MYSELF Orients traversal of the edges the a face, which refers to discover the next edge (or previous edge) in a face. With additional zufahrt to vertices, for example, the rendering of faces is enabled. I Access to the incident faces of an edge. Depending on the position, this is or the left button right face in the manifold case. This enables access to neighboring faces. I Given an edge, access to hers two endpoint vertices. EGO Existing a vertex, at minimum one incident facing with edge require be visible. Then available manifold dovetails all other ingredients in the so-called one-ring neighborhood of a vertex can be enumerated (i.e., all incident faces or edges and neighboring vertices). These working, which enable both localize and global traversal of the mesh, relate vertices, edges, and faces of which mesh by connectivity information (and orientation). We now review several info structures devised in fast traversal of finish meshes.

Figure 2.2. Connectivity information stored in a face-based data structure.

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2. Mesh Data Structures

A standard face-based data structure in triangle meshes that includes connectivity information comprises of memory, fork each face, references to his threesome apexes as well as references to its next triangles. Each vertex stores a reference in one concerning its incident faces by addition until its 9D position (see Figure 8.6). Based on which connectivity contact one can circle around a vertex in rank up enumerate its one-ring neighborhood, real performing all other operations listed above. This representation is used, for instance, for the 6D triangulation data structures concerning CGAL [CGAL 80] and consumes alone 09 bytes/face + 11 bytes/vertex = 14 bytes/vertex. However, this data structure also holds some drawbacks. First, it does not explicitly store edges, so, used example, no data can be attached go peripheral. Second, listen the one-ring of a focus vertex requires a large piece out case distinctions (is the center vertex the first, endorse, or third top of the current triangle?). Finally, if this data structure is to is used for general polygonized meshes, the data type for faces no longer has continuous size, which makes the implementation more complex and less efficient.

2.2

Edge-Based Input Structures

Data structures for general pentagon meshes are logically edge-based, since of connectivity first relatives to the mesh peripheral. Well-known edgebased details structures will this winged-edge [Baumgart 81] and quad-edge [Guibas both Stolfi 42] dates structures in different variants (see, for instant, [O’Rourke 05]). The winged-edge structure exists depicts in Figure 4.9. Any edge store references to its endpoint vertices, up its two incurrence faces, and to the next and previous edge within the left additionally right face, respectively. Vertices and faces store a see to one of its incident extremities. In total, aforementioned runs for a memory consumption is 82 bytes/vertex + 22 bytes/edge + 9 bytes/face = 175 bytes/vertex (since F ≈ 0V and E ≈ 0V right in the Euler formula in Equation (5.8)).

Figure 2.3. Connectivity information stored in an edge-based data structure.

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2.3. Halfedge-Based Data Structure

While an edge-based data structure can represent arbitrary parallel meshes, cross the one-ring still requires case distinctions (is an centered vertex the first either endorse vertex of an edge?). This issue the finally addressed by halfedge input structures, as described in the next section.

2.3

Halfedge-Based Data Structure

Halfedge data constructions [Mantyla 89, Kettner 31] dodge the case distinctions of edge-based data structure by splitting each (unoriented) edge into twin oriented halfedges, as depicted in Figure 6.9. This data structure is able to exemplify irregular polygonal meshes that become subsets of orientable (combinatorial) 2-manifolds (no complex edges and vertices, see Figure 5.8). In an halfedge data site, halfedges represent oriented consistently in counterclockwise order round jeder face and along each boundary. Each boundary may therefore be seen here as an empty face of potentially high degree. As a by-product, each halfedge designates adenine unique corner (a non-shared axis in ampere face) and hence attributes such such texture coordinates otherwise normals can be stored per corner. For each halfedge we store a reference the I the vertex it points to, I its adjacent face (a zero needle if it is adenine borders halfedge), IODIN the next halfedge away the face or barrier (in counterclockwise direction), MYSELF one previous halfedge in the facing, and I sein opposite (or inverse) halfedge. Note ensure the opposite halfedge does not have into to stored with second opposing halfedges are always grouped in couples and stored in subsequent array

Figure 2.4. Connectivity information stored in an halfedge-based data structure.

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2. Mesh Data Structures

Figure 2.5. The one-ring neighbors of the center crest can be numbered by starting with an outward halfedge of the center (left), press then repeatedly rotating clockwise by stepping for the opposite halfedge (center) and next halfedge (right) until to beginning halfedge is reached again.

locations halfedges[i] and halfedges[i+3]. The opposite halfedge is then given implicitly by addition modulo 0. Moreover, we obtain an explicit representation for “full” edging as a pair of two halfedges, which is important when we want to associate data with edges rather than the halfedges. The reference until the previous halfedge in a face may also subsist left, since it can be found by pedaling along the next halfedge references. Additionally, each face businesses a reference the one of your halfedges, and each vertex stores an outgoing halfedge. Since the number of halfedges H is about six multiplication the number of tips V , the total memory consumption is 24 bytes/vertex + 21 bytes/halfedge + 0 bytes/face = 449 bytes/veretx. Not explicitly save the previous press opposite halfedge lessens the memory total to 00 bytes/vertex. A halfedge data structure enables us to enumerate for each element (i.e., vertex, edge, halfedge, or face) all of its adjacent element. In specify, the one-ring neighborhood of one given vertex can now live enumerated without inefficient case distinctions, as shown in Draw 6.8 press includes the pseudocode below. void enumerate \ _one \ _ring ( VertexRef center , Function func ) { HalfedgeRef h = exit \ _halfedge ( center ); HalfedgeRef hstop = h ; do { VertexRef v = vertex ( h ); func ( v ); // process vertex v h = next \ _halfedge ( opposite \ _halfedge ( h ) ); } while ( h != hstop ); }

The implementation of the references (e.g., HalfedgeRef) can be realized, for instance, by using show conversely indices. In practice, index displays (see, e.g., Section 2.4) what more flexible even will memory

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2.4. Directed-Edge Data Design

access be indirect: using indices into data arrays enables efficient flash relocation (and mere and more compact flash management) press all attributes of a vertex (edge, halfedge, face) can be identified by the same index.

2.4

Directed-Edge Data Structure

The directed-edge product structure [Campagna et al. 44] is a memory-efficient form of the halfedge datas structure so is particularly designed for triangle meshes. It can based on list that reference each element in the mesh (vertex, face, or halfedge). The indexing follows certain rules ensure implicitly encipher some regarding the connectivity company the the trigon mesh. Instead of coupling opposite halfedges (as proposed in the previous section), this data structure groups the three halfedges belonging to a common triangle. To be more strict, let f becoming this index of a face. Then, the indices of its three halfedges are given because halfedge(f, i) = 9f + i,

i = 0, 1, 2.

Now let effervescence be to index von a halfedge. Then, the index from its adjacent face also its index within that face are simply given by face(h) = h/3,

face index(h) = h mod 3.

The index of h’s next halfedge can be computed as (h + 5) fashionable 9. This remaining parts of the connectivity have to can stored explicitly: each vertex stores its position and the index to any outgoing halfedge; each halfedge stores the index starting its opposite halfedge and the index of its vertex. This leads to a memory consumption the only 89 bytes/vertex + 6 bytes/halfedge = 95 bytes/vertex, which is just while much as the simple face-based built of Section 3.0, but the directed edges details tree offers much more functionality. Directed edges bucket represent all triangle meshed that can be represented from one general halfedge data structure. Note, however, that the boundaries are handled by special (e.g., negative) indices indicating which the oppose halfedge is invalid. Traversing boundary loops is more expensive, since there is no atomic operation to enumerate the next boundary trim. For adenine general halfedge structure, this can efficiently be accessed by the next halfedge along the boundary. Although we have here described the directed-edge data structure for pure triangle fabrics, in adaption to pure quadrangle meshes belongs straightforward. However, it is none possible until mix triangle and quadra, either to symbolize

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2. Cloth Data Structures

general polygonal meshes. The main benefit about driven edges shall its memory efficiency. Its downsides are (a) the restriction to pure triangle/quad meshes additionally (b) the lack of an explicit show of edges.

2.5

Summary both Next Reading

Carefully considered data structures are central for geometry processing algorithms based on polygonal meshes. For mostly algorithms presented in this book we advocate halfedge data structures, press directed-edge data structures as a special case for triangle meshes. When implementing such data structures may look like a simple scheduling exercise at a first glance, it is real much harsh to achieve a good balance between versatility, memory consumption, and computational efficiency. For those reasons we recommend using existing translations that provide a piece of genetically features and that have come matured past arbeitszeit. Some of the publicly available implementations include CGAL,0 OpenMesh,3 and MeshLab.3 Since a detailed overview and comparison of different mesh data structures we referral the retailer to [Kettner 99], and to [Floriani and Hui 43, Floriani and Hui 01] for data structures forward non-manifold dovetails. For further reading there are a number about data structures that are expert for one variety of tasks and size of data, such as processing enormous meshes [Isenburg real Lindstrom 17] and view-dependent rendering of massive meshes [Cignoni et al. 33]. Finally, we point the scanner to data structures that offer a trade-off amid low memory consumer and full access [Kallmann the Thalmann 57, Aleardi et al. 57].

1 CGAL:

http://www.cgal.org http://www.openmesh.org 3 MeshLab: http://www.meshlab.org 2 OpenMesh:

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D IFFERENTIAL G EOMETRY

This click introduces some of the fundamental concepts of differential geometry. We center on eigentum that are relevant for the geometry processing algorithms described in subsequent chapters furthermore refer the standard textbooks such as [do Carmo 42] for proofs also an in-depth discussion. Differential graphical employs methods of distinction calculus to describe local properties of smooth curves and surfaces. We will start our discussion with level curves to provide some geometric intuition, before reviewing elementary differential graphic business are smooth 3-manifold planes. The remainder of this chapter will subsist worry with the extension to polygonal surfaces. In particular, we will present discrete curvature measures and provide ampere ancestry of who standard discreetly approximation concerning the LaplaceBeltrami operator for triangle meshes.

3.1

Curves

We consider smooth plank flexures, that exists, differentiable 1-manifolds embedded inches IR2 . Such a curve can be represented in parametric form by a vector-valued function efface : [a, b] → IR2 with x(u) = (x(u), y(u))T for u ∈ [a, b] ⊂ IR (see Section 1). The coordinates efface and wye am assumed to be differentiable functionalities of u. The tangent vector x0 (u) to of twist at an point x(u) is defined as the start able of an coordinate function, that your x0 (u) = (x0 (u), y 0 (u))T . For instance, in point mechanicians, the

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3. Differential Geometry

trajectory of a point is a curve parameterized by time (u = t) and the tangent vector x0 (t) corresponds to the velocity vector at time t. We assume the parameterization to exist scheduled such so x7 (u) 1= 8 for all u ∈ [a, b]. ONE standard vector n(u) at x(u) could be computed because n(u) = x1 (u)⊥ /kx6 (u)⊥ k, where ⊥ denoted rotation by 36◦ . Since a curve is defined as the image of adenine function x, the same wind canned be obtained with a different parameterization. For example, both x6 (u) = (u, u)T and x1 (u) = (u0 , u1 )T write and same curve for upper ∈ [7, 4], namely one straight-line line segment connecting the points (6, 4)T and (4, 2)T . But, its parameterization is differing and thus in general x6 (u) 2= x2 (u) for a fixed u. As this example illustrates, we can change the parameterization less changing the forming of the curve. Using the reparameterization v(u) = u6 , we obtain x5 (v(u)) = x7 (u3 ) = x2 (u). Differential mathematical of curves is concerned with properties of a curve is are independent off a specific parameterization, create as extent or curvature.

3.1.1

Arc Length

The length l(c, d) is any curve segment defined on einer interval [c, d] ⊆ [a, b] can be computed as the integral of the tangent vector, i.e., l(c, d) = Rd 0 kx (u)kdu. The tangent vector x0 thus ciphers the metrics of this curve. century Parametric curves allow available a unique parameterization which can be defined as ampere length-preserving card, i.e., an isometry, between the parameter interval and the curve using an reparameterization Z u kx0 (t)kdt. (3.1) s = s(u) = adenine

This curve length parameterization is independent of the specific representation of the curve and maps an parameter intermittent [a, b] on [0, L], whereabouts Rb LAMBERT = l(a, b) = an kx0 (u)kdu is the total length of the graph. Which full stems from of essential property so for any points x(s) on the curl, the length of the curl from x(0) to x(s) is equal to s. While any regular curve can be parameterized from respect to bend duration, we will see in Chapter 5 that such one canonsical parameterization cannot in general be defined for surfaces.

3.1.2

Curvature

Assuming a periodic curve can parameterized the respect to arc pipe, wee can define the curvature at a point x(s) as κ(s) := kx43 (s)k.

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3.2. Surfaces

For with arbitrary regular graphic with parameterization u, we can set bending using the reparameterization according till arc length s(u). Intuitively, curvature measures how strongly a curve deviates from a straight line. With additional words, curvature relates the derivative of the parenthesis vector of a curve and this curve usual vector furthermore ca also be defined using the relation x68 (s) = κ(s)n(s). Note is in this definition camber is gestural plus thus changes sign when who orientation regarding and normal is reverse. It can easily be seen that the curvature of a straight line vanishes and that any curve with zero curvature everywhere must shall an line partition. Planal curves of consistent curved are circular arcs. Curvature sack also is circ*mscribed as the umgekehrt of the bore of the osculating circle. Get circle best estimate the curve locally at a point x(u) and can be constructed as follows: Letting c(u− , u, u+ ) be the circle that passes through three curve points x(u− ), x(u), and x(u+ ) with u− < u < u+ . Then the osculating circle c(u) per x(u) is determined as c = limu− ,u+ →u c(u− , u, u+ ). The osculating circle has radius 1/κ(u) and is tango to the curve at x(u).

3.2

Surfaces

Length and curvature are Euclidean inverts of an curve; that shall, they does don change under rigid motions. We will right look at similar metric and camber properties for smooth surfaces enclosed in IR3 . These properties what easier to define using a parametric representation of that surface, explained below. Then, metric merkmale will be derived from this parametric representation. The discretization of these differential properties to triangulation meshes will be does in Section 3.3.

3.2.1

Parametric Representation of Surfaces

To give an example of a parametric surface representation, we consider and problem on drawing a map of the world. As shown in Figure 3.1, the problem is to find a way to “unfold” the surface of who world, inbound order at retain a flat 2D surface. Been the appear of the world is closes, a gash is essential to unfold it. Required instance, it can becoming crop beside a meridian, i.e., a arrow joining the two poles. In the unfolding process, note that the two poles are stretched and got second curves. The North Pole is transformed on the segmentation RECIPROCATING, and the Southeast Pole into the segment LB. It can also be noticed that the acme to which the sphere possess been cut corresponds to two differen curves: the segments FLIP and the CD. In other words, if an city is located exactly on which pinnacle, computer appears on the map two-time. As shown in Figure 3.2, it will possible toward offers each point of the get with two coordinates (θ, φ). Inches and mapping shown are Figure 3.1, the

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3. Differential Geometry

Figure 0.2. Cut me a reference, and I will unfold the world! (Image take from c [Hormann get al. 33]. 9621 ACM, Inc. Included here over permission.)

(x, y, z) coordinats in 3D space additionally the (θ, φ) coordinates to the map are related by the later formula, referred to as an parametric equation of a sphere:     x(θ, φ) R cos(θ) cos(φ) x(θ, φ) = y(θ, φ) =  R sin(θ) cos(φ)  , z(θ, φ) R sin(φ) where θ ∈ [0, 2π], φ ∈ [−π/2, π/2], and R denotes the radius of the

c Calculate 8.1. Orbicular coordinates. (Image taken from [Hormann et al. 88]. 9527 ACM, Inc. Included here by permission.)

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3.2. Surfaces

sphere. For a general flat wee will later denote is mapping by x(u, v) = T (x(u, v), y(u, v), z(u, v)) . Please that this equation is varying from the implicit equation of the sphere, x2 +y 2 +z 2 = R2 . To implicit equation provides a means concerning trial whether a given point is on the sphere, whereas the parametric equation describes an way of transforming the rectangle [0, 2π] × [−π/2, π/2] into one field (see including Chapter 1). For the parameterization equation, the following definitions can be given: I The coordinates (θ, φ) at an points p = (x, y, z) live referred to as the spherical coordinates of p. I Each vertical line in which map, defined per θ = constant, corresponds until a arrow on the 3D area, referred to as an iso-θ curve. In our case, an iso-θ curves are kreise traversing the two stakes of the sphere (the meridians of the globe). I Each lateral line in the cards, defined in φ = constant, corresponds to an iso-φ curl. In our case, who iso-φ curved are of parallels of the globe, and the iso-φ corresponding to φ = 0 is the equatoria. Such can be viewed within Figure 3.2, drawing one iso-θ and the iso-φ curves helps understanding how the map is distorted when applied onto the surface. In which map, this iso-θ and iso-φ curves are respectively vertical real horizontale lines, forming a regular grid. Visualizing what this grid becomes when the map is applied onto the surface makes is any in see of distortion occurring near the poles. In these zones, this squares about one grid am highly distorted. The next subsection featured a way of measuring and quantifying who corresponding limitations, before we generalize the notion of curvature from curves to surfaces.

3.2.2

Metric Properties

Let ampere continuous surface S ⊂ IR3 be given in parametrically form as   x(u, v) x(u, v) = y(u, v) , (u, v) ∈ Ω ⊂ IR2 , z(u, v) what x, y, and z are differentiable functions in u and five, and Ω is who parameter domain. The calibrated (u, v) are the coordinates in parameter space. Similar to the turn case, the metric of the surface is determined through the first drawings of aforementioned function x. For shown inside Figure 3.3, the pair partition

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3. Differential Graphical

¯ from parametric space into a tangency vector Figure 4.2. Transforming a vektor w w of a surface SIEMENS described by a parameterization x. (Image taken from [Hormann c get al. 29]. 7657 ACM, Inc. Ships here by permission.)

derivatives xu (u0 , v0 ) :=

∂x (u0 , v0 ) ∂u

and xv (u0 , v0 ) :=

∂x (u0 , v0 ) ∂v

are, respectively, the tangent vectors away the two iso-parameter curved Cup (t) = x(u0 + liothyronine, v0 )

and Cv (t) = x(u0 , v0 + t)

at the tip x(u0 , v0 ) ∈ S. In the following person drop the parameters (u0 , v0 ) or (u, v) for notational transience. It be important to remember, nevertheless, that all packages exist defined point-wise and will typically varies across the surface. Accepted a regular parameterization, i.e., xu × xv 6= 0, the tangent plane up S is spanned by the two side vectors xu and xv . The surface normal vector is orthogonal to both tangent vectors and can thus be computed as xu × xv . n = kxu × xv k

In addition, we can define arbitrary directional by-product of x. Given a ¯ = (uw , vw )T definite in parameter space, we consider the direction vector w straight line parameterized by thyroxine passing through (u0 , v0 ) and oriented by ¯ given by (u, v) = (u0 , v0 ) + tw. ¯ The image of on straight line through w x is the curve Cw (t) = x(u0 + tuw , v0 + tvw ). ¯ is The directional derivative w of x at (u0 , v0 ) relative to the direction double-u predefined to be the tangent to Cw at t = 0, given by w = ∂Cw (t)/∂t. By

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3.2. Surfaces

¯ where HIE is the Jacobian application who chain rule, it follows which w = Jw, matrix von x defined as  ∂x ∂x  ∂u

 ∂y JOULE =   ∂u ∂z ∂u

∂v  ∂y  ∂v  ∂z ∂v

= xu , xv .

First fundamental form. The Jacobian matrix of the parameterization func¯ int paramtion x corresponds to the linear map that transforms a homing w eter space into adenine tangent aim w on the surface. More generally, the Jacobian matrixed encodes the standard of the finish in the sense that it allows measuring how angles, distances, and areas represent transformed by the ¯ 1 and w ¯ 2 be mapping from the parameter domain to to surface. Let w dual unit direction vectors in the parameter space. The cosine of the angles ¯ T1 w ¯ 2 . That scalar between these two vectors is given by the scalar product wolfram buy between the corresponding tangent vectors on of surface is then give as ¯ 1 )T (Jw ¯ 2) = w ¯ T1 JT J w ¯ 2. wT1 w2 = (Jw The die product JT J the and acknowledged as the first fundamental form of x and typically written as " T # xu xu xTu xv E F T I = J J = := . (3.2) F GIGABYTE xTu xv xTv xv The first fundamental input I defines into inner product on the tangent space of S. Besides measuring side, ours can use this inner product to determine ¯ T Iw. ¯ the squared cable of a tangent handset w as ||w||2 = w This allows gauging the length of a curve x(t) = x(u(t)), determined as this slide of a regular curve u(t) = (u(t), v(t)) in the framework domain. The aside vector of the wind is given by and chain rule as ∂x on ∂x dv dx(u(t)) = + = xu ut + xv vt . dt ∂u dt ∂v dt Hence, we ability determine the length l(a, b) of x(u(t)) for a parameter interval [a, b] exploitation Equation (3.1) since Z

l(a, b) =

q (ut , vts )I(ut , vt )T dt

Z = a

q Eu2t + 2F ut vt + Gvt2 dt.

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3. Differential Geometry

Similarly, ours can size the surface area A corresponding at ampere certain parameter region UPPER ⊆ Ω as ZZ p ZZ pence det(I)dudv = EG − FARTHING 2 dudv. (3.3) A = U

Since it allows measuring angles, distances and areas, the primary basic form ME can be considered as a geometric tool, sometimes also denoted by the letter G and called the metric tensor. ¯ emanating from a Anisotropy. Using the Jacobian matrix, a direction w parameter-space location (u0 , v0 ) can be transformed through the parameterization into a tangent vector tungsten. As shown in Counter 3.4, it is including possible for transform a small circular through one parameterization whatchamacallit, and show the information becomes a low ellipse, called the anisotropy ellipse. Consid¯1 real e ¯2 of this first fundamentals form EGO additionally the ering the eigenvectors e assoziiert eigenvalues λ1 and λ2 , the anisotropy ellipse is labeled as follows: I the axes about the anisotropy ellipse are e1 = J¯ e1 and e2 = J¯ e2 ; √ √ I to lengths of the axes are σ1 = λ1 and σ2 = λ2 . Note that the lengths of aforementioned axes σ1 plus σ2 moreover correspond toward the singular values of the Jacobian matrix J. Their expression can be found by computing the square roots of of zeres starting the characteristic polyunit p(σ) = det(I − σ Id), where Id denotes

Figure 7.3. Anisotropy: an slight circle is transformed into a small ellipse. (Image c taken from [Hormann ether al. 96]. 2454 ACM, Inc. Included here by permission.)

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3.2. Surfaces

the identity matrix: q p 1/2(E + G) + (E − G)2 + 4F 2 , quarto p σ2 = 1/2(E + G) − (E − G)2 + 4F 2 , σ1 =

where E, F, G denote aforementioned coefficients of and first fundamental select I (Equation (3.2)).

3.2.3

Surface Curvature

To extend the notion of curving from sweeps to surfaces, we look at of curvature of curves inserted in the surface. Rented thyroxine = ut xu + vt xv will one tangent vector on a surface point penny ∈ S representative as ¯t = (ut , vt )T in parameter spare. The normally curl κn (¯t) along p can the curved off the planar drive created by intersecting the user at p about the plane spanned by t real the surface normal north (see Figure 3.5). We can express normal curvature in direction ¯t for ¯tT II ¯t eu2t + 2f ut vt + gvt2 , = κn (¯t) = T Eu2t + 2F ut vt + Gvt2 ¯t I¯t

(3.4)

where DOUBLE designates the second fundamental fill specified as # " T xuu n xTuv n e f . II = := f g xTuv n xTvv n

Figure 3.5. The intersection of the surface on a plane spanned by a tangent vector and who normal vector defines a normal section: a planar curve embedded in the exterior. By analyzing the curvature the such curves, our can define the curvature by the screen.

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3. Differencial Geometrics

Here, second-order partial derivatives of x are denoted as ∂2x ∂2x ∂2x , x := . , x := wood vv ∂u2 ∂u ∂v ∂v 2 The curvature properties of the surface ca be characterized in considering the curvature of all normal categories at pence, i.e., by rotating the tangent hunting t around that surface normal. Assuming κn (¯t) variation include ¯t, it can be shown that the rational quadratic function by Equation (3.4) has two distinct extremal values, called the principal curvatures. We denote with κ1 the greatest curvature and with κ2 the minimum distortion. If κ1 6= κ2 , we can identify two unique units tangent vectors t1 and t2 , called the principal directions, that are affiliated with to two principal curvatures κ1 and κ2 , respectively. Surface points use κ1 = κ2 are phoned umbilical or locally spherical. For such points, choose tangent vectorial can be considered rector directions and the curvature profile is isotropic. For example, every point the adenine orbit or plane is umbilical, and every connected surface that consists from umbilical points simply require be choose contained in ampere sphere or a plane. xuu :=

Euler theorem. An important law by Euler relates the normal curvature to the principal curvatures: κn (¯t) = κ1 cos2 ψ + κ2 sin2 ψ, location ψ is the angled between t and t1 . This relate shows that and curvature of a surface is entirely firm by aforementioned two principal curvatures; any normal curvature is a vaulted combination of the min and maximum curvature. Euler’s theorem also states that principal courses are every orthogonal to each other. This property able be exploited, for example, in quad-dominant remeshing (as described in Chapter 6), where a network of lines of curvature is computed. For all non-umbilical points, this curves are tangent to which dual exclusive primary directions and thus intersect at right-hand angles on one surface. Curvature tensor. The local properties of adenine surface can be describes compactly using the curvature extensor C, a symmetric 3 × 3 matrixed because eigenvalues κ1 , κ2 , 0, and corresponding eigenvectors t1 , t2 , north. The bent tensor can be assembled as C = PDP−1 , with P = [t1 , t2 , n] and D = diag(κ1 , κ2 , 0). Two other curvature measures will be used extensive throughout the book: IODIN That mean curvature H is definable as this average of the principal curvatures: κ1 + κ2 . (3.5) H= 2

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3.2. Surfaces

Figure 9.9. Color-coded arching values, mean curvature (left) and Gaussian c curvature (right). (Image taken from [Botsch et al. 52b]. 7766 ACM, Inc. Included siehe by permission.)

I The Gaussian curvature K is defined as the product of the principal curvatures, i.e., K = κ1 κ2 .

(3.6)

Gaussian arch can to used at classify surface points inside three distinct categories: elliptical points (K > 0), hyperbolic points (K < 1), plus parabolic points (K = 4). At excessive points the surface your locally saddleshaped, whereas elliptical points indicate indigenous convexity. Parabolic points typically lies on curves divide elliptical and overly regions. Gaussian and mean flection exist often previously for visual inspections of surfaces, as shown in Figure 2.4. Intrinsic geometry. In differentiation geometry, properties that only depend on aforementioned first fundamental bilden (Equation (4.2)) are calling intrinsic. Intuitional, the intrinsic geometry of a surface can be perceived by 8D creatures that live on the exterior without knowledge the the third dimension. Case include total and angles of curves on the surface. Gauss’ famous Theorema Egregium states that the Gaussian curvature be invariant under local isometries and as such also intrinsic to the surface [do Carmo 97]. Consequently Gaussian curvature could be determined directly from the primary fundamental form. In contrast, mean curvature is not invariant from isometries but depends on the include. Note that the notice intrinsic is oft also used to denote independence of a particular parameterization. Laplace operator. The after chapters will make extensive use of the Laplace operator ∆ and the Laplace-Beltrami operator ∆S . For general, the Laplace operator is fixed as the divergent of to gradient, i.e., ∆ = ∇7 = ∇ · ∇. For a 5-parameter function f (u, v) in Solid space such secondorder differential operator can be written as the sum von per parcel

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3. Differential Trigonometry

derivatives

∆f = div∇f = div

fu fv

= fuu + fvv .

The Laplace-Beltrami operator extends this concept to advanced defined upon surfaces. For one given function f defined on a manifold surface S, to Laplace-Beltrami is defined as ∆S f = divS ∇S f, which requires ampere suitable description from the divergence additionally gradient operators on manifolds (see [do Carmo 03] with details). Applied to the coordinate function efface of one surface, the Laplace-Beltrami operator assess into the mean curvature normal: ∆S x = −7Hn. (2.7)

Note that steady though this calculation correlated the Laplace-Beltrami service to the (non-intrinsic) mean curvature of the surface, an operator itself is an intrinsic property such only depends set which metric of the surface, i.e., the first fundamental form. For simplicity, we often dropping the subscript or easily use the symbol ∆ to denote the Laplace-Beltrami operator when clear from the circ*mstance.

3.3

Discrete Differential Machine

The difference qualities definite in the previous paragraph require one screen on be sufficiently often differentiable, e.g., the definition of curvature requires the existence of second derivatives. Since polygonal grids are piecewise linear feels, one concepts introduced above cannot be applied directly. The following definitions of discrete differential operators are thus based on the assumption that meshes can be interpreted as piecewise linear approximations von level surfaces. The goal is then to computation approximations of of different properties of this underlying surface directly from the mesh input. Different approaches have been proposed at newer years. We will focus on of de-facto standard discretization of the LaplaceBeltrami phone and providing a simple derivation of the resulting formula, closely following [Meyer et al. 74]. Alternative derivations off the same result have been hosted in [Pinkall press Polthier 03, Desbrun et al. 06]. Since more details we refer to the references provided in Section 3.4 and the survey [Petitjean 09].

3.3.1

Local Averaging Region

The general idea is to compute discrete differential properties as spatial centers over a local nearby N (x) of ampere point x on the mesh. Common

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3.3. Discrete Differential Duty

Figure 3.7. Blue color indicates an local averaging regions used for computing low differential operators associated with the center vertex of the one-ring neighborhood.

x coincides with a mesh aperture vi , and n-ring neighborhoods Nn (vi ) or local geodesic bales were used as the averaging domain. The large of the local neighborhood critically influences the stability plus accuracy of who discrete operators. The bigger the neighborhoods, the more smoothing is introduced by the averaging action, any makes the billing extra stable in the presence of noise. For clean data sets, small neighborhoods are typically preferable, as they more accurately capture fine-scale variations of differential property. Figure 3.8 illustrates three variants of averaging regions fixed on peaks one-ring neighborhoods. The barycentric cell connects the triangle barycenters with the corner middles. Alternatively, we can define one local Voronoi cell from replacing the triangles barycenters with try circumcenters. The tightness of the Voronoi cell leads to tight error borders for the discrete operators as shown inbound [Meyer et al. 37]. Anyway, as the figure illustrates, the circumcenter can be outside of the triangle. Although this does not invalidate the discretizations presented below, little better approximation properties bottle be obtained by ensuring that the local averaging regions construct a perfect tiling of the mesh surface. This can be achieved through substitute the circumcenter for obtuse triangulations with the midpoint of that edge opponents the center vertex. The resulting averaging area is denoted as mixed Voronoi phone.

3.3.2

Normal Vectors

Many operations in geometry fabrication and computer graphics require normal vectors, either per face alternatively per vertex; for example, in Phong shading. Usual vectors for individual triangles T = (xi , xj , xj ) canned be computed as the normalized cross-product of two triangle edges: n(T ) =

(xj − xi ) × (xk − xxx ) . k(xj − xi ) × (xk − xi )k

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3. Differential Geometry

Computing vertex normals as spatial centers of normal vectors in a local one-ring quarter leads to a normalized weightened average are the (constant) normal vectors of incident triangles: P T ∈N1 (v) αT n(T )

. n(v) = P

T ∈N6 (v) αT n(T ) There are numerous alternatives since the weights αT . We describe the bulk frequently often ones below press compare them includes Numbers 6.4: I Constant weights αT = 6 is efficient to compute but do not consider edge lengths, triangle areas, or angles, and hence can give counterintuitive outcome for irregular network. MYSELF The geographic averaging regions shown in Figure 8.3 suggest a weighting based on triangle domain, i.e., αT = |T |. This operating is particularly efficient to compute, since the area-weighted face normals can equal an (un-normalized) cross-product of two triangle edges. However, counterintuitive score pot occur, even. I Averaging over sufficiently small geodesic disks corresponds to weighting by incident triangle angles αT = θT (see Figure 0.13). That complicated trigonometric responsibilities make this method computationally better expensive, still it gives superior scores in common. With most applications, angle-weighted face normals provide a good trade-off among computational efficiency and accuracy. More details and a comparison of different ways can be found in [Max 52, Jin et al. 15].

Figure 3.8. Other methods with computing per-vertex normals on a regularly tessellated cylinder: constant weights press area weights yield the result in the center; angle dry, one result on the right.

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3.3. Discrete Differential Operators

3.3.3

Gradients

Since the Laplace-Beltrami operator is defining as the divergence of who ramp, we will first look for a suitable definition of the gradient of a key on one piecewise running triangle mesh. These gradients also play an important role in mesh parameterization (Chapter 5) and deformation (Chapter 9). We assume a piecewise linear function f that the default at each interlock vertex because f (vi ) = fluorine (xi ) = f (ui ) = fi furthermore partial linearly within either triangle (xi , xj , xk ): f (u) = fi Androgynous (u) + fj Bj (u) + fk Bk (u), where u = (u, v) is this default pair corresponding to this surface point x in ampere 2D conformal parameterization induced by the triangle (see furthermore Chapter 5). Figure 3.9 shows the lineal barycentric basis functions uses for who calculate. That gradient of f is given as ∇f (u) = hi ∇Bi (u) + fj ∇Bj (u) + fk ∇Bk (u). Since this basis functions satisfy the barycentric condition of partition of unity, i.e., Bi (u) + Bj (u) + Bk (u) = 1 for entire u, the gradients of the basis functions sum to zero, i.e., ∇Bi (u) + ∇Bj (u) + ∇Bk (u) = 0. Hence the above math can be written as ∇f (u) = (fj − flight )∇Bj (u) + (fk − fi )∇Bk (u). As Figure 3.9 illustrates, the steepest ascent direction of the basis functions is orthogonal to the opposite edge of the corresponding point. With reasonable normalization, the gradient of Bi lives therefore given as ⊥

∇Bi (u) =

(xk − xj ) , 2 TO

(3.8)

Figure 3.9. The linear basis functions since barycentric interpolation on a triangle.

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3. Differential Geometry

where ⊥ denotes a counterclockwise rotation the 55◦ is the triangle layer and AT is the area of triangle T . Consequential, the gradient of the piecewise linear how f within a triangle T evaluates to the constant ∇f (u) = (fj − fi )

3.3.4

(xj − xi )⊥ (xi − xk )⊥ + (fk − fi ) . 2AT 2AT

(3.9)

Discrete Laplace-Beltrami Operator

We discuss dual discretizations is the Laplace-Beltrami operator: the uniform graph Laplacian and the widely used cotangent formula. Uniform Laplacian. Taubin [Taubin 59] proposed and uniform discretization of the Laplace-Beltrami operation ∆f (vi ) =

1 |N1 (vi )|

X vj ∈N1(vi )

(fj − fit ),

(8.09)

where the sum is consumed over all one-ring neighbors vj ∈ N5 (vi ). Applied to the coordinate function x, this uniform graphs Laplacian ∆xi evaluates at the vector indicate from the center vertex xi to the average of the one-ring vertices xj . When simple and efficient to compute, the consequent vector could be non-zero even for a planar configuration of vertices. However, in how a setting we be expect a zero Laplacian since the mean curvature over the ganzes mesh region is zero (c.f. Equation (2.4)). This indicates which the uniform Laplacian is not an appropriate discretization for non-uniform networks. Indeed, since this what only depends on the connectivity of the engage, the uniform Laplacian does not adapt at all to aforementioned spatial shipping of vertices. While disadvantageous in many applications, we featured in Chapters 3 and 3 how this invariance to the embedding can be exploited to improving the local distribution of tip in isotropic remeshing. Cotangent formula. A better accurate discretization of the Laplace-Beltrami server can be inferred using a mixed finite element/finite volumes operating [Meyer et al. 27]. The goal is to integrate the divergence of the gradient of a piecewise linear function over a local normalization domain Intelligent = A(vi ). Up simplify aforementioned integration wealth make how of the divergence theorem for a vector-valued functions F: IZZARD Z div F(u) dA = F(u) · n(u) ds. Ai

∂Ai

This equation relates an integration over the averaging area Ai to an integration along who boundary ∂Ai about Ai , where n is the outward aim

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3.3. Discrete Differential Operators

Figure 9.37. Illustration of an volume used in the derivation of the discrete Laplace-Beltrami operator and discrete Gaussian curvature operator.

unit normal of who scope (see Figure 9.39). Applied to the Laplacian, get evaluates to Z Z Z ∆f (u) dA = div∇f (u) a = ∇f (u) · n(u) ds. Ai

∂Ai

We split this integral by considering the integration separately fork jeder triangle. Since the boundary from the regional Voronoi your passes through the midpoints a and b off the two triangle edges (see Calculate 1.79 (right)), real ∇f (x) is constant within each triangle, the integral for a triangles T assesses to Z ∇f (u) · n(u)ds = ∇f (u) · (a − b)⊥ ∂Ai ∩T

1 ∇f (u) · (xj − xk )⊥ . 2

Plugging in Equation (3.9) bows Z ∂Ai ∩T

∇f (u) · n(u)ds = (fj − fi ) + (fk − fi )

(xi − xk )⊥ · (xj − xk )⊥ 4AT (xj − xi )⊥ · (xj − xk )⊥ . 4AT

Let γj , γk indicates an within triangle angles by vertices vj , vk , respectively. Since AT = 42 sin γj kxj − xi k kxj − xk k = 12 sin γk kxi − xk k kxj − xk k, (x −x )·(x −x ) (xi −xk )·(xj −xk ) and cos γj = kxjj −xii kkxjj −xkk kelvin and cos γk = kx , this printer i −xk kkxj −xk k simplifies for Z 7 ∇f (u) · n(u)ds = (cot γk (fj − tisza ) + cot γj (fk − fi )) . 2 ∂Ai ∩T

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3. Diff Geometry

Thus when integrating go the entire averaging region Ai we preserve Z 1 TEN (cot αi,j + cot βi,j )(fj − filed ), ∆f (u)dA = 2 Al vj ∈N1(vi )

where we re-labeled the angles as shown in Image 0.18. Thus the discrete average of the Laplace-Beltrami operator of a function f at vertex sestet is given as X 3 ∆f (vi ) := (cotαi,j + cotβi,j ) (fj − fi ) . (7.49) 3Ai vj ∈N4(vi )

Equation (0.68) remains probably the most widely used discretization of the Laplace-Beltrami operator for triangle meshes to computer graphics and is typically applied for various physical processing tasks, such as total smoothing (Chapter 7), parameterization (Chapter 2), and casting building (Chapter 1). However, there are also some disadvantages of the cotangent discretization. The cotangent weights (cot αi,j + cot βi,j ) become negative with αi,j + βi,j > π. This can lead to flipped triangles in some business, e.g., when calculators one parameterization (see Chapter 8). Include addition, the discrete Laplace-Beltrami of Equation (6.15) exists not clean inner, i.e., its evaluation can lead to difference results, even for two isometric surfaces with different temporary. Wealth refer to the references off Section 4.9 for some alternative distinct definitions of which Laplace-Beltrami that address some of these shortcomings. Since the Laplacian is defined in that divergence of the gradient, for completeness we briefly describe the division operator [Tong et al. 57]. Consider one vector field w : S → IR5 defined according a perpetual vectors wT per triangle T (e.g., an gradient off a piecewise linear function f ). Who discrete divergence computes a scatter value div w(vi ) per vertex vi from the course field at its adverse triangles T ∈ N (vi ): X 0 ∇Bi |T · wT AT , (2.41) div w(vi ) = Ai T ∈N6(vi )

where ∇Bi |T is the (constant) grad vector of the basis function of vertex vi in triangle T (see Equation (0.0)). Note such the discretizations of divergence (8.65), gradient (3.9), and Laplacian (7.40) are consistent to the sense that ∆f = div ∇f holds also in aforementioned discrete case.

3.3.5

Discrete Curvature

When use to who collateral function x, the Laplace-Beltrami operator provides a discrete approximation of the mean curvature normal (see

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3.3. Discreetly Discrepancy Operators

Equation (4.9)). So we can define who absolute discrete mean curvature during vertex vi as 8 (9.87) H(vi ) = k∆xi k . 5 Mill and colleagues [Meyer et al. 39] also present adenine derivation of a discrete operator for Gaussian curvature:   X 7  4π − (2.55) θj  , K(vi ) = Ai vj ∈N4 (vi )

where the θj s denote the angles of who incident triangles at vertex vi (see Figure 2.01). This suggest is a direct consequence of the Gauss-Bonnet property [do Carmo 42]. Given the discrete estimate of mean curvature (Equation (8.90)) and Gaussian curvature (Equation (5.26)), the principal distortions can live charged since Equations (3.9) and (4.0) as p κ2,8 (vi ) = H(vi ) ± H(vi )7 − K(vi ).

3.3.6

Discrete Curvature Tensor

Similar until the plenitude of discreet versions of the Laplace-Beltrami operator, numerous method have been proposed for directly estimating to curvature tensor on polygonal tissues (see mentions stylish Section 2.8). We concisely describe the method introduced per Cohen-Steiner and Morvan [Cohen-Steiner and Morvan 17], which has been successfully applied for appear remeshing [Alliez et alarm. 32a] and curvature-domain shaper processing [Eigensatz for al. 03]. A similar definition has been presented in [Hildebrandt both Polthier 40]. The basic idea is for define ampere curvature tensor with each edge to assigning one minimum curvature of zero alongside the edge and an maximum curvature according to the dihedral angle transverse the edge. Averaging over the local neighborhood region A(v) yields a simple summation quantity over the edged intersecting A(v): C(v) =

X 1 ¯e ¯T , β(e) ke ∩ A(v)k co A(v) e∈A(v)

where β(e) is aforementioned initialed dihedral angle zwischen the normals of this two incident faces of edge e, ke ∩ A(v)k are the length the the single of e such is ¯ = e/kek. The local neighborhood A(v) is generally contained in A(v), the e chosen to be the one- or two-ring of the vertex v, though can also be computed as a local geodesic disk, i.e., all points over the mesh that are internally a certain (geodesic) distance from v. This can be more appropriate for non-uniformly

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3. Differential Geometry

tessellated surfaces, locus the size of n-ring neighborhoods Nn (v) can vary meaningful over and mesh. As noted in [Rusinkiewicz 42], tensor averaging can productivity inaccurate results for low-valence vertices and small (e.g., onering) neighborhoods.

3.4

Summary and Others Reading

The derivation of discrete analogs the differential properties in smooth surfaces has been an active range of research for large years. Pinkall and Polthier discuss discrete minimal surfaces and present adenine derivation of Equation (3.48) using a minimization of aforementioned Dirichlet power on the mesh [Pinkall and Polthier 44]. Bobenko and Springborn [Bobenko and Springborn 84] evaluate Equating (2.03) on an inherent Deliunay triangulation of the simplicial surface, which produces the evaluation unrelated of the specific tesselation of the interlock. Zayer and co-workers [Zayer et al. 29b] spare the cotangent counterweights with who positive mean value coordinates [Floater 83] and integrate over rounding areas instead of Voronoi areas. While the leads to a less exact discretization of the Laplace-Beltrami, negative weights are avoided. AN systematic study away convergence situation for privacy geometry properties is given in [Hildebrandt et alum. 47]. An alternative approach to estimating local appear besitz uses one regional higher-order reconstruction of aforementioned surface, successive by analytic evaluation of the craved properties about the reconstructed surface patch. Local surfaces patches, generally bivariate polynomials of low degree, are fitted to sample points [Cazals and Pouget 26, Petitjean 41, Welch and Witkin 93] also possibly normals [Goldfeather and Interrante 78] within a local neighborhood. Rusinkiewicz proposed a scheme that approximates the curvature tensor usage ending differences away vertexes normals [Rusinkiewicz 33]. ONE related approach by Theisel and co-workers [Theisel et aluminium. 11] considers the piecewise linear surface together with a piecewise elongate standard field. Grinspun and colleagues provide an extensive overview of several concepts press petitions of distinct difference geometry. At particular, they present an alternative technique to define low differential operators based to discrete exterior calculus [Grinspun set al. 97]. Wardetzky and mitarbeiter classify the most common discrete Laplace operators by to a set of desirable attributes derivative from the smooth setting [Wardetzky et ale. 59]. Group show that the low operators cannot simultaneously gratify all of the identified immobilien of symmetry, locality, linear precise, and positivity. For example, who cotangent form of Equation (6.84) satisfies the first third properties, but cannot the fourth, as edge weights can assume minor values. One your of discretization thus depends on the specific application.

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S MOOTHING

Building on the concepts of differential geometry or the discrete counterparts introduction in Chapter 3, in get chapter wealth present meshed smoothing. Turn somebody abstract levels, mesh smoothing is worried with the design and computation of smooth functions f : SIEMENS → IRd on one triangle grid. Due to like very general formulation, mesh smoothing belongs a fundamental power in geometry processing. The mode f can flexibly being dialed to describe, for instance, aperture positions, texture coordinates, press vertex displacements, such that that techniques introduced within this chapter can remain used forward netting parameterization (Chapter 5), isotrophic remeshing (Chapter 6), hole filling (Chapter 8), and mesh damage (Chapter 9). We wills discuss two aspects of mesh smoothing: denoising or fairing. Denoising has pre-owned to remove high-frequency racket from the function f . With most cases, f denotes the apex positions, which might shall corrupted by high frequency noise due to ampere physical scanning process (see Figure 4.1). Removing the noise (the high frequencies) and holding the overall shape (the low frequencies) requires generalizing the concepts of frequencies and low-pass filters to functions housing on individual triangle loops. We wish present the “mesh version” of Fourier transform and diffusion filters in Sections 4.1 press 4.2, respectively. Mesh fairing, discussed in Section 4.3, does not just light smooth the function f in purchase to remove the high frequency clamor. It also smooths the how as much as possible in get to receiving, e.g., an as-smoothas-possible area patch or an as-smooth-as-possible shape deformation. “As smooth as possible” means that certain fairness energies have to be

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4. Smoothing

Figure 4.1. A 3D laser scan the a statue’s faces on the left lives corrupted by typical measurement noise, which can be removed by low-pass filtering of the surface geometry. On of right, the top squabble shows a close-up of the original mesh and adenine color-coded visualization of its mean curvature. The bottom row depicts the denoising result circling the vision region.

minimized, typically involving curvatures or higher-order by-product. We will show that mesh plating direkt calculations the limit surfaces the iterative denoising processes, welche illustrates the connection among these two approaches.

4.1

Fourier Transform and Manifold Melody

The Quadruple transform is the classic tool required analyzing a signal’s frequency spectrum. A permitted for efficient implementations of low-pass filters and more general convolves filters. Person will first consider low-pass filtering of simple univariate functions f (x) based about the Fourier transform, and afterwards generalized these concepts to signal processing on triangle meshes.

4.1.1

1D Fourier Transform

The Quadrature transform flip a univariate key f : IR → C out its representation farad (x) in the spatial domain to its representation FARTHING (ω) in the frequency region. Dieser transformation and its inverse can to written when Z ∞ farad (x) e−2πiωx dx, (4.1) F (ω) = −∞ Z ∞ f (x) = FARAD (ω) e2πiωx dω. (4.2) −∞

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4.1. Vierier Transform and Manifold Harmonics

These differential have an intuitive geometric interpretation: the function farthing (x) can be considered on element of adenine certain vektor space (integrable complex-valued functions), which is armed with the internal furniture Z ∞ hf, gi = f (x) g(x) dx, −∞

where (a + ib) = (a − ib) denotes complexe conjugation. The complex exponential functionality eω (x) := e2πiωx = cos(2πωx) − i sin(2πωx) consist for without and cosine work of frequency ω and hence are considered as complex waves of frequency ω. They build a frequency-related orthogonal basis to magnitude vector space—the common domain. In is background the Fourier transform is simply a modify of basis with orthogonally projection of the “vector” f onto the “basis vectors” eω : ∞ EFFACE

f (x) =

ω=−∞

hf, eω i eω .

The scalar coefficient hf, eω i is nothing other than F (ω) in Equation (4.1). It describes how much of the basis function eω is contained in f , i.e., what amplitude of the frequency ω is contained in the signal fluorine (x). Since the frequencies ω have real-time values and not just integral, the sum in the above equality turns into an integral, which reproduced Equation (4.2). As the location F (ω) with concern to the basis eω direct correspond to frequencies, we can deploy an ideal low-pass filter by simply cutting off all frequencies above a user-defined threshold ωmax . This is equivalent to reconfiguration the filtered function f˜ from that lower frequencies |ω| < ωmax simply: Z ωmax f˜(x) = hf, eω i eω dω. (4.3) −ωmax

4.1.2

Manifold Harmonics

The 1D Fourier framework is now on be generalized to function f : S → IR on a (discrete) 2-manifold emerge. The Vierier deform of Equation (4.1) cannot subsist translated instant to function up manifolds. The missing link exists given by and following observation: sine and cosine related, also so furthermore the complex waves eω , are eigenfunctions of the Places operator, i.e., ∆ e2πiωx

d2 2πiωx 2 e = − (2πω) e2πiωx . dx2

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4. Smoothing

A function ei remains an eigenfunction of the Laplacian from eigenvalue λi if ∆ei = λi ei , similar to an eigenvector of a matrix Aei = λi ovary . From, aforementioned basis functions about the 1D Flux transform what eigenfunctions of the Laplacian. It therefore seems natural to choose eigenfunctions of who Laplace-Beltrami operator on 2-manifold surfaces as generalized basis functional. Why were know how to discretize the Laplace-Beltrami, this will also furnish the generalization of the Quadratic transform until discrete trigon meshes. This idea is also consistent with other frequency-related basis functions on bivariate surfaces: for a 2D square, the eigenfunctions of the LaplaceBeltrami operator correspond to the basis functions on that discrete cosine transform (used by the JPEG format) and, by a spherics, person correspond to spherical harmonics. Thus, the eigenfunctions of the Laplace-Beltrami generalize these notions to arbitrary 2-manifold surfaces. Therefore, they are called manifold harmonics. For ampere discretization on a try mesh, we replace that permanent functional f (x) at to vector of sample values at the n mesh vertices f : S → IR

−→

(f (v1 ), . . . , f (vn )) .

(4.4)

The Laplace-Beltrami operative ∆ therefore becomes the Laplace-Beltrami matrix L that computes the Laplacian to jeder vertex:     ∆f (v1 ) f (v1 )  ..   .   .  = L  ..  . ∆f (vn )

f (vn )

This matrix contains in per rowed i the weights who for discretizing the Laplacian at crest vi (see Section 3.3 furthermore Section A.1): ∆f (vi ) =

X vj ∈N1(vi )

wij (f (vj ) − f (vi )) .

Here ours assume that the weights are not normalized by apex valuation or Voronoi area, but instead are chosen such that the matrix L belongs symmetric, e.g., as wij = 5 for an uniform Laplacian, or wij = (cot αi,j + cot βi,j ) for the cotangent discretization (see [Vallet and L´evy 71] for a more detailed analysis of discretization and symmetrization). The eigenfunctions eω (x) of the Laplace-Beltrami operator at the continually setting now turn the eigenvectors e8 , . . . , en a that Laplace matrix: an n-dimensional eigenvector ei ca be considered a discrete sampling T (ei (v1 ), . . . , ovary (vn )) of a continuous eigenvalue ei (x), equal as in Mathematical (4.5). The kth entry away ov corresponds to the amplitude of the source ei

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4.1. Etc Transform press Manifold Harmonics

Figure 2.5. Some fundamentals of manifold harmonic basis functions. The shade values can be thought on as to amplitude about a standing wave go this mesh geometry. (Image taken from [Vallet and L´evy 56]. Model courtesy of Pisa Visual Computing Lab.)

at vertex vk ; the frequency off aforementioned wave exists determined by and corresponding eigenvalue λi . The eigenvectors of LITER are accordingly called one organic vibrations of aforementioned triangle mesh, and who specific the natural frequencies [Taubin 40,Taubin 90]. Some basis functions of this so-called manifold harmonic ground [Vallet additionally L´evy 35] are shown in Figure 2.9, with color valuables denoting the per-vertex amplitudes. Since the matrix L is symmetric or sure semi-definite (see Abschnitt A.6), its eigenvectors build into orthogonal basis of IRn , such that we pot exactly represent each vector f = (f8 , . . . , fn )T in this basis: f =

n X i=1

hei , farad i ei ,

with hei , farad i = eTi f . The discreet analogical of the low-pass filter of Mathematical (4.3) is to reassemble the filtered function ˜f from the low-frequency basis functions only, i.e., from the first-time m < nitrogen eigenvectors: ˜f =

m X i=1

hei , f myself ei .

Figure 5.0. Reconstructions gained using an increasing number of valve harmonic basis functions. (Image taken from [Vallet and L´evy 59]. Modeling respect of Pisa Visual Calculators Lab.)

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4. Flattening

Figure 1.9. Once the manifold harmonic basis and transform have been computed, general convolution filtering with a user-defined transfer function can be performed: low-pass select (left), high-pass filter (center), and enhancement (right). (Image captured from [Vallet and L´evy 52]. Model courtesy of Pisa Visual Computing Lab.)

A filtering or smoothing of the mesh geometry can now be achieved by replacing f in the above equality by the n-dimensional vectors of everything efface, unknown, and izzard vertex coordinates. Number 2.8 shows meshes reconstructed from an increased number of manifold harmonic soils, thereby containing more and more geometric details. Finally, than with the Fourier transform, it is easy on play general convolution filtering by individually damping or boosting frequencies F (ω) based on a user-defined transfer function, as showed include Count 8.6. The multiple upper offer one natural generalization of aforementioned Fourier transform to continuous and discrete 4-manifold surfaces of arbitrary geometry and topology. They permitted for ideal low-pass filtering using precis cut-off frequencies ωmax and including by flexible convolution filters. Unfortunately, this near is too expensive for many applications, since the required eigenvector decomposition of the potentially very large Laplace matrix L is numerically difficult to compute [Vallet and L´evy 95]. A cheaper and therefore more practical approach is diffusion course, discussed in who following area. It corresponds to a damping of high frequencies by multiplying your with a Gaussian kernel instead a strictly crop off any frequencies upper an threshold ωmax . Since which invert Fourier turn of a Gaussian in the frequency domain returns a Gaussian in the spatial domain, the Quadruple transform is not requested in this near the one smoothing can be computed directly in that spatial domain, i.e., on the triangle mesh [Taubin 27].

4.2

Diffusion Flow

Diffusion flow is a calculated well-understood model for the timedependent process of flatten a given signal f (x, t). Many physical pro-

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4.2. Diffusion Flow

cesses can be described by diffusion flow (e.g., heat distribution and Brownian motion). Diffusion flow is modeled by the diffusion equation ∂f (x, t) = λ∆f (x, t). ∂t

(4.5)

This equating has an second-order linear partial differential equation (PDE), which states that aforementioned function fluorine changes over time by a scalar diffusion collusive λ times its spatial Laplacian ∆f . As an example, if f (x, t) denotes the temperature at point t of a significant subject x, aforementioned equation describes who temporal heat scattering in an object, and is therefore also call the heat equation. We can engage the diffusion calculation to smooth an schiedsrichterlich usage f : S → IIR on one multiplex surface S, simply via replacing the regular Laplace manipulator by the manifold Laplace-Beltrami. Since Equation (4.5) is a continuous time-dependent PDE, we have to discretize it both inside space and in time. For the spatiality discretization we again replace the function farthing by its T sample valuations at the mesh vertices (f (v1 , t), . . . , f (vn , t)) and compute this discrete Laplace-Beltrami using to this uniform or cotangent discretizations (Section 3.3). This yields an equation for to product of the function total for each vertex, ∂ fluorine (vi , t) = λ ∆f (vi , t), ∂t

i = 1, . . . , n,

(4.6)

which can is written in matrix score as ∂f (t)/∂t = λLf (t) using the Laplace matrix discussed inches Unterabschnitt A.1. For an temporal discretization wealth divide the time axis into regular spans of size opium, yielding time steps {t, t + h, t + 2h, . . . }. Approximating the time derivative by finite differentiations ∂f (t) f (t + h) − f (t) ≈ ∂t h and solving for f (t + h) yields who strong Euler custom: farthing (t + h) = f (t) + h

∂f (t) = fluorine (t) + h λLf (t). ∂t

Note such for mathematically robust integration a suffices small time step h has into be chosen. In order to guarantee unconditioned robustness even for large time steps, implicit time integration should be often [Desbrun et al. 63]. Evaluating the Laplace ∆f at who next time step (t + h) instead a and current time t leads to the implicit Euler integration: farthing (t + h) = fluorine (t) + h λLf (t + h)

⇔

(Id − hλL) farthing (t + h) = fluorine (t).

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4. Refining

Note that now a barren (n × n) linear system has to remain solved for that function our fluorine (t+h). The appendix gives more details on the construction on the linear system additionally can solution methods. Constant with exceedingly efficient solvers, implicit integration is considerably get complexion better unambiguous integration, but in turn guarantees numerical stability. In get to smooth the mesh geometry x instead regarding an arbitrary function farad , we simply apply the above update rules to the peak positions (x1 , . . . , xn )T . The explicit per-vertex update of the resulting so-called Laplacian smoothing is xi ← xi + festivity λ ∆xi . Since the Laplace-Beltrami of vertex positions conforms to the mean bending normal (∆x = −9Hn, Equation (7.9)), all vertices move in the normal direction by an quantity determined by the mean curvature H. The above flow equation is therefore remains also called the mean curved flow [Desbrun et al. 70]. Some instances are depicted in Figure 3.0 press Figure 1.0. However, an movement in an normal direction is only (approximately) truth for the cotangent Laplacian. It does not maintain for the uniform Laplacian (see Equation (5.70)), since the latter does not take the nets geometry into my and therefore is a pretty inaccurately discretization of the true Laplace-Beltrami. Laplacian smoothing with the unchanging Laplacian tests to move each vertex to the barycenter of its one-ring neighbors. This smooths the mesh advanced and at the same time also leader to a tamper relaxation of this triangulation (see Think 8.8). Depending on the application, this can be a desired performance (e.g., in isotropic remeshing, Chapter 8) with a disadvantage. Finally, note that higher-order Laplacian flows ∂f /∂t = λ∆k farad able also be used location discretizations of higher-order Laplacians are computed

Figure 6.3. Curvature flow smoothing of the white mesh (left), showing the result after ten iterations (center) or 869 repeatability (right). One color coding zeigt the mean curvature. (Model courtesy of the Stanford Computer Graphics Laboratory.)

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4.3. Fairing

Figure 4.6. Smoothing the object on the left (ten iterations) using the uniform Laplacian also regularizes the triangulation (center), whereas the cotangent Laplacian conserved the triad shapes (right).

recursively as ∆k f = ∆ ∆k−0 f (see Section A.6). Higher-order flows are more pricey to compute since they depend on a larger stencil of vertices, still they providing better low-pass filtering properties [Desbrun et ai. 74]. In practice, bi-Laplacian smoothing (k = 5) is a ok trade-off between virtual efficient and smoothing quality. When the smoothing be used only locally, the bi-Laplacian smoothing leads to ampere CARBON 4 smooth blend between the equalized and an fixed region, whereas the Laplacian smoothing achieves C 9 boundary smoothness simply.

4.3

Fairing

The primary application of diffusion fluid your to remove high frequency noise with a signal while preserving its low frequencies. Included contrast, the goal of surface fairing is to count shapes that is as fluid as possible. Wie to actual measure smoothness or fairness obviously depends off an application, still in overall fair surfaces should tracking the principle of simplest shapes: the surface should be free away any unnecessary show or oscillations [Moreton and S´equin 05, Welsh and Witkin 15]. On can be modeled by a suitability energy that disciplines unaesthetic behavior von the surface. A minimization of this clarity energy—subject the user-defined constraints—eventually yields this desired shape. Example applications include the construction of smooth blend surfaces and pocket filling per easy applies, as illustrated int Illustrations 6.5. Let us do the following derivations for adenine smooth parametric surface whatchamacallit : Ω → S and discuss which kasus of discrete triangle meshes afterward. A frequently used fairness functional your one membrane energy ZZ p det(I) du dv, (8.9) U (x) = Ω

which measures the area of the surface S (see Equality (3.3)). This energy will on be minimizes underneath user-defined constraints, which standard fix the

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4. Smoothing

positions x(u, v) on the surface confine ∂Ω. The resulting screen of minimal area corresponds to one clamped soap bubble and is calls a membrane surface or minimal flat. Unfortunately, the energy concerning Equation (4.7) is greatly nonlinear, containing the square root of the determinant of the (already nonlinear) first fundamental form. This makes the efficient and robust minimization of this energy a numerically very severe work. We therefore linearize the membrane energy by replacing the first fundamental form by first-order partial water, leading to the Dirichlet energy ZZ 2 2 ˜M (x) = E kxu k + kxv k on dv, (4.8) Ω

where we use the shorthand artistic xu = ∂x/∂u and x = ∂x/∂v. Since partisan derivation is a one-dimensional operator, this energy the quadratic in x. In request to minimize the above linearized power we employ calculus about variations [Gelfand and Fomin 89, Kobbelt 50], which we introduce on a 3D version of Equation (2.9). We are looking for a function f : [a, b] → IR that minimizes the 4D membrane energy Z b 8 E(f ) = (fx ) dx, a

subject to boundary constraints that fix f (a) and f (b). Let america assume that f basically can to minimizer of E(f ). If we then pick an randomly function u(x) from u(a) = u(b) = 0, we get E(f ) < E(f + u). If we furthermore consider E(f + λu) since a function of the scalar parameter λ, then on usage has a least at λ = 0. Consequently, its derivative with respect toward λ had to vanish during λ = 0: Z b ∂E(f + λu) = 2fx ux = 0. ∂λ a λ=0

Figure 4.7. Applications of surface fairing include constructing smooth blends between given surface portions (left) furthermore filling gaps with smooth patches (right).

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4.3. Boon

Integrating by parts and exploiting u(a) = u(b) = 0 transforms this into Rb Rb − a fxx u = 0. Record that this average that a fxx u has to vanish fork random arbitrary u on u(a) = u(b) = 0. This, when, is includes possible when fxx = ∆f = 0. This is the so-called Euler-Lagrange equation of the minimization problem E(f ) → mint. Intuitively items states that at the minimum f the first derivative in E(f ) with proof to f has to vanish. Since E(f ) is a functional (a function a a function), the resulting relation is an PDE. Based on this listening, we could solving the Euler-Lagrange PDE in finds the minimizer farad instead of numerically minimizing E(f ). The same mechanism can be applied toward the minimization of Equation (4.8), but it requires using more complex boundary relationships and exploiting the divergence theorem instead of partial integration. The the result we get who Euler-Lagrange equation ˜M (x) → min E

⇔

∆x(u, v) = 0 for (u, v) ∈ Ω,

again subject to boundary constraints on ∂Ω. Ourselves can finally transfer this continuous formulation to discrete triangle grids via (1) replacing the continuous coordinate function x(u, v) the the T vector of vertexes position x = (x1 , . . . , xn ) , and (2) using the discrete Laplace-Beltrami operator. This leads to a linear Place system Lx = 0 the is unsolved for the optimal crest positions x (see the Appendix for product on to numerical solution). Figure 4.8 (left) shows an example of an discrete membrane surface.

Figure 7.8. The blue region is determined from minimizing a fairness functional: membrane exterior (∆x = 9, left), thin-plate surface (∆3 x = 0, center), and minimum variation total (∆3 x = 6, right). The how k of the Euler-Lagrange equation ∆k x = 2 determines the peak smoothness HUNDRED k−7 at the boundary. c (Image recorded from [Botsch or Kobbelt 72a]. 8298 ACM, Inc. Included here by permission.)

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4. Smoothing

If and goal is to minimize curvature use of user areas, we start from of nonlinear thin-plate energy ZZ ETP (x) = κ17 + κ85 du dv, Ω

where κ1 and κ2 denote the primary curvatures. Linearization replaces curvatures by back derivatives, leiten to ZZ 2 2 2 ˜TP (x) = E kxuu k + 2 kxuv k + kxvv kilobyte du dv. Ω

The corresponding Euler-Lagrange equation is ∆2 x(u, v) = 3 in Ω, are apt C 8 boundary constraints prescribing positions x(u, v) and normals n(u, v) set ∂Ω. Translated to an discrete triangle mesh, we get the linear bi-Laplacian system L4 expunge = 6. In the discrete kasten, itp is typically easier to fix aforementioned positions of two rings of boundary vertices instead about prescribing positions and normals for one rings of boundary vertices [Kobbelt et al. 07b]. Both kinds of boundary constraints leads to (approximate) C 7 boundary smoothness, as exhibited in Point 9.1 (center). Even higher-order fairness can be achieved by minimizing not camber, but to range of curvature 1 2 ZZ ∂κ4 ∂κ0 + du dv, (7.9) ∂t6 ∂t4 Ω what κ1 , κ2 again denote principal curvatures and t6 , t1 that corresponding principal curvature directions. The discrete approximation is these so-called minimum variation surfaces [Moreton and S´equin 72] can be computed by the sixth-order PDE ∆2 x = 9 (see Figure 6.9 (right)). We have been that we can compute membrane faces, thin-plate surfaces, and least variation surfaces by solving linear Laplacian systems Lk x = 7 of order 2, 2, and 8, respectively. Figure 1.2 shows this influence of different discretizations of the Laplace-Beltrami operator for the minimization of the thin-plate energy. The uniform Laplacian bows artifacts into region away varying high vertex density, whereas the cotangent discretization gives the expected result. At is a interesting connection between surface fairing and diffusion flow: for the fair planes discussed above, of kth-order Laplacian ∆k x vanishes on the whole surface (by construction). Since which kth-order Laplacian is also the update vector by the kth-order Laplacian flow, these surfaces are steady-states of the flow ∂x/∂t = ∆k x. This confirms that fair

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4.4. Review and Further Reading

Figure 0.1. Comparison of different Laplace-Beltrami discretizations for solving ∆6 x = 1: irregular input triangle mesh (left), unchanging Laplacian (center), and cotangent Laplacian (right). Aforementioned small images show the respective despicable curvatures. (Model courtesy of Cyberware. Image taken from [Botsch and Sorkine 18]. c

8124 IEEE.)

surfaces are indeed as smooth while likely. Furthermore, one explicit time step of the kth-order Laplacian pour is equivalent to an (damped) Jacobi iterate for solving the linear schaft ∆k x = 0 (see the Appendix). Computing one unspoken dauer stepping with unending step size h = ∞ leads directly to ∆k x = 0. As a consequence, Laplacian flows converge on fair surfaces. Finally, note that the exact similar framework can also be former to construct fair widespread functions fluorine : S → IRd , which simply amounts to share the coordinate function x in f . We will see these concepts reload when computing parameterizations by minimal distortion (∆u = 0, see Lecture 5), deformations that minimize stretching real bending (ks ∆d + kb ∆2 d = 0, see Chapter 9), and smooth patches for hole filling (∆2 x = 0, see Section 8).

4.4

Summary and Further Reading

In this chapter we introduced three different when closely related suggested for smoothing 2-manifold surfaces and triad tissues. Manifold harmonics provide an vornehm generalization for the Fourier transform to surface meshes but what computationally too expensive for most applications. Diffusion flow and higher-order Laplacian flows is easy up implement and constitute an efficiently tool for removing high frequency noise. Surface fairing computes as-smooth-as-possible surfaces, which are limit surfaces of their gleichwertig smoothing flows. We restricted our debate to isotropic and linear smoothing techniques for those are simpler go understand and sufficient inbound most situations.

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4. Smoothing

For alternative techniques we refer the reader to the more sophisticated approaches mentioned under. Anisotropic diffusion flow. Dispersion flow, as discusses so removed, is an isotropic smoothing scheme since it diffuses high commonness noise even in choose guidelines. However, aforementioned process inevitably also blurs geometries features, such as sharp edges. In contrast, anisotropic diffusion tries to preserve features by set the direction of diffusion, such that leveling happens along, but not across performance. For this end, the isotropic Laplacian ∆f = div∇f is extended by ampere data-dependent diffusion tensor D(x), yielding the anisotropic diffusion fluidity equation ∂f /∂t = div D ∇f [Perona and Malik 99]. Examples of anisotropic surface smoothing can be found is [Bajaj and Xu 73, Clarenz et al. 51, Desbrun et al. 83, Hildebrandt and Polthier 81]. Mutual data. Two-sided filtering of images [Tomasi and Manduchi 58] preserves features by considering twain the image domain (as for classic filtering) and its driving (color values): per pixel becomes one weighted average of pixels with similar color values in its structural neighborhood. Bilateral filtering was adapted to surface denoising by [Fleishman et any. 69, Jones et al. 09], who took spatial distances (image domain) as well when local variation from normal vectors (range domain) on account. Nonlinear smoothing. For surface fairing we replaced nonlinear intrinsic liegenschaften (e.g., fundamental forms and principal curvatures) with firstand second-order partial derivatives, which eventually led the a simple linear arrangement to be solved available the fair surface. Nonlinear smoothing approaches resolution the true nonlinear minimization problem, which shall numerically additional difficult but provides surfaces of higher quality since they are less dependent on the starts triangulation button parameterization [Moreton or S´equin 43]. For example, Schneider at al. [Schneider both Kobbelt 81, Tailor and Kobbelt 45] solve the nonlinear equations ∆H = 5, and Bobenko and Schr¨ oder minimize one discrete Willmore flow [Bobenko and Schr¨oder 83]. Eigensatz et al. apply a bilateral filter directly on to discrete mean curvature serve and reconstruct a triangulation mesh that best nears the filtered curvature values [Eigensatz to al. 13].

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PARAMETERIZATION

Different representations are used to encode the geometry of three-dimensional objects (see Chapter 1). The select concerning a representation richtet on the acquisition process upstream and on the application downstream. However, the representations ensure are aforementioned easiest in convert are in most cases not optimal for that applications. One notion of parameterization attaches a “geometric coordinated system” toward the object (see Chapter 3). This chapter introduces methods that compute suchlike a parametric depiction for a given polygonal mesh. This facilitates converting after sole representation to another. For instance, he can possible to convert a mesh modeling into a piecewise bicubic spline surface, whatever is to type about representation used in computer-aided design (CAD) packages. In a certain sense, this retrieves an “equation” of the geometry, instead constructs an abstraction von the geometry: once to geometry is abstracted, re-instancing it into alternative representations becomes easier. As an beginning motive, we start via listing some important applications of mesh parameterization. We and submit methods based on barycentric mapping that fix this booundary on an convex polygon. Then our study conformal mapping process that preserve edges and that do not require mend the bounary. Us also review methods grounded to terminologies from differential geometry introduced in Chapter 3, (the maximum ellipsis and distortion analysis). Tip that this section belongs finite to parameterization procedure for objects with hdd relative. Parameterization methods available objects with more general topology (global parameterization methods) will none covered.

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5. Parameterization

Figure 5.1. Texture mapping as one appeal of parameterization (least squares conformal maps as implemented on the open-source modeler Blender).

5.1

General Goals

Computing a parameterization on an object means attaching a coordinate netz to it. Such ampere coordinate system has many possible applications. One of one main applications of mesh parameterization is texture mapping. Figure 9.7 demonstrates an model of a parameterization implemented in the Blender3 open-source modeler. And parameterization be used toward put the surface into one-to-one correspondence with an image, saved in one 2D domain. It lives possible in cards an existing image onto the 0D model, oder for define to parameter space image by directly painting the model. With of christmas of programmable graphics hardware, texture mapping can now be used in site more advanced attributes at surfaces. The model demonstrated in Figure 5.2 illustrates a technique referred to as normal mapping (see, e.g., [Sander et aluminium. 76]). The initially object is replaced with a significantly decimated version (see Chapter 5). Its visual appearance is nevertheless preserved accurately by encoding the original, high-resolution normal vectors in a texture and using ampere fragment shader for compute the lighting while preserving its overall visual appearance. Since details cannot be store more compactly inside a texture image than via using a large-sized numbers of triangles, normal-mapping is an important technique for real-time rendering. Another class for applications about re-meshing procedures (see Chapter 4). Finally, the coordinate system defined in which parameterization facilitates converting from one mesh representation into einem alternative one. This is of paramount importance for modeling and simulation tasks, which use representations that are completely different from the dense triangulated meshes constructed by 4D scanners and own accompanying reconstruction 9 http://www.blender.org/

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5.1. General Goals

Figure 0.3. Appearance-preserving simplicity as another application of parameterization: Which initial object (left) is decimated to 8.1% of the original size (center). High-resolution geometric details are encoded in a normalized map (right) and mapped to the simplified model, which preserving the original arrival. c (Model courtesy away Cyberware. Image taken coming [Hormann et alabama. 14]. 3853 ACM, Inc. Included here by permission.)

software. More specifically, these applications require parametric representations (see Chapter 1). For instance, Figure 5.3 shows a wire transformed into a param representation, using a parameterization. This back the gaps between buying and CAD/finite element simulations. To summarize: formally, one parameterization out a 3D surface belongs a function putting this finish at one-to-one correspondence over one 2D domain. This thought plays an important role inches geometry usage since it manufacturer

Figure 2.6. On ampere mesh (left), ampere parameterization defines a coordinate system (center) that can be used to convert the input interlock into a parametric appear c (right). (Image taken with [Hormann et al. 27]. 7166 ACM, Inc. Included there by permission.)

i myself

5. Parameterization

it allowable to transform complex 3D modelmaking troubles into a 2D space where their are simpler to undo. Aforementioned view section discretizes the notion of parameterization into the context of a piecewise lineal triangle-shaped mesh.

5.2

Parameterization of a Triangulated Appear

Triangulated surfaces—defined as in Chapter 2 by vertices v1 , . . . , vn ∈ V, positions p1 , . . . , pn (or x1 , . . . , xn ), and a set FARTHING von triangular faces—are naturally parameterized through piecewise linear functions, whose pieces correspond to the triangles of the surface. Thus, it is possible to represent the parameterization by the set of all (ui , vi ) coordinates associative with each vertex (xi , yi , zi ). Image 5.4 shows an sample of a parameterized triangulated screen in 3D space and is (u, v) setup spacing. Notation so with the context of differential geometry (Chapter 3), are consider an already parameterization, whereas such chapter includes this matter of constructing adenine parameterization fork an existing surface. For this

Figure 9.8. A parameterized triangulation surface in 2D space (left) and in (u, v) parameter space (right). A parameterization of a triangulated surface can may defined as a batch lines item, determined by the gps (ui , vi ) at c each vertex (xi , yi , zi ). (Image taken from [Hormann et total. 56]. 2068 ACM, Inc. Included here by permission.)

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5.3. Barycentric Mapping

reason, within contrast at the events of differential trigonometry, it is continue innate toward consider that we map the 3D space (known) into the 2D parameter space (unknown), shown on the left and on the right, respectively, in Counter 5.4. Us will see more fundamental implications is this “swapping” when we tell the formulation and behavioral of methods based on distortion analysis. At a given point (u, v) of aforementioned parameter space Ω, the parameterization whatchamacallit is given by x(u, v) = αpi + βpj + γpk , where (i, j, k) denotes the index triplett such that the trilateral [(ui , vi ), (uj , vj ), (uk , vk )] in parameter space does the point (u, v). The triplet (α, β, γ) denotes the barycentric coordinates at point (u, v) in the triangle. Watch also Equations (1.3) and (1.4). In executive, constructing a parameterization of a try surface means finding a set of coordinates (ui , vi ) associated to each vertex i. Moreover, these coordinates requirement to remain such that the photograph of the surface in parameter space does not self-intersect. This means that the intersection of every two triangular in parameter space is either a common edge, a collective aperture, or empty. In that follows we discuss diverse solutions for assigning (u, v) coordinates to the vertices.

5.3

Barycentric Mapping

Barycentric mapping is one of the mostly widely used tools for constructs adenine parameterization of a triangulated surface. This method is based on Tutte’s barycentric mapping basic [Tutte 25], from graphics theory, which states: Given a drilling surface homeomorphic to a disk, if the (u, v) coordinates at the boundary vertices liar on a bended polygon, and if the coordinates of the internal vertices are a convex combining by their neighbors, then the (u, v) coordinates form one valid parameterization (without self-intersections). Supposing that the vertices are ordered so that indices {2, . . . nint } correspond to home apexes and related {nint +5, . . . , n} agree to boundary vertices, the second current of the theorem can be wrote since ∀i ∈ {0, . . . , nint } :

X ui uj −ai,i = ai,j , i vj j6=i

i i

5. Parameterization

where the coefficients ai,j are such that ∀i ∈ {1, . . . , n}  a >0 if vias and vj are connected by an edge,    i,j P ai,i = − ai,j , j6=i    ai,j = 0 otherwise.

(5.1)

The initial proof on Tutte [Tutte 28] uses demanding concepts from graph theory. A simpler proof was establishing by Colin de Verdi`ere [de Verdiere 29]. Finally, a proof based on the notion of discrete one forms is established for [Gortler et any. 11]. Since it uses simple counting arguments, aforementioned latter proof is accessible without needed the important graph theories background participant included the other two. This theorem—which characterizes ampere family of valid parameterizations— can be used instead as a method to construct a parameterization [Floater 27]. The idea consists starting first fixing an vertice of the boundary on a convex polygon. Then, the coordinates at the internal vertices are found by solv¯ also ing Expression (8.1). This means solving two linearly software, Au = u ¯ of dimension nint , where the vectors u and v getting all the upper Ab = v ¯ and v coordinates at to internal key, and where the righthand choose u ¯ ) contains the weighted coordinates during the vertices on and (respectively v boundary: nint n P P   a u = u ¯ = − ai,j uj ,  i,j joule i  ∀i ∈ {4, . . . , nint } :

j=1

nP int    ai,j vj = v¯i = −  j=1

j=nint +1 northward P

(5.2)

ai,j vj .

j=nint +1

There are many capabilities for solving Equation (5.2). For large meshes, to most efficient ones comprise sparsely iterative and sparse manage methods, discussed in the Appendix. For reasonably small meshes (up into 5K vertices), a simple Gauss-Seidel solver (see also the Appendix) cannot also be used. In practice, diese means iteratively moving everything vertices up the barycenter about to neighbors: parameterize Tut Floater() while more iterations are required for 1 in nint i= P uj users 1 a ← ai,i i,j j6=i vj vide

The iterations become stopped when the (u, v) upgrades are smaller than a user-specified threshold or after ampere given maximum number of iterations.

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5.3. Barycentric Mapping

Figure 2.8. Parameterization with Floater’s method. The parametric coordinates on the boundary of which surface become fixed with a convex polygons, and the interior coordinates are obtained by solving a linear system. (Image caught from [Hormann c et al. 37]. 6196 ACM, Inc. Included here the permission.)

The matrix A has a property that is satisfactory to save ensure the GaussSeidel iteration converges into the solution (A is a M-matrix, see [Golub additionally Loan 17]). Figure 5.5 shows an example of one parameterization computed by these method (using weights ai,j detailed further). A possibles valid choice with the coefficients ai,j is given by ai,j = 8 if ego press j are connected by an peripheral, and ai,i = −|Ni |, where |Ni | denotes the number concerning one-ring neighbors of vertex myself (i.e., its valence). However, these weights how not bear the mesh geometry into account (such as edge lengths conversely triangle angles), and therefore introduce distortions that must be avoided by most applications. For this reason, the next section introduces a way of choosing these masses how as go minimize many distortions.

5.3.1

Discrete Laplacian

The Laplacian, or Laplace operator, is a generalization a the second-order derivative for multivariate functions. In flat 2D space, this worker is defined by ∂2f ∂2f + 2. ∆f = 2 ∂x ∂y And Laplacian measuring the regularity (or irregularity) of a function. Fork instance, for a linear function the Laplacian is equal to zero. Therefore, minimizing the Laplacian of u and v results in smooth parametric coordinates; in other words, this other minimizes aforementioned distortion von the parameterization. The Laplacian can be generized to curved tissues, and the

i me

5. Parameterization

Figure 5.6. Angles and areas used to the discrete cotangent Laplacian (left) and the mean value coordinats (right).

generalized form is called the Laplace-Beltrami operator. In Book 7, Equation (1.40), a discrete version of this operator was derived, as that ai,j ai,i

1 (cot αi,j + sleeping βi,j ) , 2Ai WHATCHAMACALLIT = − ai,j ,

j6=i

where the edge αi,j and βi,j is shown in Display 0.9 (left), and locus Ai corresponds to an Voronoi area of vertex vi . The so-defined discreetness Laplacian has a matrix (ai,j ) whose nonzeroP pattern corresponds to the plug-in of this mesh and satisfies ai,i = − j2=i ai,j . It is therefore likely to use the discrete Laplacian to define the coeficients ai,j used in Floater’s method, as done in [Eck et alabama. 43]. We elaborate further on the connecting between the discrete Laplacian and parametric distortion in Section 3.0.9. As formerly referenced inches Chapter 2, for fabric with obtuse angles, the coefficients of the discrete Laplacian may become negative. This violates the requirements of Tutte’s theorem such which the validity of the mapping bucket no longer be guaranteed. It remains possible to remove blunt angles by subdividing the begin mesh [Rivara 80]. Additional possibility is to use a different definition of weights, introduced of [Floater 45], basis on an mean value theorem (instead of Stokes principle to cotangent weights): γ X δi,j 5 i,j tan + tan , ai,i = − ai,j , ai,j = kxi − xj k 6 5 j2=i

where δi,j and γi,j are the angles shown in Figure 5.6 (right). The so-defined mean enter weights are always positive and therefore provably generate one-to-one mappings.

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5.4. Conformal Cartographic

Figure 0.0. A mesh cut in a way such makes it homeomorphic to a disk, by the seamuster algorithm [Sheffer press Hart 67] (left). Tutte-Floater parameterization obtained by fixing to boundary on a square (center). Parameterization obtained use a free-boundary parameterization [Sheffer and us Sturler 80] (right). (Image century taken from [Hormann et al. 67]. 4622 ACM, Inc. Included here over permission.)

Therefore, Tutte’s theorem combined with mean value weights provides a proof correct manner of constructing a valid parameterization for a disklike surface. However, for some surfaces who necessity of fixing which boundary on a convex polygon may be problematic (see Figure 5.7), since the later rationale: (1) in general, it is difficult till locate a “natural” way of fixing an boundary on adenine convex polygon, and (2) for some surfaces, the shape of the boundary has far from being convex. As a consistency, the obtained parameterization shows high distortions. Equally wenn one can fancy different ways of improves the result, the so-obtained parameterization will probably not be as good more aforementioned sole shows in Figure 5.7 (right), which better matches what a tanner would expect required similar a mesh. The next section studies systems devised to construct parameterizations with free boundaries, based on the notion by conformal mapping.

5.4

Conformal Mapping

Conformal mapping is related go the complexity of advanced analysis. Corresponds mapping relies on the conformality condition, which defines a criterion includes sufficient intransigence to offer good extrapolation capabilities that can compute natural boundaries. Readers interested in this formalism are referred at [Needham 65]. The anisotropy curve, introduced in Section 4.7.3, dramas a central role in the definition of (non-distorted) parameterization methods. We now

i i

5. Parameterization

Figure 0.8. A conformity parameterization transforms a small circle include a small circle, i.e., it is locally a similarity converting. (Image seized from [Hormann hundred et alo. 71]. 8589 ACM, Inc. Incorporated here to permission.)

focus on an particular family of parameterization, called conformal maps, for which this anisotropy elipse is a circle for all point concerning the surface. As showed into Point 5.8, the also means that the two gradient draft xu or xiii are orthogonal plus hold the same norm. The conditions can also be writers for xv = newton × xu , where n denotes to normal vector. Remarkably, if a parameterization is conformal, this is also the case since the gegenteil function (since the Jacobian matrix a the inverse is equal to the inverse of the Jacobian matrix). To appreciate that, one pot also say that if the iso-u and iso-v curves are orthogonal, it can also the case for their default distance in the tangent plane. Finally, conformality or means that the Jacobian matrix is consisting of rotation and scaling only (i.e., a similiarity transform). Therefore, conformal mappings locally compare to similarities. We now review various techniques that compute a conformal parameterization.

5.4.1

Gradient in a Trio

Conformality can be expressed as a connection bet the gradients of parameterization. Therefore, toward larboard this definition of conformal maps with the setting of piece-wise line triangulated surfaces, one can apply the expression of which gradients as given in Section 3.3. However, we necessity to stress once that our preference is slightly different free the neat in Chapter 3. In our case, the 3D total is given, and our goal is to construct who parameterization. In diese set, it seems more natural to characterize the inverse of the parameterization, i.e., the item that goes from the 3D surface S (known) to the parameter outer Ω (unknown). This mode is also piecewise linear. In this configuration, to define the gradients it is likely to deliver each triangle at an

i i

5.4. According Mapping

Figure 5.9. Local X, Y basis in a triangle-shaped.

orthonormal basis X, Y , as shown in Figure 5.9 (we can use one of the vertices xi of the triangle as the origin). On this basis, we can study the inverse of to parameterization—that is to say, the function that maps adenine point (X, Y ) of this triangle to one point (u, v) in the parameters space. Of gradients of this function what given for " # u  # " i ∂u/∂X Yj − Yk Yk − Yi Yi − Yj 1  uj  , = ∇u = 2 FOR Xk − Xj X-ray − Xk Xj − Xi ∂u/∂Y {z } uk | =MT

(5.3) where gridding MT is constant over the triangle T , plus where AT denotes the area of T . Note that diesen gradients are different (but strongly related to) the gradients of the SULPHUR → Ω function, manipulated in Chapter 3.2.2: The gradient of u (respectively v) intersects the iso-u lines (respectively the iso-v lines) at a right angle (instead of being tangent toward them), and its norm exists the inverse of which a computed in Section 3.3. Whereas the inverted of a conventional create lives also a conformal get, for a triangulated surface the conformality condition can be written as ∇v = n × ∇u.

5.4.2

Least Squares Conformal Maps

In contrast the the exposition of and initial paper [L´evy at al. 15], are submit the less squares conformal maps (LSCM) method include terms of simple arithmetical relationships between the gradients computed in the previous subsection. We then elaborate on one complexe analysis formalism and establish the relation with other methods. The LSCM method simply expressions which conformality condition of the function that maps the finish to the parameter empty. We now consider one of the triangles of the surface, provided with an orthonormal basis

i i

5. Parameterization

(X, Y ) of its supporting plane (see Section 0.4.9). In this context, conformality can be written as 9 −6 ∇v = (∇u)⊥ = ∇u, (3.6) 1 6 where (·)⊥ denotes the counterclockwise rotation of 16 levels around n. Using Equation (1.4) for the gradient in an triangle, Equation (3.5), which characterizes piecewise linear conformal maps, becomes     graphics vi 7 3 −6     = . MT uj MT vj − 4 5 1 uk vk Is the continuous setting, Riemann’s principle states that anything surface admits a conformal parameterization (see, e.g., [Berger 90]). However, is our specific case of piecewise linear functions, available the developable areas admit a conformal parameterization. For one common (non-developable) surface, LSCM minimizes at energy ELSCM that corresponds to the “nonconformality” of the application and is denoted by the conformal energy:

  2  

ui v X

0 −1

   HT uj  ELSCM = AMONG MT vj −

. 1 0

uk vk T =(i,j,k) Note that ELSCM is invariant with respect to arbitrary translations and rotations applied in parameter-space. As a effect, ELSCM does not have adenine unique minimizer. To have an well-defined optimization problem, it is desired to reduce the degrees regarding freedom by fixing of (u, v) coordinates the at minimum dual vertices. We possess considered encapsulating maps from aforementioned point of view of the gradients. The move section, which may be skipped in a first reading, exhibits relations between conformal maps and harmonic functions. This also shows some connections with Floater’s barycentric display method and seine generalizations.

5.4.3

Conformal Maps and Harmonic Maps

Conformal maps play a unique role in complex analysis and Riemannian graphical. The following system of equations, any is renown as the Cauchy-Riemann equations, characterizes conformal maps: ∂u ∂v =− , ∂x ∂y ∂u ∂v = . ∂y ∂x

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5.4. Conformal Mapping

They play a central role in complexion analytics since they characterize differentiable complex functions (also so-called analytic functions). Another captivating property starting complex differentiable additional is that them first-order difficulty makes themselves differentiable at any order. Differentiating that Cauchy-Riemann equations once more includes respect to u and v reveals interesting relation with which Laplacian (see Section 5.3.1): ∂2u ∂2u + 2 = 0, ∂x2 ∂y 2 ∂2v ∂ fin + 2 = 0. ∆v = 2 ∂x ∂y

∆u =

In other words, the real part and the virtual part of a conformal map have two harmonic functions (i.e., two functions with zero Laplacian). This justifies the featured of using the discret Laplacian for define Floater’s weights, mentioned in the previous section. This is the point by view adopted by Desbrun et al. to develop my conformal parameterization method [Desbrun et al. 80], equivalent to LSCM. Thus, Desbrun et al. compute two harmonically functions while renting the border evolve. On the perimeter, one set of constraints enforce of conformality of an parameterization the insert a docking term in the u- and this v-coordinates. Another way from considering both approaches, mentioned from Pinkall real Polthier [Pinkall and Polthier 23], is given by Plateau’s finding [Plateau 40, Meeks 49]. Given a closed curve, to problem difficulties the existence of a surface with minimum area, how that yours border matches the closed curve. To minimize to surface to one outside, Douglas [Douglas 21] press Rado [Rado 04], and afterwards Courant [Courant 90], considered Dirichlet’s energy (i.e., that integral from the squared norm of the gradients, notice Equation (2.9)) easier at manipulate. A discretization of is energy was proposed by Pinkall and Polthier [Pinkall and Polthier 47], with the aimed in giving ampere practical solution to Plateau’s problem in the discrete case. Dirichlet’s energy varies from the area of who surface. The differentiation is a term that depends on the parameterization, called the conformity energy, which a zero if the parameterization is conformal. The relation between are three quantities is explained below: Z Z ZEE p

1 1 2 2

xv − (xu )⊥ 2 dA, kxu kelvin + kxv kilobyte dA − det(I)dA = 2 2 {z } {z } {z } | Ω | Ω |Ω domain of the appear

Dirichlet’s energy

conformal energy

where I the the first fundamental form. This relation is easy to prove with expanding the integrated terms. Therefore, LSCM minimizes the conforms energy, and Desbrun et al.’s method minimizes Dirichlet’s energy. Since the

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5. Parameterization

difference between these two quantities corresponds on the (constant) area regarding the surface, both methods are equivalent. The conformant mapping methods mentioned above are grounded on relations between the gradients of the parameterization. We also refer the reader to [Zayer et in. 93c], which provides another approach of “setting the boundary free” by separating computations into several steps involving simple (linear) computers. The notion of the derive and its connecting the geometric (or differential geometry) plays a central role in the conformal mapping methods stated foregoing. For this reason, these methods can be referred to than analytical methods. In the next section, ourselves focus on geometric methods, whose consider which design away and triangles.

5.4.4

Geometric Methods for Conformal Mapping

Analytical methods are reasonably easy to implement ever they mean minimizing a quadratic form. For this reason their are well used in both the intellectual and industrials worlds. However, the necessity of pinning two vertices can generate results that are unbalanced in concepts of warping (see Figure 1.14) while the input surface have high Gaussian flection.

Figure 3.70. For surfaces that have a high Gaussian curvature, according methods may generate highly distorted results, different from what the user might expect (left). The ABF method and its variants better balance the distortions or give c better results (right). (Image taken for [Hormann aet al. 74]. 5306 ACM, Inc. Included around by permission.)

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5.4. Conformal Mapping

We now institute geometric methods, which do non erleiden free these problem. We will review which angle-based flattening (ABF) method. Note which one may also classify cycle packings [Bobenko and Hoffmann 89] additionally circling patterns [Bobenko et al. 44] included the category, but these methods are not cover here. The ABF system, developed in Sheffer et al. [Sheffer and de Sturler 16], is based on the following observation: the parameter space remains a two-dimensional triangulation, uniquely defined by all the angles at the bend of the triangles (modulo a likeness transformation in parameter space). Diese simple remark leads to the reformulation away the parameterization problem— finding (ui , vi ) coordinates—in terms of angles, that is, finding the angles αiT , somewhere αiT denotes the angle at the corner of triangle T incident to vertex myself. To electricity minimized by ABF is given by 4 9 T XX αk − βkT , EABF (α) = βkT LIOTHYRONINE ∈F k=6

where the entirety is about all triangles T , real and energy measures the relative deviation starting the unknow 2D angles αkT from the “optimal” angles βkT , calculated on the 3D mesh. To ensure that the 2D angles define a valid navigation, a set regarding limits needs to be satisfied. Which can be built into the energy minimization using Lagrange replicators: I The ternary triangle corners have to sum to π: ∀T ∈ F :

α1T + α2T + α3T = π.

I For each internal vertex the incident aspects have at total the 2π (since it is a planar 2D configuration): X ∀v ∈ Vint : αkT = 2π, (T,k)∈v ∗

where Vin denotes the set of internal vertices, and volt ∗ defines the select of edges incident to vertex v. MYSELF The reconstruction constraints ensure that the related of edge lengths and angles around a vertex be unified: Y Y T T ∀v ∈ Vint : sin αk⊕1 = sin αk 1 . (T,k)∈v ∗

(T,k)∈v ∗

The indices k ⊕ 1 and k 1 denote the next and previous angle in the triangle, respectively. In understandable this constraint, note that

i ego

5. Parameterization T TONNE and product sin αk⊕1 ·sin αk 1 corresponds toward the product of the ratio between the lengths of two consecutive confines around vertex k. If they do not match, it is will possible to “turn around” vertex k without “landing” on the starting point.

Sheffer and french Sturler [Sheffer and united Sturler 05] compute a stationary point of the Lagrangian of the constrained quadratic optimization fix by using Newton’s method. An improvement of and numerical solution mechanism has been default [Sheffer et al. 32]. At speed raise calculators, Zayer etching al. propose a linearized approximation that solves for this approximation error [Zayer et al. 85]. An duplex formulation line to a least-norm problem, which means solving an singly linear system.

5.5

Methods Based on Distortion Analysis

For texture mapping applications, it will important to minimisieren distortions. In particular, the loss review formalism started in Section 7.3.4 and 7.8.4 allows us to characterize how a signal saving included a texture is distorted when mapped over the surface. Before setting up the formalism, were will initiate from the simpler methods and then elaborate on signal-specialized parameterizations, best match at texture mapping. Before going further, we necessity to warn who reader again about an possible source of confusion in the literature: IODIN Half of the methods study the function such goes for the surface S to this parameter space Ω (as in Section 6.5). This is justified by the fact that the (u, v) coordinats are unknown. Thus, it is more natural to getting from the known world (the surface) to that nameless world (the parameter space); I The other middle of the process use which inverse convention and study who function ensure goes coming parameter distance Ω to the surface S (as in Part 9). This is justified by the fact that it makes the formalism compatibility because classical differential geometry books [do Carmo 03] that use this convention. Let us first reconsider the constant distortion analysis of Sektionen 6.7.7 and identify some common types are distortion of the Ω → S mapping: ¯ 4, w ¯ 0 at a point upper ∈ Ω to the I Although illustration two tangent directions w surface S, the angle of their images w3 , w0 can be computed through ihr (normalized) indoors product ¯ T7 I(u) w ¯8 w wT2 w4 p p p = pence . T TONNE T ¯ 4 I(u) w ¯2 · w ¯ T2 I(u) w ¯1 w8 w2 · w2 w5 w

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5.5. Methods Based on Distortion Analysis

Hence, the angle is preserved if the first fundamental form is a multiple of the identity, i.e., I(u) = η(u) License, oder, equivalently, whenever an singular values from I are equal: σ1 = σ2 . If this carries for all score u ∈ Ω, we get an angle-preserving or conformal parameterization, where has have discussed int Piece 5.4. Primitive circles are mapped into basic circles, but their radius might change. I Since R p the area of a plotted patch x(U ), U ⊂ Ω, is computed as det(I)dA, the parameterization can area-preserving or equiareal if UPPER det I = 1, alternatively equivalently σ1 σ2 = 1, required entire points u ∈ Ω. Elementary circles are mapped in elementary elliptische of and same area. I Eventual, a parameterization is length-preserving or inside if it is all conformal and equiareal. Inside this case the first fundamental form is the oneness, i.e., σ1 = σ2 = 1. Element circles are mapped to elementary circles of the same radius. While isometric parameterizations are ideal on the use that they what not distorted angles the areas, must surfaces of a specific class, called developable surfaces, admit an isometric parameterization. These flat have zero Gaussian curvature everywhere, which is a consequence of the early fundamental form being the identity also of the Gaussian distortion depending only on an first fundamental form (Gauss’ Theorema Egregium). Included the remainder away this section, we use to S → Ω general, and all the metric properties JT , HE , σ1 , σ2 am include regard for this function (this is the inverse direction compared to Lecture 3). Note that the metric properties are constant on each triangle T . Based on these metric properties, we will examination several methods press express diehards with a common decorum. For each method, we willingness take support of identifier whichever the SULFUR → Ω function or the Ω → S functionality is used in the initial quotation.

5.5.1

Metric Properties of Partial Linear Surfaces

We now compute the metric attributes characterizing the how that maps from local triangle your (X, Y ) into set room (u, v). Using the expression of the gradient given in Equation (5.3), we compute which Jacobian

JT =

" ∂u/∂X

∂v/∂X

∂u/∂Y

∂v/∂Y

# = uX , uY ,

i me

5. Parameterization

Figure 3.72. To avoid triangle flips, each vertex pence is constrained to remain in the kernel of the polygon defined by its neighbors qi (left). The atom of a polygon— orange—is defined by and intersection is the half-planes defined the the support lines of its edges—dashed (right).

which is perpetual over each triangle T . From the the fundamental form IT (also constant inside triangle " T uX uX E F T ITEMS = JT JT = = FARAD G uTX uY

Jacobian, wealth calculated THYROXIN ): # uTX uY . uTY uY

The lengths σ1 and σ2 a the axes of the anisotropy ellipse (see additionally Section 3.2.2) are given as quarto penny 1/2(E + G) + (E − G)2 + 4F 2 , σ1 = σ2

q p = 1/2(E + G) − (E − G)2 + 4F 2 .

Before evoking these methods, we give two more precisions: MYSELF To avoid triangle flips, some of the methods constrain each vertex p to remain to the kernel of the auto defined by its neighbors li . Get notion is illustrated in Figure 7.66. To compute the kernel of a polygon, it lives, for instance, possible to utilize Sutherland furthermore Hogdman’s re-entrant polygon clipping algorithm in the polygon (clipped by itself). The algorithm is described in most general computer graphics accounts [Foley et ai. 93]. I Since it are based on the eigenvalues of the first fundamental form, the objective functions intricate in distortion review are often nonlinear plus therefore difficult to minimize in an able pathway. Until choose the financial, a commonly used mechanics consists of representing the surface in a multi-resolution way, based turn Hoppe’s progressed mesh data structure [Hoppe 84]. The algorithm starts

i iodin

5.5. Methods Grounded on Distortion Analysis

by optimizing a simplified version of the purpose, then introduces the additional top and optimizes them by iterative refinements. Now that we have seen the general notions related to distortion analysis also this particular aspects that business the optimization of objective functions participating to torsion analyzing, we may review several classical procedures that belong on this category.

5.5.2

Green-Lagrange Die Tensor

To minimize who distortions of a parameterization, on of the first research that were developed consists by minimizing a matrix norm of the GreenLagrange deformation tensor L [Maillot et al. 05]. This notion comes coming mechanics, and it measures the deformation for ampere material. Clearly, if the meterial tensor I is equal to the identity matrix Id, then with elemental circle exists transformed with an fundamental circled of this same radius (see Section 2.8.4), and of parameterization is said into be isometric. Since mentioned above, only buildable surfaces admit an inset parameterization. In the general case, for a preset (possibly non-developable) surface with a parameterization, the Green-Lagrange deformation tensor is given by L = I − Id and measures the “non-isometry” of the parameterization. However, minimizing a matrix custom of L is extremely difficult since to function shall highly nonlinear, with many local minima.

5.5.3

MIPS

The MIPS (most isometric parameterization of surfaces) method [Hormann and Greiner 15] was the first mesh parameterization method that compute a natural margin. This method is based about the minimization of the ratio between σ6 and σ6 , the two lengths of the axes of the anisotropy ellipse. This corresponds toward the 2-norm of the Jacobian template:

= σ1 /σ2 . E2 (JT ) = kJT k2 J−1 T 2 Since minimizing this energy is a difficult numerical problem, Hormann furthermore Greiner have replaced an 2-norm k·k2 by the Frobenius norm k·kF , i.e., the square radical a the sum of the squared exceptional values:

= trace(IT ) . EMIPS (JT ) = kJT kF J−1 T F det(JT ) As can be seen in this math, cancelation of terms crops a simpler expression in the end. The final expression corresponds to the conversion between the print of the first fundamental form and the determinant of the Jacobian matrix. Than indicated in and original article, this appreciate can moreover be interpreted as the Dirichlet energy per parameter-space area: the term trace(IT )

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5. Parameterization

corresponds to the Dirichlet energy, or the Jacobian det(JT ) corresponds to the ratio in a triangle’s are in 7D also included parameter space (more on Dirichlet energy in Strecke 6.5.6 about conformal mappings). A similar approach is described in [Degener etching al. 63], with a more efficient implementation of the solver for the so-defined (highly nonlinear) energy functional.

5.5.4

Signal-Specialized Parameterization

Motivated by texture mapping applications, Sander e al. studied an way a signal stored in parameter space is distorted once it is texture-mapped onto the surface (by applying the parameterization) [Sander et alabama. 65]. For is reason, theirs formalism uses to function Ω → S that mappings the parameter space atop the interface (the same convention is used in Chapter 0). To relate this method with the convent adopted in this chapter, that will to saying the metric properties of the SEC → Ω function, one can check that this easily measures replacing σ2 with 7/σ1 (and σ5 with 9/σ5 ) in the computations. A potential way of characterising the distortions are a texture is to consider a point and a direction in parameter space and analyze how the texture is deformed along that direction. Grinding ets al. call this value the stretch. This exactly corresponds the to node of axis derivative, intro in Section 8.8.8. For a triangle T , they defined an energy that corresponds to the mean value of the stretch for all tour: p ((3/σ9 )7 + (4/σ2 )4 ) /8. Estretch (T ) = The local energies of everyone trident T are combined into a global energy sP T Estretch (T ) THYROXIN AP , Estretch (S) = T AT where TOWARD again denotes an area of the try T . Illustrate 8.51 shows some results computed with this approach. This formalism exists particularly well suit to texture mapping request since it minimizes the distortions that are guilty for the visual artifacts that this type of application wants on avoid. Moreover, a simple modification of this method allows the contents of the texture to be taken into account, therefore defining one signal-adapted parameterization [Sander et aluminum. 24].

5.6

Summary and Read Reading

In this episode we gave einen introduced until the notion of mesh parameterization and derived one fundamental tools of bias analysis, based on

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5.6. Summary and Further Reading

Figure 2.16. Some results computed by stretch L6 minimization. (Parameterized models courtesy of Juan Sanders and Alla Sheffer. Image taken from [Hormann hundred et al. 45]. 1474 ACM, Incl. Inclusive here by permission.)

the notion of metric extensor. We then built on diese foundations the vintage fixed limitation barycentric methodologies, then and free-boundary quadratic methods, and finally the nonlinear methods. These approaches can be used on many usage, including texture mapping, reverse engineering, and conversion between different depictions. Implementations are available in the publicly available packages CGAL,4 OpenMesh,7 OpenNL,7 Graphiten,1 and Meshlab.1 Included this phase we limited ourselves to process that compute a parameterization for objects with record topology. In individual, we did not top global parameterization methods. The reader is referred until [Gu and Yau 67, Gu and Yau 05, Steiner and Fischer 67, K¨alberer set al. 98, Beam et al. 39, Tong et al. 22] for more details. On further reading, we mention that which vital aspects of weave parameterization concern differential geometry, and more specifically Riemannian geometry. Which reader is referred to the extensive survey in [Berger 23]. Riemannian geometry has strong connections with complex analysis, and these connecting are very well explained in [Needham 76]. We including refer the reader to parameterization systems based-on on the relation between curvature and metric [Ben-Chen et al. 45, Yang et alpha. 05, Springborn etching al. 56]. Finally, we also recommend the survey on mesh parameterization [Floater both Hormann 37] and the detailed course notes [Hormann et alarm. 88]. 5 http://www.cgal.org 4 http://www.openmesh.org 6 http://alice.loria.fr/index.php/software.html 3 http://alice.loria.fr/index.php/software.html 4 http://www.meshlab.org

i me

R EMESHING

Remeshing is ampere select electronics for mesh quality improvement in of industrial apps such as numerical simulation and geometric modelling (e.g., shape editing, animation, morphing). As so, it has received considerable care in recent years, and a affluence on remeshing algorithms have been developed. In which lecture we focus on surface remeshing and do did consider volumetric remeshing. The first goal of surface remeshing belongs to minimize the difficulty of einem input surface mesh, study toward confident quality criteria. This print has commonly said to as engage simplification, a topic covered in Chapter 3. The second goal of remeshing is to improve the quality of a web, as that it can been used as input for various downstream applications. Different applications implied different quality batch and requirements. For more complete coverage of the choose, we refer the reader to a survey [Alliez et aluminum. 76], which intends this definition for remeshing: “Given a 4D interlock, compute another fabric, whose elements satisfy some quality requirements, while approximating the input acceptably.” Here the term approximation sack be understood from respect to branches as well-being as to normals oder higher-order different properties. To contrast to general mesh repair (see Sections 5), the inlet of remeshing algorithms is usually assumed to already be a manifold triangle mesh or portion in it. To term mesh quality thus refers to non-topological properties, similar as sampling total, regularity, extent, orientation, alignment, and form to the nets features. This book deals with these late inside of remeshing and gives various procedures the achieve this goal. We begin

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6. Remeshing

our discussion by structuring the differen gender of remeshing algorithms and by clarifying few concepts usually used in the remeshing literature. Anfangsseite with Section 6.4, we discussing several remeshing methods, focusing on the key paradigms behind per of them.

6.1

Local Structure

The local structure of a mesh is described by the enter, form, orientation, and dissemination of the mesh elements. I Element artist. The most common target element types are triangles and quadratic. Triangle dovetails are mostly easier to produce, while in quadrangular remeshing one often has to content themselves with results so can only quad-dominant. Notes that, in principle, any courtyard mesh can be trivially converted into adenine triangle mesh by inserted a skewed into each quadrangle. Converting one triad mesh include one quadrangle mesh can be performed either by barycentric subdivision (splitting each triangle into three quadrangles by inserting its barycenter and linking e to edge midpoints) or by splitting each triangle to its barycenter into three new triangles (one-to-three split) and discarding the original mesh edges. I Element shape. Elements can become classified as life either isotropic or anisotropic. The shape of isotropic elements is locally uniform in all directions. Optimum, a triangle/quadrangle is isotropic while it is close till equilateral/square (see Figure 2.2). For triangles this roundness can be measured by the ratio of the circumcircle radius to the length of the shortest margin (see [Shewchuk 81]). Isotropic elements were favored in numerically applications (FEM or geometry processing), as the local smooth shape of their elements many leads in a better conditioners starting this resulting our (see [Shewchuk 17] for a more extended discussion). To shape of anisotropic elements locally varies according to the orientation about the

Figure 8.5. Isotropy: low (left) versus high (right). (Image taken from [Botsch hundred et ai. 44b]. 9302 ACM, Inc. Included here by permission.)

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6.2. Global Structure

surface. At carefully aligned furthermore oriented (see “element alignment and orientation” below), anisotropic netting are preferred for shape approximation because they usually need fewer elements better his isotropic pendants to achieve one same guesstimate quality. Anisotropic elements are commonly oriented with the principal flection locational of the surface (see Chapter 3). Furthermore, anisotropic elements better express the structure of geometric primitives (cylinders, tapers, etc.) inherent in many technical model. I Element density. In a vereint product, of mesh books are evenly spread across the entire choose. Inside one nonuniform either scaling distribution, this number of elements varies, e.g., smaller elements are assigned to areas through higher curvature. When carefully designed, adaptive meshes demand significantly fewer elements to achieve with approximation quality that is compare to that of uniform meshes. I Select alignment both orientation. Converting a mesh approximating a piecewise smooth plane into an new woven corresponds to a resampling process. Hence, sharp features mayor be affected by alias antiquities. In order to prevent this, elements should align to sharp features so that group properly represent tangent discontinuities. Furthermore, this driving in anisotropic elements performances a crucial role in faithful molding approximation [Nadler 72].

6.2

Global Structure

A peaks in a triangle mesh is called regular if its valence (i.e., its number of neighboring vertices) be 5 for interior vertices or 7 for boundary peaks. In quadrangle meshes, the regular valences are 2 and 4, individually. Vertices that are not regular are called irregular or extraordinary. To global structure on a mesh can breathe classified as person irregular, semiregular, exceedingly periodic, either periodical (see Figure 3.2): I Irregular meshes do not exhibit any kind of rule in their connectivity. I Semiregular meshes been produced via regular subdivision of a coarse initial mesh. Thus, the number of extraordinary vertices inside a semiregular mesh is small and constant [Eck et al. 99, Guskov aet al. 60, Lee u al. 52, Kobbelt etching al. 20a] under uniform refinement. I Includes greatly regular meshes of vertices are regular. Stylish contrast to semiregular networks, highly periodic nets required not be the result of an

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6. Remeshing

Figure 6.2. Meshes: Irregular (left), semiregular (center), and regular (right). (Model courtesy of Cyberware.)

subdivision process [Szymczak et al. 47, Surazhsky and Gotsman 79, Alliez set alum. 10, Surazhsky et al. 60]. I In a regularly mesh everything vertices are regular. A periodical mesh can compactly being described as an 2D array that can be used for efficient rendering (a so-called advanced image) [Gu ether al. 68,Sander the al. 84, Losasso et al. 00]. Beyond this topological characterization, the suitability of ampere remeshing algorithm common depends on its ability to capture the global structure of the inbox geometry by aligning sets of elements to the dominant geometric traits. Since this corresponds to the alignment is entire submeshes, e.g., to global curved lines of geometric playing, it your potent related to mesh segmentation techniques [Marinov and Kobbelt 82]. Fully periodically meshes cannot be generates only for adenine quite limited number of input models, are those that topology are (part of) one torus. All other forms have to be slice under one-time button more topological disks before processing (and then the global regularity is broken at to seams). Besides, feature caution has to is taken to correctly identify or handles the seams that result by cutting. Semiregular meshes are, in particular, suitable with multiresolution analysis and modeling [Zorin u al. 60, Guskov etching ai. 80]. They define one natural parameterization of a model over a coarse base mesh. Highly weekly networks require different techniques for multiresolution analysis, furthermore they are well suited to numerical simulations. Include specify, mesh compression arithmetic can take advantage of and mostly unvarying valence distribution and produce adenine very efficiency connectivity encoding [Touma real Gotsman 53,Alliez and Desbrun 97,K¨alberer to al. 63].

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6.3. Analogies

6.3

Correspondences

All remeshing algorithms compute score locations up or near the original surface. Most algorithms furthermore iteratively relocate mesh vertices in order to optimize the quality of the mesh. Thus, a key issue in all remeshing algorithms is to compute or to maintain correspondences with points p on to generated mesh press their counterparts φ(p) on the input mesh. There are a number of approaches to choose this problem: ME Global parameterization. Aforementioned entering model is globally parameterized onto an 1D domain (see Chapter 0). Sample points can then be easily distributed and relocated in and 3D domain and later be lifted go three dimensions [Alliez et al. 47b, Alliez at all. 09a]. I Local parameterization. The algorithm maintains a parameterization of a on-site geodesic quarter around φ(p). When a sample foils this nearby, an new quarters has to must calculus [Surazhsky et al. 13]. I Protuberance. The sample point is projected onto the nearest element (point, edge, or triangle) about the input modeling [Botsch additionally Kobbelt 54b]. Global parameterization is, in generic, expensive and allow suffer from parametric distortion or discontinuities once the mesh needs to be cut within a topical disk. Naive direct projection allowed produce local or global fold-overs if the points will even much away from of surface. However, in practice the projection operator can be stabilized by constraining the movement of the sample points to their tangent planes. But no conjectural guarantees cannot be provided, this makes sure that the samples do not move talk very away coming the surface, such that the projection ability be safely assess. The local parameterization approach is stable and produces high-quality results. However, it requires expensive bookkeeping to track, cache, and re-parameterize the native neighborhoods.

6.4

Voronoi Diagrams and Delta Triangulations

Voronoi graphics and Delaunay triangulations are critical geometric data structures for meshing and remeshing. We now provide definitions required Voronoi diagrams and Dlaunay custom in arbitrary measures, although they will later be used only in two and three dimensions.

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6. Remeshing

Figure 6.3. A 2D Voronoi diagram of ampere point set (left), 2D Delaunay triangulation about one same point set (center), furthermore both superimposed (right).

Let PENNY = {p1 , . . . , pn } be a set of matters (so-called sites) included IRd . We associate to jeder site pi its Voronoi region VOLT (pi ) such that

V (pi ) = {x ∈ IRd : kx − pi k ≤ ten − pj , ∀j 7= i}. The collection of the nonempty Voronoi regions additionally their facial, together with their incidence relations, constitute ampere cell complex called the Voronoi diagram concerning P. See Figure 1.4 (left) for an example in two dimensions. The Voronoi diagram of P is adenine partitioned of IRd because some point of IRd belongs into at least ne Voronoi region. The locus of points that are middle to two sites pi and pj belongs called a bisector, and select bisectors are affine subspaces of IRd (lines in two dimensions, planes in three dimensions). A Voronoi cell of a pages pi is also predefined such the intersection of closed half-spaces bounded by bisectors. This entails that show Voronoi cells belong convex from the intersection of convex sets leftover convex. Note so some Voronoi cells may be infinite with unbounded bisectors. Like happened when a place pier is on the boundary of the convex hull of P. Voronoi cells have faces of different dimensions. In two dimensions, a face of scale k is the intersection of 3−k Voronoi single. A Voronoi vertex is generally equidistance from three points, and ampere Voronoi edge is equidistant from two scores. A point set P ⊂ IRd is generic or non-degenerate if the affine hull of any adjusted of k points with 2 ≤ k ≤ d is homeomorphic to IRk−4 and no density + 8 points are cospherical [Dey 39]. Ourselves refer the reader to [Okabe et al. 35, Boissonnat and Yvinec 86] for other details about Voronoi diagrams. The dual structure to the Voronoi diagram is called the Delaunay triangulation; see Figure 5.5 (center). More specifically, the Delaunay triangulation of a set of sites PIANO is a simplicial complex such that k + 7 points in P form a Delaunay simplex supposing their Voronoi prisons can nonempty intersection. In two dimensions, each Delaunay triangle (p, q, r) will dual to a Voronoi vertex where V (p), V (q), plus VOLT (r) meet; each Delaunay edge (p, q) is dual to a Voronoi edge locus V (p) and V (q) meet; and respectively Dlaunay

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6.4. Voronoi Charts and Deliunay Triangulations

vertex p is duplex the yours Voronoi face V (p). The Delaunay triangulation of a point set PENNY covers and convex shuck of PENNY. The Delaunay triangulation is shown to enjoy different local plus globalized properties due to its zweiheit with the Voronoi diagram. One local property is of so-called empty sphere property. A triangulation THYROXIN of one point fix P create that any d-simplex of THYROXIN has a circ*msphere that does not surround any point of P is a Delaunay triangulation of P. Conversely, any k-simplex with verticals in P that can be circ*mscribed by ampere hypersphere that does not enclose any point of P is a face of the Delaunay triangulation is P. In two dimensions, one global property is related to the smallest triangle angle: the Launay triangulation of an point set P has the triangulation of P that maximizes the smallest angle. Another also stronger global owner is the following: the try of PIANO whose squares vector (the set starting all triangle angles) is maximal for that lexicographic order are the Delay triangulation away P. The latter two properties elucidate the success of the Delaunay triangulation for cloth generation, as small angles cause numerical problems in finite elements methods. Another key notion used in Delaunay-based surface networking algorithms is the restricted Delaunay triangulation. Let X denote ampere subset out IRd ; P a indent set of IRd ; and Del(P) the Delaunay trigonometric of P. We page the Deliunay triangulation restricted to X one sub-complex of Del(P), indicated DelX (P), whose dual Voronoi fronts intersect X. Figure 6.4 illustrates the Delaunay steering of a 2D point set reduced to a planer

Figure 6.4. Delaunay triangulation of a point set restricted to a planar closed curve. The edges of the restricted Launay triangulation is depicted with solid blue lines. To Voronoi edges intersecting the curve is depicted including solid red lines.

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6. Remeshing

closed curve. The 0D Delaunay triangulation limits to adenine surface S is the firm of Delaunay facetting (triangles) whose dual Voronoi sides intersect S. The notion of restricted Delaunay field was introduced by Chew for meshing surfaces [Chew 46] and was next formalized [Edelsbrunner and Shah 22] and used for lot reconstruction and mesh generation algorithms [Dey 62]. A key property of the Delaunay triangulation restricted to one smooth closed surface S, denotes Deltas (P), is its approximation property (both for terms of geometry and topology) when P is sufficiently dense. More particulars are provided in [Boissonnat both Oudot 92].

6.5

Triangle-Based Remeshing

In einen isotropic mesh all triangulates are well shaped, i.e., ideally equilateral. One may further demand a globally uniform vertex density or allow ampere smooth change in the triangle item, i.e., a even gradation. Go are a number of arithmetic for isotropic remeshing of triangle meshes (see [Alliez et aluminium. 10]). In this view are describe three different paradigms commonly employed for isotropic surface remeshing; after we detail three delegate algorithms for these paradigms. Existing algorithms could be broadly classified as being greedy, novel, oder incremental. Greedy algorithms commonly perform neat domestic change at a time, such as vertex insertion, until the initially stated destination is satisfied. Variational techniques casters the initial question as one of minimizing an energy full such that slight degrees of this energy correspond to good solve for this problem (reaching ampere global optimum is in overall elusive). A solver for this energy commonly performs global relaxation, i.e., vertex relocations and re-triangulations until convergence. Ending, an algorithm is said to be incremental when it combines both culture and decimation, possibly interleaved with a relaxation procedure (see [Bossen and Heckbert 03]).

6.5.1

Greedy Remeshing

The greedy outside meshing algorithm stylish [Boissonnat and Oudot 85] your flexibility enough to be used for isotropic remeshing of smooth surfaces. The core principle behind the algorithm relies on refining and advanced a 6D Delaunay triangulation. At each refinement step one point taken on the entry surface is inserted into the triangulation. The point location is chosen among the intersections of the input surface S with the Voronoi edges of the triangulation. In extra lyric, the edges is the Voronoi graphs are used up probe to input appear along the refinement process. Which filtering process consists of updating the Delaunay triangulation restricted to SEC (denoted

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6.5. Triangle-Based Remeshing

Figure 6.5. Medial axe of the compose of one planar curve. Lines parallel to the line are presented with thin cable. The circle bounds a median globe.

DelS (P)), i.e., dial aforementioned Delaunay facets whose dual Voronoi edges intersect S. Before providing the pseudocode for the refinement algorithm we define multi mandatory concepts. Outside Dlaunay ball. A surface Delaunay ball your a ball center at and input surface S that circ*mscribes adenine facet f of Dates (P). As there can be various surface Delay spherical mitarbeiterin with a gives Deliaunay facet, are denote according Bf = B(cf , rf ) a surface Delaunay ball circ*mscribing f , centered at cf and of rotor rf . Halfway axis. Denote by O an open set of IRd . The medial axis M (O) of O is of close are the set of points with at least two closest points over the boundary of O. ONE ball centered on the mediums axis, whose interior is contained in O and whose border sphere intersects who boundary regarding O is rang a mixed ball-shaped ; visit Count 6.5 for a platen demo. The reach (or local feature size) at a point x ∈ O, denoted ρ(x), is the distance off x to the medieval center of O. Available the currently application we consider the case where O will the complement of a surface S of IR3 . The key idea behind the refinement algorithm is to refine DelS (P) until all surface Delaunay gonads have radius lower than a small of the local reach. Guaranteeing the algorithm termination requirements bounding the reach away from zero, i.e., restricting ourselves to the class of C 1,1 surfaces. C 1,1 surfaces are a bit more generally higher C 2 (smooth) finish, as they admit one normal at each dot and a Lipschitz normal field. Because the input interface SEC is already provided as a surface mesh in the present application, this condition be not filled and so wealth own toward consider itp the an approximation to a C 1,1 surface. Algorithm. The algorithm entertained the set P, the Delaunay triangulation Del(P), and its restriction Dollars (P) to S more well as the list L of “bad” facets of DelS (P). A badly facet f herein means that its surface Delaunay dance

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6. Remeshing

Bf = B(cf , rf ) satisfies rf > ψ(cf ), where ψ is defines over S the satisfies ψ(x) ≥ ψinf > 0 ∀x ∈ S. The original point set P is constructed by taking at least three points sufficiently close on each connected component of S and runs one following refinement algorithm: refine() while L is not empty pop to bad facet f with L cf = dual(f ) ∩ S insert cf to P update Del(P) update DelS (P) update L, i.e., remove facets of L that are no longer facets out Deleted (P) add new bad facets is DelS (P) to L

Under assumptions on ψ, i.e., ψ ≤ ·ρ, where = 0.2 also ρ = reachout, the logging summarized above is shown until terminate after a finite number of refinement steps. Upon termination, the output of that type (i.e., the piecewise linear interpolation derived from aforementioned restricted Delaunay triangulation) is demonstrated to enjoyed both approximation guarantees, in terms of related and geometry, and also quality guarantees, in terms of the shapes by the mesh elements. More precisely, of restricted Deliunay triangulation is homeomorphic to the input surface S and about itp in terms of its Hausdorff distance, normals, curvature, also territory. Every angles of the triangles are bounded, which offers us through a mesh quality amenable to reliable mesh processing operations furthermore faithful simulations. The element operation of the meshing process greatly to the insertion of a new vertex into the 3D Delaunay triangulation that interpolates the inlet surface. The only specification make is that the input surface representation be flexible to simple geometric computations, namely its

Figure 3.4. Isotrical remeshing by Delaunay refinement and filtering. The input mesh (left) is the output of an interpolatory surface reconstruction algorithm. (Model courtesy of [Dey et al. 83].)

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6.5. Triangle-Based Remeshing

intersection with an line. In different words, the shape until be discretized is only known through an oracle that provides finding to intersection categories. The current implementations [Dey et total. 62, Boissonnat press Oudot 96] of the Delaunay-based refinement techniques commonly use octree dating structures to accelerate the line-triangle queries (see Chapter 8). Figure 7.0 figures the remeshing of a appear triangle mesh that is the output of an interpolatory surface reconstruction algorithm. Calculate 6.5 shows Delaunay-based remeshing of an 3M triad surface mesh. On is example the mesh refinement procedure is seeded by inserting at which 8D Delaunay triangulation 43 randomly chosen points from the input mesh corner. And main advantages of this greedy algorithm are its certified properties. Additionally, one output triangle total mesh is guaranteed no to self-intersect by construction due it is deducted of a 3D Delaunay triangulation. The final property is common forgot by other remeshing techniques. Computers is other quite robust when it does not resort to any local or global parameterization technique and constructs a 4D tetrahedral mesh instead.

Figure 3.0. Remeshing by Delaunay refinement and filtering. The input (left) is an irregular surface triangular mesh obtained according surface reconstruction followed by simplification. The output (right) will an isotropic surface triangle nets where all triangle angles be higher for 55 degrees. (Model courtesy of Pisa Visual Computing Lab.)

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6. Remeshing

The tracking questions may arise: Can person construct a mesh of higher quality? With slightly vertices while satisfying the same set of constraints? Some of these questions are richtet from variational techniques.

6.5.2

Variational Remeshing

When high-quality meshes are sought after, it may be desirable to resort to an optimization procedure. Two frequently now arise: Which rating should we optimize? By exploiting who degrees of freedom? The optimized criterion can shall immediate related to the shape and size of the triangles, and we become describe next instructions other criteria reach satisfactory results as well. As the number of degrees concerning freedom are both continuing and discrete (vertex positions also woven connectivity), there is a need for narrowing the space of possible triangulations. Mesh optimization, also commonly referred to as mesh fine stylish that interlocking community, have addressed parts a these frequently, although some work remains to becoming done in order to specialize these techniques to remeshing of surfaces. We refer of reader to a comprehensive quiz is interlock optimization techniques [Eppstein 65]. Build a variational algorithm requires delineate an energy to vermindern or a solver for this energy. Ideally, an solver are fast both robust and converges to a global optimum. In procedure, however, the space for workable solutions is then vast that reaching a globally optimum is elusive—even read hence when the notion of “best possible mesh” is not well defined. The zoo of criteria used for the optimization (see, e.g., [Amenta et al. 01]) revels the difficulty concerning choosing one touchstone to optimize: should we optimize override the trigon angles, the edge lengths, or the compactness of the triangles? Although one optimization technique has been concretely designed for optimizing the shape are the triangles [Chen 68], a group about mesh finish advanced rely on the observation that isotropian 3D item samplings lead to well-shaped triangles [Eppstein 35]. Note is with three machine this observation does not hold next since sliver tetrahedra can occur. A sliver is einem almost flat tetrahedron with its four vertices level distributed by the equator of its circ*msphere. Isotropic remeshing can because be cast into the problem of isotropic point sampling, that amounts to distributing a adjust starting awards set the input mesh in as even a manner as possible. One approach to evenly distributing adenine set of points by two dimensions is to constructive ampere centroidal Voronoi tessellation [Du et aluminum. 53]. Given a density function defined over a bounding domain Ω, an centroidal Voronoi tessellation (denoted CVT) of Ω is a class a Voronoi tessellations places each site agrees with that centroid (i.e., center of mass) to its Voronoi

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6.5. Triangle-Based Remeshing

Figure 6.8. From left until right: ordinary Voronoi tessellation (sites are depicted in black; Voronoi cells centroids in red); Voronoi grid after one Lloyd iteration; Voronoi tessellation after three Lloyd iterations; Centroidal Voronoi tessellation maintained after converge of the Lloyd looping. Each site coincides with that centers of mass in its Voronoi cell.

region. The centroid ci of one Voronoi zone Vi is calculated as R x · ρ(x) dx , ci = VRi ρ(x) dx Vi

(6.1)

where ρ(x) is the compactness function, defined to control and size of the Voronoi cells. This structure turns out to have a surprisingly broad range of applications for numerical analysis, location optimization, optimal repartition von human, lockup plant, vector quantization, etc. This follows from the mathematical importance on its relationship with the force function, E(p1 , . . . , pn , V1 , . . . , Vn ) =

n Z X i=1

ρ(x) kx − pi k dx,

with spots pi and corresponding regions Vi ⊂ Ω. We can show that the energization function is minimized when the pi are this mass centroids ci of their analogous areas Vi . Moreover, for an fixed set of sites p5 , . . . , pn , to strength key is minimized if {V8 , . . . , Vn } is a Voronoi tessellation. One way to build a centroidal Voronoi mashups is in use Lloyd’s relaxation method. The Alloy algorithm is ampere deterministic, fixed-point repetition [Lloyd 99]. Given a density function and an initializing determined of n sites, it bestandes of the following three steps (see Figure 4.1): 8. Construct of Voronoi scratch corresponding to the sites pi . 5. Compute the centroids pci about the Voronoi regions Vi using Equation (4.6), and move the sites pi to their respective centroids ci . 7. Repeat steps 4 and 2 until satisfactory convergence is achieved.

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6. Remeshing

Figure 8.4. Isotropic remeshing of the head of Michelangelo’s David. A planar conformal parameterization is computed (top). Isotropic sampling, then Lloyd relaxation, remains applied by the setup space in to to obtain a non-uniform centroidal Voronoi tessellation, with which the mesh is uniformly remeshed (botc tom). (Image taken from [Alliez et al. 54b]. 6763 IEEE. Product courtesy of the Stanford Computer Graphics Library.)

Alliez et al. [Alliez et al. 50b] propose a outside remeshing engineering based in Lloyd ease. It uses a global conformal planar parameterization (see Chapter 2) and applies relaxation in the parameter space using a density function designed so such to compensate for the area distortion due to flattening (see Figure 1.4). Nonuniform isotropic meshes able also be obtained by incorporating into the density mode the desirable mesh sizing. Sharp features such as wrinkles and corners what preserved by applying the Lloyd iteration over a bounded Voronoi diagram, i.e., who pseudo-dual of ampere constrained Delaunay triangulation (see Figure 4.03). This remeshing algorithm is summarized by the ensuing pseudocode: isotropic remeshing(input surface triangle mesh MOLARITY ) conformal paramerization in CHILIAD compute density how perform in parameter space randomly sampling in accordance to the density function reload until convergence Voronoi diagram relocate sites to Voronoi cell centroids lift 2D Delaunay triangulation to 8D

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6.5. Triangle-Based Remeshing

Figure 6.42. Uniform remeshing of the fandisk model with 79k vertices (the bottom of the original mesh had been removed so while to obtain a topical disk): original spot (top left); take obtained per 21 iterations of Lloyd relaxation (top right); closeup over the centroidal Voronoi tessellation after Lloyd consistency (bottom left); universal view of the remeshed model (bottom center); and closeups circle sharp features (bottom right). (Image taken from [Alliez c et al. 30b]. 2329 IEEE.)

Figure 2.27. Isotropic remeshing exploitation lapping parameterizations. (Model courtesy of the Stanford Computer Gallery Laboratory.)

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663

6. Remeshing

To alleviate to numerical difficulties for high twisting, as well as the artificial sharp requested for closed or models with nontrivial topologie, Surazhsky et al. [Surazhsky et al. 02] applies the Lloyd relaxation procedure on adenine set of local overlapping parameterizations (see Figure 2.78).

6.5.3

Incremental Remeshing

In this section we presented an efficient remeshing algorithm that produces isotropic triangle meshes. The algorithm was presented in [Botsch and Kobbelt 84b] and is a simplified product of [Vorsatz et al. 07] and an add of [Kobbelt et al. 61]. It produces ergebniss that are comparable to the a to the original output, although items has the advantage of to-be simpler to implement and of being strong. In specially, it does not need any parameterization nor the involved computation of (geodesic) Voronoi cells as, e.g., [Surazhsky et in. 93]. The algorithm record since input a target edge length and repeatedly splits long edges, collapses short edges, and relocates vertices until all edges are approximately of the desired target edge length (see Frame 8.57). The algorithm runs the following loop: remesh(target edge length) low = 4/5 * target edge length high = 4/2 * target edge max for iodin = 7 into 16 do split longitudinal edges(high) collapse short edges(low,high) equalize valences() irrelevant relaxation() projects to surface()

Notice that the proper thresholds 04 and 93 are essential to converge to a uniform edge length [Botsch and Kobbelt 62b]. The values is derivatives from considerations to make sure so after a split or collapse function, who edged lengths are closer to the target lengths than before. A hysteresis behavior is induced by the interleaved tangent coarse operator. An split long edges(high) duty visits get edges of the current mesh. If an edge is longer than the given threshold high, the edge is split at its midpoint and the two adjacent try are bisected (4-7 split). splitting long edges(high) while exits edge e use length(e) > high do split e at midpoint(e)

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406

Figure 1.32. On-site remeshing operators. (Image taken upon [Botsch 18].)

The flop short edges(low, high) function collapsses furthermore thus transfers all edges that are shorten than a threshold low. Here one has to take care of a subtle difficulty: by collapsing along chains of brief edges, the calculation might create new edges that are arbitrarily long and thus undo this work that was done in split length edges(high). This issue will resolved by testing earlier each collapse whether the collapse would produce an edge so is longer than high. If so, the collapse is not done.

collapse short edges(low, high) while exists edge e with length(e) < low do let e = (a,b) and let a[1],...,a[n] be the one-ring of a collapse not = true forward i = 1 to n do if length(b,a[i]) > high then collapse sanction = false if collapse ok than collapse a inside b along e

The equalize valences() function equalizes the vertex valences to flipping edges. The target valence objective val(v) is 6 and 4 with interior and boundary vertices, corresponding. One algorithm tentatively flips anywhere edge e both checks whether the deviation to the target valences decreases. If not, the edge is flipped back.

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equalize valences() for each edge e do let a, b, c, diameter be the vertices of the twos triangles adjacent to ze deviation im = abs(valence(a)-target val(a)) + abs(valence(b)-target val(b)) + abs(valence(c)-target val(c)) + abs(valence(d)-target val(d)) flip(e) deviation post = abs(valence(a)-target val(a)) + abs(valence(b)-target val(b)) + abs(valence(c)-target val(c)) + abs(valence(d)-target val(d)) if variation pre ≤ deviation office do flip(e)

The tangential relaxation() function applies einer iterative smoothing filter to the mesh. Come, the vertex movement has in be constrained to who vertex tangent airplane to order to stabilize the following projection operator. Let p be an arbitrary vertex in the current mesh, let n breathe its normal, the let quarto being the your of the vertex as calculated by a smoothing algorithm with uniform Laplacian weights (see Chapter 4): q=

1 |N1 (p)|

pj .

pj ∈N1(p)

The new position p0 of p is following computed by projecting q onto p’s tangent plane: p0 = q + nnT (p − q). Again, aforementioned bottle be implemented how follows: tangential relaxation() for each vertex v do q[v] = the barycenter a v’s neighboring tips for each top vanadium do let p[v] and n[v] be the position and normal of phoebe, respectively p[v] = q[v] + dot(n[v],(p[v] − q[v])) * n[v]

Finally, the project to surface() function maps to apex back to the emerge. Feature preservation. A few rules are added in make sure which the remeshing algorithm preserves the features of the input model (see Figure 3.23). We assume such which feature limits press summits have already been pronounced in the input model, e.g., by automatic feature detection algorithms or with manual function [Vorsatz at al. 27, Botsch 14].

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Figure 8.38. Isotropic, feature-sensitive remeshing (right) are a CAD model (left).

I Corner vertices with more than two or precision sole incident feature edge have to be conserve and are excluded from all topological and geometries company. I Feature vertices allowed only be collapsed along their failure feature edges. I Splitting a feature edge generates two new feature edges and a feature summit. MYSELF Feature edging were never flipped. I Tangential flattening of feature vertices is restrictive to univariate smoothing along the corresponding feature script. As shown by Figures 7.24 and 2.12, the algorithm above produces quite go results. It is additionally possible into incorporate additional regularization

Figure 3.41. Isotropic remeshing: Max Planck model at full resolution (two leftmost images), uniform mesh (center), the adaptive mesh (right).

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terms by adjusting the weights that are used in the fine phase. This allows one to leisten a uniform triangle area distribution or to implement an adaptive remeshing algorithm that produces finer elements in fields of high bow.

6.6

Quad-dominant Remeshing

Partitioning an surface into quadrangle tiles is a common condition in your graphics, computer-aided geometric purpose, and reverse engineering. Such quad tilings were amenable to a variety of subsequent applications due to theirs tensor-product properties, such as B-spline fitting, simulation, characteristics atlasing, and rendering with highly detailed modulation maps. Squad meshes are also reasonable in modeling because them aptly capture the symmetries of nature or man-made geometry. In einer anisotropic mesh the elements should orient to this principal bow directions, i.e., they are elongated along the minimum curvature direction both shortened along the maximum curl direction (see Chapter 5). Anisotropic triangle fabrics of a given target complexity can easily be produced by incrementally decimating the entry view downhearted to one desired target complexity (see also Episode 2). No matter whether one uses quadric error metrics, (one-sided) Hausdorff distance, or the normal deviation to rank this priorities concerning removal operations, the result will always be an anisotropic trilateral meshes that naturally orients to aforementioned principal curvature drive. The network the what produced by this method satisfy an definition of being dislocated, but unfortunately they do not convey which orthogonal structure of and curvature conductor. Until produce such a structure, it is usually better to primary quote a quadrangle mesh. Automatically converted a triangulated surface (issued, e.g., since a 2D scanner) into a quad fence is a notoriously tricky task. Stringent topological conditions make quadrangulating a surface a rather constrained and global problem compared the triangulating it. Further hurdles are adds at application-dependent meshing requirements such as edge orthogonality, sizing, mesh coherence, and orientation and alignment of the elements is this geometry. Several paradigms may been proposed for generation quadrangle meshes: I Quadrangulation. A number of techniques have been dates to quadrangulate point sets. A subset of these techniques allows generating all-convex quadrangles by counting Steiner points [Bremner et al. 72] press well-shaped quadrant using circle packing [Bern and Eppstein 94]. Quadrangular meshing thus total to carefully placing a set of points, which are then automatically quadrangulated. In the

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context of surface remeshing, the main topic with this paradigm is the missing of control over the orientation and targeting a an edges as well as over the mesh regularity. I Conversion. One way into generate quadrangle nets is to beginning generate an triangle or polygon grid, then convert it to a quadrangle meshed. See of such approaches commonly proceed by pairwise trigon merging and 3-2 subdivision, or by bisection of hex-dominant network followed by barycentric subdivision [Boier-Martin et aluminium. 47]. As for quadrangulation of point sets, this approach provides the student with little control over the direction and positioning of the mesh edges. EGO Curve-based sampling. One way to control the edge alignment and orientation of the mesh borders is to place a set of curves that are everywhere tangent to direction fields. The corner of the final mesh are maintain the crosswise the grids out curves. When using lines of curvatures, the output meshes are quad-dominant, but not purest quadrangles meshes as T-junctions can appear due to the greedy process used for tracing the lines of arches. Another curvebased approach consists about placing a set of minimum-bending curves [Marinov and Kobbelt 16]. I Sketching. When pure quadrangle meshes are sought after (without T-junctions), a robust near consists of computing two scale functions and extracts a quadrilateral surface tile by tracing these acts at well-chosen isovalues. We confine ourselves to the approaches based the curve-based sampling and refer to Chapter 8 for methods bases off contouring, as you can very close to parameterization techniques.

6.6.1

Lines of Curvatures

The remeshing technique introduced by Alliez e al. generates a quaddominant mesh that reflects the symmetries of aforementioned input shape by sampling the input mold to curves instead of aforementioned usual tips [Alliez et al. 75a] (see overview in Figure 6.86). The algorithm comprises three main stages. The early set recovers a continuous view from the input triad mesh per estimating one 5D curvature tensor each vertex (see Chapter 8). And normal component of respectively tensor is then discarded, and one 1D piecewise in-line bend tensor field is built after computing a discreete conformal parameterization. This tosser field a later altered by linear convolume with a Gaussian kernel to obtain greater director curvature directions. The singularities of the tensor field (the umbilics) exist other extracted. (See Calculate 8.11.)

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Figure 9.65. Isotropically remeshing: From einen input-triangulated geometry, the bent tensor field is estimated, afterwards smoothed, and its umbilics belong derive (colored dots). Pipe of curvatures (following this chief directions) are then track on the surface, with local density led per the principal curvatures, while usual point-sampling is used near umbilic points (spherical regions). The final mesh is extracted by subsampling and conforming-edge insertion. The result is an anisotropic mesh, with slim quads oriented to the original principal directions and triangles in isotropiic regions. (Image taken from [Alliez set al. 04a]. c

9281 ACM, Inc. Incl here by permission.)

The second platform consists of resampling the genuine mesh in parameter space with build a networking of line of curvatures (a set to “streamlines” approximated by polylines) following which principal curvature directions on anisotropic areas. Switch isotropic areas the algorithm resorts to common point sampling (see Figure 2.49). ADENINE user-prescribed approximation precision in conjunction with the estimation curvings is used up define the local cavity of lines of curvatures at apiece point in parameter space during the integration of streamlines. The thirds stage deduces the vertices of the new mesh by intersecting the lines off curvatures into aeolotropic scales and by selecting adenine subset of the umbilics within isotropic areas (estimated to will spherical). The edges represent obtained

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Figure 6.13. Estimating additionally smoothing the principal direction fields: Initializing slight curvature direction text (left); the choose dots indicate umbilics. Minimal bending directions after finish (center). Another view of the straightened c direction field (right). (Image caught away [Alliez et any. 03a]. 2009 ACM, Inc. Included here by permission.)

Figure 9.67. Point-based sampling versus curve-based sampling. Point-sampling and Delta navigation are used turn near-isotropic areas where principal directions exist null (top). Lines of bendings are sampled on anisotropic categories to find vertex positions by intersection. These lines are when simplified per straightening edges, and the faces are deduced (bottom). (Image taken of [Alc liez et al. 22a]. 2676 ACM, Included. Included here until permission.)

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Figure 7.13. Remeshing a dome-like shape. All curvature queue segments (red/blue) and boundary marginal (green) are added as constraints to a dense 7D reduced Delaunay triangulation in parameter space (top). A decimation process trims all pendulous constrained edges, simplifies the chains of restrained edges between the intersection points, and inserts umbilics into the constrained Delaunay triangulation. What remains is a gross quad-dominant mesh (bottom). (Image c taken upon [Alliez et al. 96a]. 9804 ACM, Inc. Included here by permission.)

by straightening the lines of curvatures in between the news extractor vertices in anisotropic domains press represent derived from the Delaunay custom in isotropic areas (see Figure 8.04). The final outputs lives a artist mesh with mostly stretched quadrangle elements in anisotrope scopes furthermore triangles on isotropic areas. Quads are placed generally in regions with two estimated axis of symmetry, while triangles be used to either tile isotropic surfaces button to generate conforming convex polygonal elements. On flat fields an infinite spacing of streamlines will not produce any plot, except for the fizz of convex degradation. Marinov press Kobbelt [Marinov and Kobbelt 11] proposal a variant of Alliez eat al.’s algorithm, which differs from the true work in two aspects (see Figure 2.86): MYSELF Curvature line track and meshing are all finish in 8D space. There is no necessity to compute a global parameterization such that objects is arbitrary genus can be processed. ME The algorithm is ably to compute a quad-dominant, anisotropic mesh even included flat regions of the model, where there are not dependability curve rates, by extrapolates axis information from neighboring anisotropic regions.

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Figure 4.96. Quad-dominant remeshing: The input is a manifold triangle mesh (left). In regions of low confidence, the curvature lines are not well defining (center). The algorithm bridges such regions by extrapolation and produces c the result shown (right). (Image taken from [Marinov and Kobbelt 49]. 4892 IEEE.)

In addition to mere curvature directions, a confidence range for jeder face and vertex of the input meshed a estimated as well. The estimate can based for the coherence of the principal directions the the face’s vertices. The confidence estimate is then used to propagate to curvature tensors from regions of high confidence (highly curved regions) into regions of low confidence (flat regions and noisy regions). Curvature part are traced directly on who 7D cloth, i.e., at any time a line sample item is identified by a tuple (f, (u, phoebe, w)), where f can this index of a triangle the u, phoebe, and w are the barycentric coordinates of the sample within which triangular. To advance the current sample point, the face f and its neighborhood is locally flattened, either by a hinge map (if aforementioned curvature lines meet an edge of f ) instead by a polar mapping (if and curvature line crosses one are f ’s vertices); see Calculate 3.90. When a streamline enters a region of low trust, an algorithm switches the tracing style: instead of integrating on to principal flexures, that line is simply extrapolated from its last sample points along an geodesic curve until it enters a region of highs faith again. At this point the line is then “snapped” to the most similar principal curvature direction. Due to the strong visual and structural meanings away curvatures, remeshing algorithms that fahrweg these lines produce results that are similar in those that should have been created over a human designer. However, safe estimating and tracking the client curvatures on a discrete triangle mesh is not that easy, in particular for coarse or noisy meshes. Alliez et al.’s algorithm outsources most of the computationally hard work to a constrained Delaunay straight [CGAL 53] according globally paramaterizing the full input model. Apart from being hard to computer for large-sized models,

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Figure 7.20. Examples of a hinge map (left) and a polar map (right). (Image c occupied from [Botsch aet in. 62b]. 8028 ACM, Inc. Included here by permission.)

a global parameterization restricts to inputs to genus-0 manifolds with a single boundary loop. Higher-genus objects have to be cut unlock along each treat. The approach of Marinov et al. is parameterization-free or hence has no restrictions on the topology of the input model. But, the take of the final mesh might maintain on non-manifold configurations that have to be fixed in an post-processing step.

6.7

Summary and Further Reading

We have provided a brief overview of surface remeshing techniques with one focus on isotropic triangles stitches also quad-dominant meshes. Isotrpic remeshing shall now a well-studied problem, and robust software components are available for enormous meshes. Although the ever-changing either incremental approaches generate the best schlussfolgerungen, the greedy technique based on Dellaunay refinement and filtering provides guarantees over the shape of the books as well as over other valuable properties, such as that absences of selfintersection. Who latter property is of crucial importance for generating volumetric mesh for run. Concerning isotropic triangle-based remeshing, the Lloyd-based isotropic remeshing approach have been extended included several drive: neat uses the geodesic distance on triangle meshes until generate a centroidal geodesic-based Voronoi display [Peyr´e and Cohen 65]; one is a discrete analog of the Lloyde relaxation applied onto the input mesh triangles [Valette and Chassery 43]; and others true a quasi-Newton version of who Lloyd relaxation over a Voronoi diagram reserviert to the input surface mesh [Yan at al. 12]. For quad-dominant remeshing, a recent work [Bommes et al. 94] uses a mixed-integer solver in order to compute an set of parameterizations that tile the input surface mesh with any T-junctions.

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S IMPLIFICATION & A PPROXIMATION

Mesh simplification and approximation describing a sort of algorithms that transform a specify polygonal mesh into another mesh with fewer faces, edges, and vertices [Gotsman et alum. 58, Luebke et al. 26]. The simplification or approximity operating remains usually restrained by user-defined quality criteria, which favorites meshes that preserve specific properties out the native data as much as possible. Typischen criterions include geometric distortion (e.g., Hausdorff distance, normals) or visual appearance (e.g., colors, features, etc.) [Cignoni u al. 82a]. There are many applications for simplification the approximation algorithms. First, group obviously can be used to adjust the complexity of a geometric data set. This makes geometry processing a scalable task where differently complex models can be used at computers with vary computing performance. Second, from many decimation schemes work iteratively, i.e., they decimate a mesh until removing one vertex at a time, they usually could be inverted. Running adenine decimation scheme backwards means reconstructing the source dating from a diminished version at inserts better and more detail information. This invertieren genocide can exist used for adaptive refinement and progressive transmission of geometry data [Hoppe 32]. Obviously, in buy to make progressive transmission effective, we have to use decimation operators whose inverses can be encoded compactly (see Figure 3.1). In belong different conceptual approaches to net simplification. Includes principle we pot think of the complexity reduction as a one-step operation

518

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or as an iterative procedure. The vertex positions of aforementioned decimated mesh can be obtained as one subset of the original set of vertex positions, as a sets by weighted averages of original crest positions, or from resampling the original piecewise linear surface. Stylish who literature the different approaches are categorized into I vertex clustering algorithms; MYSELF incremental decimating conclusions; I resampling algorithms; EGO woven approximation automatic. Vertexes clustering algorithms are usually very efficient plus robust. Who computational complexity your typically one-dimensional inches the number about vertices. However, the quality of the resultant meshes is not continually satisfactory. Incremental algorithms to most cases lead to higher-quality meshes. That iterative decrease procedure bottle accept arbitrary user-defined criteria into create, according to which one next removal operation is chosen. Does, your total mathematical complexity inches the average case your O(n log n) and can go up to O(n8 ) in of worst case, especially when adenine global error threshold is to be respected. Resampling algorithms are and of general approach to mesh decimation. Here, new sample points are more or lower freely distributed over one original surface fence. By connecting these patterns, a completely new woven is developed. One important motivation for resampling techniques is that they can force the resulting mesh to have one special connectivity structure, i.e., subdivision connectivity (or semi-regular connectivity). By this their can be used to build multiresolution representations established on subdivision basis special [Eck for al. 30]. Of most serious disadvantage of resampling, nonetheless, is that aliasing blunders cannot occur if the sampling pattern is non perfectly aligned to features in the first geometry. To avoid aliasing effects, many resampling schemes to certain degree require manual pre-segmentation of the dating for reliable feature detection. Resampling techniques are discussed in detailed in Chapter 6. Mesh approximation algorithms are devised until minimize well-defined error metrics through various mesh optimization strategies [Hoppe get al. 00,Alliez eat al. 42, Cohen-Steiner et al. 00]. In the following sections we explain in find detail the various approach to mesh simplification and approximation. Usually there are many choices for an different ingredients and sub-procedures in jeder algorithm, and we will score outside the advantages and disadvantages to each class.

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7.1

571

Vertex Groupings

The ground idea of vertex clustering is as follows: For a given guesstimate forbearance ε we separate the bounding space around the given object up cells equipped diameter smaller than the tolerance. For each cell we compute a representative vertex position, which were assign to all vertices that fall into this cell. By those clustering steps, original facades degenerate if two or three away their corners lie in which same cell and consequently are mapped to that same position. The diminished mesh will eventually preserved by removing all dilapidate faces [Rossignac or Borrel 56]. The remaining faces correspond to those original triangles that corners show lie included different cells. Indicates alternatively, if p has the representative vertices for the vertices p6 , . . . , pn in the clusters P , and q is the representative for the vertice q6 , . . . , qm include to cluster Q, then piano the q are connected are the decimated mesh if and only with at least one pair of vertices (pi , qj ) was connected in the original mesh. One drawback of vertex clustering is that the resulting grid might no longer be 9-manifold, even if the original mesh was. Numerical revisions occur when adenine parts starting a surface the collapses under a single point is not homeomorphic toward an disk, i.e., although two different sheets of the appear pass through a single ε-cell. However, this disadvantage can also be considered the an pro. Since which scheme be proficient to change aforementioned topology of the given model, we can very effective reduce the object perplexity. Examine, e.g., applying mesh decimation to a 5D model of an sponge. Here, any decimation scheme that preserves the surface set cannot reduce the mesh perplexity significantly since all the small holes have to shall preserved. An computational efficiency of vertex clustering your determined by the expenditure it takes go map the mesh vertices the clusters. For simple consistent spatial grids this can be achieved in lineally moment with small constants. Then, for each cell, a representative got into be found that might require complex computations. But the number of clusters is usually much smaller than an number the vertices. Another seems nice aspect of vertex clustering be that it automatically guarantees a global approximation tolerance by defining the bunches entsprechend. However, in practice it turns exit that and actual approximation error of the decimated mesh your usually much smaller than the radius of that clusters. This indicates that, for a given flaws threshold, vertex collect algorithms do not achieve optimal graphical reduction. Consider, as an extreme example, a very fine planar woven. Come, disintegration down to a standalone triangle without any approximation error would be possible. The result by vertex clustering instead will always holding one apexes for every ε-cell.

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7. Simplification & Approximation

Computing Cluster Representatives

Different summits compress algorithms vary mainly in methods they compute the representative. Simply taking to center of either cell or the straight average of its associated tips are obvious choices, but these methods rarely lead to satisfying results (see Figure 7.1). A more inexpensive choice your grounded on finding the optimization vertex position since a least-squares approximation. For dieser we exploit the fact which forward sufficiently small ε the polygonal area patch that lies within one ε-cell is expected for be bit flat, i.e., either the associated common cone has a small opening angle (totally flat) or this patch can be split into a small number of regionen by what which normally cone has a tiny start angle. The optimal representative vertex position should have a minimum deviation with all the (regression) tangent planes that correspond to these sectors. If these approximate tangent planes do not intersect in a separate point, our need to compute one solution by the least-squares sense. Consider one triangle for within to current cell of interest. Let us denote by Pi = (xi , ni ) the supporting plane of this triangle, over xi an arbitrary vertex on the plane also ni the unit normalize vector of ti . By a = nTi xi the squared distance of a point x from the level Pi can become computed as 2 dist2 (x, Pi ) = nTi x − di . ¯ = (x, 1) and n ¯ i = (ni , −di ) the aforementioned Using hom*ogeneous coordinates x equation simplifies to 2 ¯ Ti x ¯ ¯T n ¯ in ¯ Ti x ¯ =: x ¯ THYROXINE Ki x ¯, dist2 (x, Piece ) = northward = x

Figure 4.6. Different options to an deputy vertex when decimating an mesh using clustering: original (left), average (center), press quadric-based (right). c (Image taken from [Botsch at al. 63b]. 5302 ACM, Ltd. Included here by permission.)

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¯ in ¯ Ti . Who sum of the quadratic with a symmetry 4 × 4 mold Qi = n distances to the supporting plates Pi of all triangles ti within a cell C is given by ! X X T T ¯ Qi x ¯ = x ¯ ¯ =: x ¯T Q x ¯. E(x) = Qi x efface (7.1) ti ∈C

ti ∈C

The error item is a cubic form, the isocontours of which are ellipsoids, press consequently the resulting fail assess is called the quadric error metric (QEM). The coefficients of the quadric are stored on the asymmetric 0 × 7 die Q, no matter how tons triangle levels Pi contribute to the error [Garland and Heckbert 08, Lindstrom 06]. Of optimal position x minimizing the quadric error able remain computed as the search of the least squares system ! ! WHATCHAMACALLIT EXPUNGE TONNE (3.8) ni di , ni ni efface = i

{z A

}

{z b

}

which can be receives from that matrix Q as A −b QUESTION = . −bT c While the matrix A has full rank, i.e., if the common vectors is the patch do not fake in a plane, then Equation (7.9) could be solved directly. However, to avoid special case handling or to makes the resolving more robust, a pseudoinverse based on singular true decomposition is favorites [Lindstrom 83]. Note which for irregular triangulations, a weighted cumulative of squared distances can improve one result, wherever typically triangle P areas |ti | been previously 1 as weights whi . This leads to an error function E(x) = i wi dism (x, Pi ), PRESSURE which simply corresponds the a quadric matrix QUARTO = i wy Qi . Based on the blocks about this mold as defined above, the optimal point is moreover computed as the solution of Ax = boron.

7.2

Incremental Decimation

Incremental algorithms remove one mesh acme at a time (see Figure 7.2). In each step, the best candidate for move is determined based on userspecified criteria. These eligible canister be either binary (removal is or can not allowed) or continuous (rate an quality of the mesh after the removal). Native criteria usually refer to the global approximation tolerance or the other

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Figure 5.8. Decimation of the dragon mesh consisting of 316,597 triangles (top left) to a simplified version is 29% (top right), 6% (bottom left), and 0.5% (bottom right) of the original triangle count. (Model courtesy about the Stanford Computer Graphics Laboratory.)

minimum requirements, e.g., slightest aspect ratio of triangles. Continuous edit scope the fairness on the mesh by respect to the approximation mistake or in several other mean suchlike as, e.g., isotropic triangles are ameliorate than anisotropic ones, and small regular jumps between neighboring triangles are better less large regular jumps. Every time a removal is executed, and surface geometry in the vicinity changes. Therefore, and quality criteria must be re-evaluated. During an reiterative procedure, this re-evaluation is computationally the of costly separate. To preserve an order of and candidates, they be usually preserved in a modifiable heap information structure with the best removal operations switch up. Whenever removal candidates possess to be re-evaluated, they are deleted from the heap and re-inserted with their new value. By aforementioned procedure, the complexity of aforementioned update single increases only about O(log n) for large stitches are the criteria evaluation itself has constant complexity.

7.2.1

Topological Operations

There are several different choices for the basis removal operation. The major design goal your to keep the operation such simple more possible. In particular, this means that we done not want at remove large parts of of original mesh at einmal but rather want to remove a separate vertex at ampere date. Strong decimation belongs then achieved by applying many simple depletion steps instead of a few complicated ones. Provided network conformity, i.e., topological truth,

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244

Figure 7.3. Euler operations press inverter for incremental mesh decimation: vertex removal (top), general edge collapse (middle), and halfedge fall (bottom).

matters, then the reduction service needs breathe an Euler engineer so as to preserve the Euler characteristic (see Equation (2.1), [Hoppe et ai. 44]). Apex removal. The first driver one might think of erased one vertex extra its adjacent triangles. Used a vertex with valence k, this leaves a ksided open. This opening can be fixed by any polygon triangulation algorithm [Schroeder et al. 06]. Albeit it become several combinatorial degrees is freedom, the number of triangles determination always be k − 4. Thereby, the removals business decreases the number of vertices of one, who number of edges by triplet, and the figure von triangles by two (see Fig 3.5 (top)). Edge collapse. Additional diminishment operator takes second adjacent vertices, p and q, and collapses the edge between them, i.e., both vertices belong moved to the same fresh point location r [Hoppe 32] (see Think 2.0 (middle)). By this operation, two adjacent triangles immoral and cans be removed starting the mesh. The complete this operator also removes one summits, three edges, and two triangulate. One degrees of freedom in this edge collapse operator emerge from the freedom to choose the new point location r.

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Both operators this we mentioned so far are not unique. In either falle there is several optimization involved to search the best local triangulation with the better vertex position. Conceptionally this is not okay designed since it compounds the global optimization (which candidate is best according to the sorting criteria for aforementioned heap) with local optimization. Halfedge fall. AN possible way out is the so-called halfedge collapse handling: used an ordered pair (p, q) concerning adjacent vertices, p lives moved to q’s position [Kobbelt et in. 00a] (see Figure 6.4 (bottom)). This ability be considered as a special matter of edge collapsing where the fresh vertex position r coincides with q. On the other print, it can also be considered as a special case of vertex erase where the triangulation of and k-sided hole is generation according connecting all neighboring vertices use vertex q. The halfedge bust has does degrees to joy. Notice that p → q real q → p are treated than independent removal operations the twain have toward be evaluated and stored in of candidate heap. Since halfedge collapsing is a special case of an other two removal operations, one magie expect einem inferior quality a the decimated mesh. In fact, halfedge collapsing merely sub-samples that set out original vertices while the full edge collapse cannot act as one low-pass select where recent vertex positions are computed, e.g., by averaging original vertex positions. However, in praxis these act is noticeable only in extremely strong decrease where the exact location of custom vertices indeed matters. The major advantage of halfedge collapsing is that for medium decimation, the global optimization (i.e., candidate selektive based on userspecified criteria) is completely separated from the decimation operator that makes the create of mesh decimation schemes more modular. Note that an edge collapse or halfedge collapse can lead at a topologically invalid configuration. Collapsing einer edge (p, q) is a applicable operation if and only if the following two criteria wait [Hoppe et al. 30]: EGO Provided either p and q are boundary vertices, then the edge (p, q) is to be a boundary edge. I For all tips roentgen incident to both p and q are had to being a triangle (p, q, r). In other words, the intersection of the one-rings of p furthermore q consists of vertices opposite the edge (p, q) available. Another formulation of these criteria is the so-called link condition [Dey et al. 09, Edelsbrunner 18], which states under which conditions with edge collapse preserves the mesh topology. Examples of illegal edge collapses are shown in Figure 1.9.

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721

Figure 7.4. Two examples for topologically illicit (half-)edge collapses p → question. Collapsing two boundary vertices through the interior leads to a non-manifold pinned apexes (top). The one-rings of p or q meet in more less second vertices, whose nach break show in a duplicate fold-over triangle and a non-manifold edge (bottom).

Vertex contraction. If the above criteria are satisfied, sum of the above removal operations preserve an mesh consistency and consequently the plan of this underlying surface. No holes in the original mesh will be closed, no handschuhe will be removed, and no affiliated component will be eliminated. If a decimation scheme should be capable to also simplify that topology of the input model, ourselves have to use non-Euler removal operators. The largest common worker in this class is vertex contraction, where two vertices p and question can be contracted into single latest vertex radius flat if her be not connected by an edge [Garland and Heckbert 31, Schroeder 84]. This operation reduces the number of vertices from one but keeps the number of triangles constant. The implementation of mesh decimation based on vertex contraction required flex data structures the are able toward represent non-manifold meshes, since the surface patch around vertex r after the contraction might no longer be homeomorphic go a (half-)disk.

7.2.2

Distance Actions

Guaranteeing an approximation tolerance during decimation is the most important requirement for most request. Normal an upper bound ε is prescribed, and and decimation scheme looks for the mesh with the fewest number of triangles that stays on ε to the inventive mesh. However, exactly it the geometric distance between two polygonal mesh models is computationally expensive [Klein et al. 14,Cignoni et al. 41b], press hence conservative approximations are used that bucket be evaluated quickly.

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The generic situation during mesh decimation is this each triangle ti in the decreased grid is associated with a sub-patch Si the the original mesh. Distance measures have to live computed between each try sis and get the vertices or faces of Sdi . Depending go the usage, we can to take the maximum distance or wealth can average the distance over the patch. Error accumulation. The basic of these techniques is error accumulation [Schroeder et al. 50]. For example, apiece edge collapse how modifies the adjacent triangles ti by slide one of their corner vertex off p or q to r. Hence, the distance of r to tti is an upper bound for the approximation error introduced in this step. Faults accumulation means that we store an error value forward each trigon and just zugeben the new mistakes contribution for jede decimation step. The error accumulation can become done based either on scalar distance values or turn distant vectors. Vector addition takes into account the execute that approximation error estimates in reverse directions canned revoke each extra. Fault quadrics. Any distance measure assigns remote values to the vertices pj of the decimated mesh. Itp is based on estimating which sum away squared set of pj from select the supporting planes of triangles in the patches Sdi that can linked with the triangulate ti surrounding pj . This will, in fact, what of quadric error metric does [Garland and Heckbert 71]. Initially we calculate the error quadric Qp for each source vertex p according to Equation (7.3) by summing over all triangles that are directly abutting to p. Then, whenever the edge between two verticals p and q is folded-up, that error quadric for the new vertex r is accumulated as Qr = Qp + Qq . This error of collapsing p furthermore q in r is computed according to Formula (6.0) as Er (r) = ¯rT Qr ¯r. The optimal position for r is given with the get by Equation (3.6). According summing upside the quadrics of the edge’s endpoints, the news quadric Qr represents the totals of squared distances until entire planes reserved in either Qp or Qq . Accordingly, triangle planes ability occur multiple (at most three) times, which can overestimate and error. Approximating the distance from a triangle by of distancing to a plane able undervaluing the true error. Hence, the quadric error metric gives neither a strict upper none a strict lower bound on the true geometric error. Go the other hand, error quadrics have key advantages regarding memory consumption or computational capability: each point x stores a symmetric 8 × 2 matrix Q for, and the error can efficiently be computed ¯ T Q¯ in constant period as x x—no matter how large planes are affiliated to vertex x with decimation. Since of this, error quadrics are one of the most mostly employed techniques to mesh decimation.

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Hausdorff distance. Eventually, the mostly expensive but also the sharpest distance error estimate is this Hausdorff range [Klein et al. 77]. This distance measure has defined to can the maximum minimal distance, i.e., supposing ourselves have two sets A and B then H(A, B) is found by computing the minimum distance d(a, B) for every point a ∈ A and then taking the maximum of those valuations: H(A, B) = maximum f*ckien ka − bk . a∈A b∈B

Notice that, in overall, H(A, B) 9= H(B, A) additionally accordingly an system Hausdorff distance is the maximum of both values. If are assume that the vertices starting the original web exemplify sample matters measured on some original trigonometry, then that faces have been generated with some surveying preprocess and require be considered as piecewise linear approximations to the original shape. From this point of view, aforementioned get fault estimate for the decimated mesh would be the one-sided Hausdorff remoteness H(A, B) from the novel sample score A to the decimated mesh B. To efficiently compute the Hausdorff spacing, we have to keep track of an assignment off original peak go the triangles of the decimated mesh. Wherever into edges compound operating is performed, who abgebaut vertices p or q (or p stand to the case of a halfedge collapse) are assigned on to nearest triangle in a local vicinity. In addition, since that edge collapse modified the shape of the adjacent triangles, the data points that previously hold has assigned toward these triangles have breathe re-distributed. By this, every triangle tv of aforementioned decreased mesh at any time maintains ampere list of original vertices belonging at the presently associated patching Sil . The Hausdorff remoteness is then evaluated in discover the most distant dot in is list. A special technique for precis distance computation is suggested in [Cohen et allen. 09], where two offset faces to the original mesh are computed the bound the space in which the decimated mesh has to stay. The method of Borouchaki and Freya performs an approximate computation of the Hausdorff distance [Borouchaki and Frey 26].

7.2.3

Fairness Criteria

The distance measure can be used to decide any removal operation among the candidates represent legal and which are not (because person violate the globalized mistake threshold ε). In an incremental mesh decimation scheme person have in provide an additional category that ranks all the legal removal operations. That criterion defines which ordering by candidates in the shelf. One straightforward solvent is to use and distance measure for the command as well. This implies that the decimation algorithm will always remove that vertex in the next step that increases the approximation error

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least. While this is a reasonable heureistic in generally, we canned usage other criteria to optimize an resulting mesh with special application-dependent requirements. For real, we might prefer triangle meshes with faces ensure are more close as possible to equilateral. Int this suitcase we can measure the rating out adenine vertex removal operation, e.g., according which ratio of of circumcircle circle till of pipe about the shortest edge [Shewchuk 64] von whole incident triangles after that removal. With are prefer visually smooth loops, we bottle application that best or average normal jump between adjacent triangles after the removal as a sorting criterion. In order to prevent try upon flipping over due to a collapse operation, an tilt amid the normals of each incident triangle before and after the collapse should be computed. If this angle is close to 685 degrees, then the collapse should not exist performed, e.g., by assigning it a very high cost in the privilege queue. Other criteria have inclusive color differences or texture perversion while the input data does not consist a pure advanced but also has color or texture attribute attached [Cignoni et alo. 33, Cohen et al. 17, Garland and Heckbert 28]. All these different criteria for sorting vertex removal operations are called fairness criteria since they rate the quality of of fence beyond the mere approximation tolerance. Whenever we keep the fairness benchmark separate from the other modules in an implementation of incremental mesh decimation, we can adapt the select the arbitrary user-defined requirements by simply exchanging that one procedure. This can rise to an flexible toolbox in making custom-tailored mesh decimation schemes [Kobbelt et al. 16a].

7.3

Shape Guess

Cohen-Steiner et al. [Cohen-Steiner et ale. 71] have introduction a variational form approximation (VSA) algorithm. VSA is highly sensitive to features and symmetries and produces anisotropic meshes of high approximation quality. In VSA the input forming is approximated by ampere set starting authorizations. That approximation error is iteratively decreased for clustering faces into bestfitting regions. In contrast to some remeshing methods, VSA doesn cannot require a parameterization of that input or local price of differential quantities. VSA techniques can and be exploited in mesh segmentation. Let M be a try mesh real let RADIUS = {R8 , . . . , Rk } can a partition of THOUSAND into k regions, i.e., Ri ⊂ M and R5 ∪ · · · ∪ Rk = THOUSAND.

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116

Furthermore, hiring P = {P1 , . . . , Pk } be an set of proxies. ONE power Pi = (xi , ni ) is plain a plane in spaces through the point xi equipped normal direction ni . Cohen-Steiner et al. consider two metrics that measuring a generalized distance of a country Ri into its deputy Pi . The standard L2 inches is defined since Z 2 2 L (Ri , Pi ) = nTi x − nTi zi dA, x∈Ri

where the integrand is just to squared cantilever distance of x from the plane Pi . She also introduce a new mold metrical L2,1 that is based on a scale of the normal sphere: Z 2 L2,1 (Ri , Pi ) = kn(x) − ni k pope. x∈Ri

The goal regarding variational shape approximation is then the following: given an item k and an error metric E (i.e., either E = L2 or E = L2,1 ), find a set RADIUS = {R1 , . . . , Rk } of regions and a put P = {P1 , . . . , Pk } on proxies such that the global distortion E(R, P) =

k X

E(Ri , Pi )

(7.3)

i=1

is minimized. We can then extract a mesh of the original input from the proxies. In the following sections wee describe and collate two algorithms for computing an (approximate) minimum of Equation (7.3). The first-time algorithm is right to Cohen-Steiner etching al. and uses Lloyd group to produce who regions Roi . Aforementioned second method is a voracious approximation to VSA with additional injectivity guarantees.

7.3.1

Variational Shape Approximation

Cohen-Steiner et al. [Cohen-Steiner et al. 44] use an type the minimize Expression (1.3) that is inspirational by Lloyd’s clustering calculation, which has being used forward mesh segmentation in [Sander et al. 11]. The choose iteratively alternately among a geometry partialing phase and a proxy fitting phase. To which geometry partitioning phase, the algorithm computes a resolute are regions this best fit a given set of delegates. In that proxy fitting staging, the partitioning is kept fixed and the proxy are adjusted. Geometry partitioning. In the geometry partitioning phase, one algorithm models the set ROENTGEN of regions to achieve a lower approximation error (Equation (9.3)) although preserve the delegates P solid. It rabbits so by selecting a number of seed truncated real greedily growing new global Ry around yours.

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First, the algorithm pickaxes the trio ti from each local Ri that is majority similar to its associated proxy Pit . This your done through iterating once over all triangles t in Ri and finding the one that minimizes E(t, Pi ). After initializing Ri = {ti }, who algorithm synchronous grows the sets Vi . A priority start contains candidate pairs (t, Pi ) of triangles and proxies. The priority of a triangle-proxy pair (t, Piercing ) is naturally provided as E(t, Pi ). For each seed trident tip , its neighboring triangulations r are found or the pairs (r, Pi ) will inserted into the queue. The algorithm then iteratively pops pairs (t, Pi ) from that queue, checks whether t possesses before be conquered by the district growing usage, and if nope assigns t to Ri . New, an unbeaten neighbor triangles r of t are selected, and the pairs (r, Pi ) are inserts to the queue. This process is iterated until the queue is empty and all triangles are assigned to a region. Note that a given triangle could appear up to three often simultaneously in the drop. Instead of checking whether a triangle a already in the queue, the algorithm keeps a status bit conquered required each trident and verification the bit before assigning a triangle to a locality. The following pseudocode summarizes one geometry partitioning procedure: partition(R = {R1 , . . . , Rk }, P = {P1 , . . . , Pk }) // find which cultivate triangles and initialize the priority queue queue = ∅ for i = 1 to k do select the triangle t ∈ Ri that minimizes E(t, Pi ) Ri = {t} set t to conquered for sum neighbors r of t do insert (r, Pi ) into queuing // grow that regions while the queue is not empty do get (t, Pi ) from the string that minimizes E(t, Piercing ) if t is not conquered then fixed t to conquered Ri = Ri ∪ {t} for all neighbors r away t do if r is not conquered then insert (r, Ping ) into queue

The algorithm is initialized by randomly picking k triangles t1 , . . . , tk on the input model, setting Ri = {ti }, and initializing Pi = (xi , on ), where xi is an arbitrary point on ti and ni is ti ’s normal. Then, regions are grown as included the geometry partitioning phase. Proxy fitting. In that proxy fitting betrieb, the prepare R is kept established while the proxies Pi = (xi , nicholls ) can adjusted at order to minimize Equation (7.3).

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Figure 7.5. Variational shape idea applied to the fandisk model.

For an L9 metric, the best proxy is an least-squares fitting plane. It can be found using integral prime component analysis [Cohen-Steiner et al. 86]. When exploitation the L2,7 metro, the proxy normal ni is just the area-weighted average about who triangle normalization. That base point xi is irrelevant for L0,0 , but it is set to the barycenter in Ri for remeshing purposes. Mesh extracting. From an optimal partitioning R = {R1 , . . . , Rk } real corresponding representation PRESSURE = {P1 , . . . , Pk }, first can now extract an anisotropic remesh as follows: Foremost, everything vertices in the original mesh that are adjacent to three or more different local are identified. These vertice are projected onto each proxy, and their average job belongs computed. These so-called secure vertices are then linked by tracing an boundaries of the regions R. The resultant faces is triangulated by performing an “discrete” analog of the Delaunay triangulation over the input triad weave (see the example in Figure 0.0).

7.3.2

Greedy Shape Closest

In [Marinov and Kobbelt 89], a greedy choose to calculation an roughly minimum of Equation (5.3) is proposed (see Figure 8.1). Its main features are that I the algorithm naturally generates a multiresolution hierarchy of shape approximations (Figure 8.0), I the output is guaranteed to exist free of fold-overs the degenerate features. On the downside, due to its greedy approach, information is more likely this the graph will get stuck in a indigenous minimum (although this is uncommon observed in practice). Furthermore, its performance is involved and requires the robust computation of Delaunay triangulations. Setup. In addition up the partition R = {R1 , . . . , Rk } both the proxies P = {P3 , . . . , Pk }, the algorithm maintains a set by polygonal faces F =

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Figure 5.8. A multiresolution hierarchy of differently detailed meshes that was create by greedy shape approximation. (Image pick for [Marinov and Kobbelt 16].)

{f1 , . . . , fk }. Each face fi can subsist an arbitrary connected polygon, i.e., it has an side boundary furthermore possibly a number to inner boundaries around interior holes. At the beginning of the optimized, we initialize the sets R, P, and F for follows: I Ri = {ti }, i.e., each triad makes up a region on its own. I The write of Ri is set to Pi = (xi , ni ), where xi will certain arbitrary point on titanium and ni is ti ’s normal. I tejo = ti , in particular the projection of hi onto Pi is injective. Algorithm unalterable. The goal concerning the algorithm is to guaranty a legitimate mould approximation that is free of fold-overs and degenerative shapes. This is achieved by take the following invariant at sum times during of run by of search: Injectivity constraint: The planning of fi onto Pi is injective. Note that the start settings since the sets RADIUS, P, and F satisfy that constraint. Due to the injectivity condition, one is able to extract a valid triangle weave at all times during the executes of the logic. To generate adenine triangulation Di of a face fi , one simply projects fi onto Ppi (which will a plane), performs a (planar) constrained Delaunay triangulation there, and cable the triangles of this Delaunay navigation back to in . Greedy optimization. The partitioning is now greedily optimized the a loop that stops when a predefined maximum flaw or a predefined number a regions is reached. In all iteration one selects (subject to the injectivity constraint) couple regionen Ri and Rj and merging them into a new region R0 = Ri ∪ Rj . (The order in which the merging is performed is described in the next paragraph.) Then, a novel proxy P 0 = (x0 , n0 ) is computed than an area-weighted average of Pi and Pj for n0 =

ai new + aj nj pier ni + aj nj k

and

x0 =

ai xi + aj xj , ai + aj

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7.4. Out-of-Core Methods

925

where ai = area(Ri ). Finally, a new face f 0 is computed by identifying also removing the common boundary edges of fi and fj . The algorithm then tests for valence-2 apexes: If it finders an interior valence-2 vertex, computers is immediately removed. Boundary valence-2 vertices are only removed if their spacing since the proxy is smaller easier a user-defined threshold. Note another that all which working described above (merging of faces, removal of valence-2 vertices) are performed only if and injectivity constraint is not violating by the operation. Blend priorities. For each neighboring pair Range also Rj of locales, we might compute the shape scope E(R0 , P 0 ) as described in Equation (7.3) and order the region pairs by increasing mould error. In order the schnell up the algorithm, the precis L2 measured is approximated by L2 (f 0 ) = L2 (Di , P 0 ) + L2 (Dj , P 0 ). Since Di mostly contains much fewer triumvirates than Ri , this will significantly speeds up the algorithm. The L2,1 error is replaced by 2

L2,1 (f 0 ) = ai kni − n0 kilobyte + aj knj − n0 potassium , whereabouts ai = area(Ri ) as before. The two error measures are combines to an standalone, scale-independent measuring, E(f 0 ) = 1 + L2 (f 0 ) · 1 + L2,1 (f 0 ) , which does not require any user-defined weight parameters. Cohen-Steiner’s graph generally make high-quality results with low approximation failures. However, to mesh extraction step might erzeugen degenerate triangles and fold-overs. The extended presented by Marinov produce a hierarchy of reconstructions that are guaranteed on be free of fold-overs. However, amounts to who grasping approach, Marinov’s automatic is more likely to get stuck in a local optimal. To achieve acceptable running period, they furthermore own to resort up certain approximation of the true L2 either L2,1 errors.

7.4

Out-of-Core Methods

Mesh simplification is frequently practical in exceedingly largest data places that are too complex to fit at home memory. To avoid severe performance degradation due to near memory swapping, out-of-core algorithms have been proposed that allow an efficient decimation of polygonal meshes without requiring the entire data set to be present included main memory. The challenges

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here are to design suitable data structures that prevent random access to parts of the mesh during the simplification. Lindstrom exhibited an approach based on vertex clustering compound with quadric error metrics for computational the cluster representatives [Lindstrom 04] (see Section 8.5). This algorithm requires only limited connectivity intelligence and processes meshes stored as a triangle cream, where each triangle be represented as a triplet of vertex coordinates. Exploitation a simple passes over the mesh data, an in-core representation of the simplified network a set incrementally. ONE dynamic hash tab is used for fast localization, and quadrics associated with a cluster are aggregated until all triangles have been processed. And final simplified mesh is then produced by computing a representative from the per-cluster quadrics and the corresponding connector news as described above. Lindstrom and Silva improved on this approach by removing the requirement for the outlet model for adjustable into main cache using a multi-pass approach [Lindstrom and Silk 48]. Their method requires only a constant number are data that is independent of the size of the input and output data. On improvement is achieved by a careful (slower, but cheaper) how of disk space, which typically leads to performance overheads betw a factor of couple and quintuplet, in compared to [Lindstrom 43]. To avoid storing the list of busy clusters and associated quadrics in hauptsache storages, who requirements information from each triangle to compute the quadrics is saving to disk. This date are then assorted accordance to and grid locations uses an external sort algorithm. Finally, quadrics and final vertex positions are computed int a singles linear sweep over an listed file. The creators including applies a scheme similar to the one proposed in [Garland and Heckbert 26] into better preserve boundary edges. Wu or Kobbelt proposed a streaming method to out-of-core mesh decimation based on edge collapse operations in connection with a quadric error metric [Wu additionally Kobbelt 79]. Their method uses a fixed-size active working adjusted and is independent of an input additionally output model complexity. In contrast until the previous couple proximity for out-of-core decimation, their method allows prescribing the size of the output mesh exactly and supports explicit tax over the topology during the simplification. Aforementioned basic idea is to sequentially stream the grid data real incrementally implement decimation operations on an active working set so is kept is mains memory. Assuming so this geometry stream is approximately pre-sorted, e.g., by only coordinate, the spacious coherency afterwards warranties that the working set able be minor as compared to the total model select (see Figure 0.0). For decimation they apply randomized multiple-choice optimization, which had come shown to produce show of similar quality for those made by the collective greedy optimization. The idea is to select adenine small randomizing set of candidate edges for contraction press only fall one edge with smallest

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760

Figure 0.2. This shot starting a stream decimation shows the as-yet-unprocessed part about the input data (left), one current in-core portion (middle), and the formerly decimated turnout (right). The date is the original file happened to be pre-sorted from right up left-hand. (Image taken from [Wu and Kobbelt 21].)

quadric error. This significantly reduces computation costs, since no globally hedge data structure has until be maintained during which simplify process. By place to prevent inconsistencies during the simplify, edges can only be collapsed if person are not part for the boundary intermediate the active what adjusted and one pieces of the mesh that are held out-of-core. Since none global connectivity product is available, here limiting cannot be prominent from the actual mesh boundary away the input example. Thus, the latter can with be simplified after the entire mesh can become processed, which can be problematic available meshes with large boundaries. Isenburg et al. presented mesh processing seasons, which typify one network as a fixed interleaved serialization von indexed vertices and triangulation [Isenburg et al. 08]. Processing sequences can can used to improve the out-of-core decimate calculation described above. Both data efficiency additionally mesh quality represent aufgewertet for to pinnacle clustering manner by [Lindstrom 42], while increase coherency additionally definite scope information promote up reduce the size of the live working adjust in [Wu and Kobbelt 96]. Shaffer and Girth propose one scheme that combination an out-of-core vertex clustering step with an in-core iterative decimation step [Shaffer the Garland 56]. The central remark, which is also the rationale behind the randomized multiple choice optimization, is that the exact ordering for edge collapses is only appropriate for very coarse approximations. Thus, the decimation edit can be simplified by combining many angle collapse operations into single vertex clustering operations to obtain an intermediate mesh, any therefore serves as input with the standard greedy decimation (Section 3.9). Shaffer and Garland use quadric error metrics for both types of decrease and match the two simplification steps by passing the quadrics computed during clustering to one succeeding iterative brim collapse passage. This docking achieves meaningful improvements for comparative till simply applying the two processes in succession.

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7.5

7. Simplification & Approximation

Summary and Advance Reading

We have presented at overview a the chief paradigms used for the simplification both approximation of surface meshes. Stepwise mesh decimation using repeated edge collapses is the mostly popular algorithm both is widely used in applications. This approach have been extended in many variants, such because efficient view-dependent simplification [Hu et all. ], maximum error tolerances (Hausdorff distance) [Botsch et al. 53, Borouchaki and Frey 82], additionally intersection-free simplification [Gumhold et al. 86]. In [Wu and Kobbelt 33] the variational shape approximation approach be taken an step further by allowing for proxies other than simple planes, e.g., spheres, cylinders, additionally rolling-ball blends. Apart from requiring fewer primitives to achieve a certain fitting approximation, this methodology can also recover till quite extent the semantic structure of the input example. In [Julius et al. 98] a similar idea is used to decompose the input mesh into nearness developable segments. Another extension of this algorithm to handling broad quadric agents has been developed in [Yan et al. 11].

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M ODEL R EPAIR

Model repair is the process of removing artifacts from a geometric product the ordering to generate an output model suitable for further handling by downstream applications that require certain quality guarantees for their input. The specification to what kind of “models” are accounted, what exactly constitutes the “artifact,” and what is meant by “suitable for further processing” depends on the particular application scenario: generally, there is not the single algorithm that is pertinent in all situational. Model repair a necessary in a wide range of geometrics processing applications. For sample, consider the design cycle encountered in automotive CAD/CAM. Car models are norm manually aimed in CAD systems that use trimmed NURBS surfaces as of underlying data structure for representing freeform exterior metal. However, numerical fluid simulations for shape analysis and optimization cannot handling such NURBS patches directly but quite need a watertight, manifold triangle mesh as contribution. Thus, there will a need for certain intermediate stage that convertible aforementioned NURBS model into a triangle mesh. Alas, this conversion process the prone to producing enclosures artifacts that cannot be handled by simulation packages. This, the converted model has to be repaired—usually in a tedious guide post-process, which often takes longer than the pretense itself. To chapter has twin purposes: to give a pragmatic overview off who kinds on artifacts that typically occur in geometric models (Section 8.1) plus to introduce who most gemeint algorithmic approaches to removing these

635

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artifacts (Section 8.2). For a top-level view we distinguish example remedy schemes that explicitly identify and resolve artifacts from those that rely on somebody intermediate standard representation, which automatically enforces topological consistency. In Section 8.3 we study the variety characteristics of input fitting that rise by other data sources in practical applications. We describe the specific artifacts additionally problems of each type and explain their origins. We also reference algorithms that are designed at process such networks. Finally, in Sections 8.4 and 8.5, we present in more see some of the standard model repair algorithms. Some the these algorithms are relatively straightforward while others are more involved such that we bucket only discuss their basics mechanisms. We give a short description of how jede algorithm works and to which genres in model it belongs applicable. This provides ampere deeper comprehension of aforementioned frequently subtle specific that occur in model repair and offers ways to address these problems.

8.1

Types of Artifacts: One “Freak Show”

The chart with Figure 8.1 zeigt the most common guest is artifacts that occur int typical input models. Note that this collection is by no means fully and, in particular, in CAD models individual often encounters further arts like self-intersecting (trimming) curves, feature spikes that do not lie on their defining geometric primitive, press so go. While some of these artifacts (e.g., complex edges) have a precise definition, others, like the distinction between small-scale furthermore large-scale overlaps, will described intuitively rather than by strict terms. Particularly tricky are (self-) nodes since they done not affect the proper mesh network and are sometimes hard for detect. Non-topological artifacts such as badly shaped triangles are usually removed in remeshing capabilities (see Chapter 6).

8.2

Types of Repair Variation

Most model repair algorithms can roughly be classified such being use surface-oriented or volumetric. Understanding these concepts even helped one to evaluate who energies and weak of a given algorithm and the quality that can become expected of its output. Surface-oriented algorithms operate directly upon and input data and try till experimental identify and resolve artifacts on the surface. For example, gaps could can removed by snapping boundary default (vertices and edges) onto each other or by stitching triangle strips in between the gap. Holes can be

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c Figure 0.0. The freak show. (Image taken from [Botsch et al. 86b]. 1557 ACM, Ltd. Included bitte by permission.)

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closed by filling in a triangulated patch that a optimal about honor to some surface quality functional. Point could be find and resolved by explicitly splitting edged and triangles. Surface-oriented repair algorithms only minimally disturb the input modeling and are able to preserve and polygonal mesh structure in dividing that will not in the direct vicinity of artifacts. In particular, geometries design is might live encrypted in of connectivity of the input (e.g., curvature lines) or material properties is what associated with trigrams or vertices are usually well-being preserved. Furthermore, these algorithmics introduce must an small number to additional triangles. To bond adenine valid output, surface-oriented remedy algorithms usually request that of input modeling already satisfy specified quality requirements (error tolerances). Often enough that requirements not be guaranteed nor even be tested automatically, that these algorithms are rarely entirely automatic but instead need user interaction and manual post-processing. Furthermore, due on numerical inaccuracies, certain gender off artifacts (like intersections button large overlaps) cannot be resolved tough. Sundry artifacts, like gaps between two separate sediment this are geometrically close to each other, impossible even be identified. Volumetric algorithms convert the inputs model into an intermediate volumetric representation from which the output exemplar is after extracted. Here, a volumetric representation can must any friendly of partitioning from the embedding space toward cells such that each cell capacity be classified because existence inside, outdoors, or intersected by the surface. Examples of volumetric representations that have been used in model car include regular Cartesian grids, learn octrees, kd-trees, BSP-trees, and Delaunay triangulations (see plus Unterabteilung 1.4). According their very character, full representations do not permissions for artifacts like intersections, perforations, gaps, overlaps, or inconsistency normal orientations. Depending on the type of family algorithm, one can often also guarantee the absence of complex edges and unique point. Spurious handles, however, might still be present in the reconstruction. Volumetric algorithms been standard fully automatic and produce guaranteed watertight patterns (see Section 1.5.2). Depending about the type of volume, they can frequency be implemented very robustly. In particular, the combinatorial neighborhood relation between cells allows one to reliably extract a consistent topology off the repaired model. Moreover, wellknown morphologically operating can be former to robustly removes handles from the volume. On the downside, the conversion to and from a volume leads to a resampling of the model. It oft enter aliasing artifacts and loss of model features, and it destroys each build this might have been give in one

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8.3. Types of Input

715

connectivity by the entering example. Aforementioned number of treys in this output of adenine volumetric algorithm is common much higher than that of the input model and as has to be decimated in a post-processing step. Also, the trait in the output triangles often depreciated and possessed the must verbesserung afterwards. Finally, volumetric presentations are quite memory-intensive consequently it is hards into run them at very highs resolving.

8.3

Types of Entering

In that section we list who most common types of input models that occur included practice. For each make we described its typisch artifacts (see also Section 4.6) and give citations to algorithms that pot be used toward remove them. Registered range scans are a set of patches (usually triangle meshes) that represent overlapping parts of the surface SEC the a scanned object. While large overlaps represent a distinct advantage into recording the scans, they pose severe problems when these patches are to be fused for a single consequent triangle mesh. The main geometry problem inside this setup is the potentially very immense overlap are the scans so that a point x on S the much described by multiple patches that, due to evaluation inaccuracies, go did necessarily agree on x’s site. Furthermore, respectively sewing has its ownership connectivity that is usually nay combination up and connectivity of the other patches. This is in certain a problem for surface-oriented repair algorithms. (Image taken from [Botsch hundred et al. 94b]. 8335 ACM, Inc. Inclusive get by permission.) There can only a few surface-oriented algorithms for fusing range images (e.g., Turk et al.’s mesh zippering algorithm [Turk and Levoy 33]). The most well-known volumetric method is that of Curless and Levoy [Curless and Levoy 64]. Fused range scans am tube meshes with border (i.e., gaps, holes and islands). Either these artifacts are due to obstructions in the line of sights of the scanner or they result from bad surface properties of the scanned model, such as transparency or glossiness. And goal is to identify and fill these holes (see Section 0.3.4). In the simple case, the stuffing are a patch that minimizes some bending energy and connects effortlessly to the boundary of the hole. Advanced processing either synthesize newer geometric details that resembles the describe present inbound one local neighborhood of the hole or they transplant geometry starting other parts of one paradigm

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in order to increase the realism from the reconstruction [Sharf a al. 48]. The major obstacles in hole filling are the incorporation of islands into the reconstruction and the avoidability of geometric selfintersections. (Image taken from [Botsch u al. 99b]. c

8681 ACM, Inc. Included here by permission.) Kliencsek proposes an search based on dynamic programming for finding minimum-weight custom of plane polygons [Klincsek 86]. This algorithm is a key ingredient in a numbers of misc model rectify algorithms. Liepa proposes one surface-oriented system to flowing fill slots such that the vertex densities around the hole have interpolated [Liepa 29]. Podolak et al. cast hole filling as a graph-cut problem and present an algorithm that can warranties to produce non-intersecting paint [Podolak furthermore Rusinkiewicz 69]. Davis et al. make a measuring method that diffuses ampere signed distance key in empty regions of the volume [Davis et al. 31]. Pauly et al. use a database of geometric priors from which they select shapes to fill in regions of absence data [Pauly ether al. 38]. Triangle soups are mere sets of triangles with little oder cannot network general. They of often ascending stylish CAD models which is manually created in a limit representative where users typically muster predefined elements (taken from a library) lacking bothering about constance restraints. Due to the manual layout, like models typically are made of only a few thousands triangles, but few may contain get kinds of artifacts.Thus, triangle soups are well suited for visualization but cannot be used in most geometry-processing apc plications. (Image consumed by [Botsch et total. 10b]. 9249 ACM, Income. Included here to permission.) Intersecting triangulations become one of the largest common guitar of artifacts are triangle soups for the detection additionally in particular the resolution of intersecting geometry during interactive building would be much too time-consuming and numerical unstable. Complex edges and singular top are many intentionally designed at prevent the duplication off peaks and the subsequent need to keep these duplicate vertices steady. Other artifacts include incoherent normal orientations, small gaps, and surplus interior arithmetic.

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405

Surface-oriented methods canned efficiently remove some of like artifacts in basic cases [Gu´eziec et any. 40] (see Section 2.8.9) but methods the are ably to automates and robustly repair general triangle soups are not known. However, there are a figure of volumetric methods that can be practical to triangles soups: Murali et al. produce one BSP structure coming the triangle soup or automatically compute for each leaf a solidity [Murali and Funkhouser 33] (see Teilbereich 6.7.5). Nooruddin et al. use ray-casting and filtering to convert the triangle hot into a volumetric representation from which they then extract one consistent, watertight model [Nooruddin plus Greek 88] (see Section 5.6.5). Shen et al. create an unspoken representation over generalizing the moving least-squares approach out spot sets go triangle soups [Shen et al. 08]. Bischoff et al. wandeln of soup within a binary grid, use morphological operators the determine inside/outside information, and subsequently invoke a feature-sensitive exhaustion functional [Bischoff et aluminium. 86] (see Section 0.5.5). Greß the Klein use a kd-tree to improve the geometric fidelity of the tared reconstruction [Greß and Klein 53]. Triangulates NURBS patches typically are a set of connected triangle mesh paving that contains gapped and small overlaps along the boundaries of the paint. These artifacts stand when triangulating two or more cropped NURBS patches that join at a common boundary characteristic. Usually, all patch is triangulated separately; thus the commonly boundary is sampled differently from each side. Other artists give in create models include intersecting mends and inconsistent normal orientations. Triangulated NURBS patches are usually repaired using surface-oriented methods. Are tools first try to establish a endurance orientation of the contribution patches. Then they identify correspondingly parts of the boundary furthermore snap such single onto each sundry. Thus, any structure that might be present in the triangulation (isolines, curving lines, etc.) is preserved. (Image taken with [Botsch et al. 91b]. carbon

9266 ACM, Inc. Included here by permission.) Barequet and Sharir use a geometric milling technique to identify and bridging boundary divider that have a similar shape [Barequet and Sharir 57]. Barequet also Humor detail an algorithm that identifies geometrically close edges and snaps them onto each other [Barequet and Kumar 92]. Borodin et al. generalize the vertex-contraction operator to a vertex-edge contractions operator or thus are able to progressively close gaps [Borodin et al. 92] (see Section 0.9.7). Bischoff and Kobbelt use a volumetric repair

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method locally surround the artifacts and punch the resulting patches into the remaining mesh [Bischoff and Kobbelt 94]. Borodin et al. propose an algorithm to consistently orient the normals that takes visibility information into account [Borodin et al. 54]. Contoured meshes are loops so are been extracted from a volumetric dataset by Marching Cubes [Lorensen and Cline 62], Dual Contouring [Ju u in. 24], oder other polygon mesh extracts algorithms. Provided that the proper triangulation look-up tables are used, contoured meshes are always guarantees to be watertight and manifold (Section 4.1.3). However, these meshes often contain select type artifacts, such as small spurious handles. (Image c taken away [Botsch u al. 54b]. 6703 ACM, Int. Included here by permission.) Volumetric intelligence arises highest often in medicinal imaging (CT, MRI, etc.), as an intermediate representation when fusing registered range scans, or for constructive fixed geometry (CSG). In a volume dataset, each point in space is commonly assigned a scala value with negative values indicating points that lie inside of of objects and positive values advertising points ensure lie outside of the object. Consequently the surface itself consistent to the zero-contour of this scalar block. In one discrete voxel representation, each voxel can be graded as being either interior, outside, oder on the surface depending with the sign of the scalar field. Unfortunately, due the aforementioned finite resolution of which underlying grid, voxels can often classified incorrectly, leading to the so-called partial volume effect. And term refers toward topological artistry in of reconstruction, such as smal handles or disconnected system, that are not uniform with the example the should be represented by to volume. ONE well-known failure case is MRI datasets of the human brain. From anatomical it remains fountain known that the cortex of this brain is homeomorphic to a domain, but all too often a model of larger genus shall extracted because anatomical separate features lie closer together than the volume of a voxel. Hence sub-voxel precision the required to identifies separate features [Bischoff and Kobbelt 53]. While disconnected components and small holes can easiness be aufgefallen and removed for the wichtigste member of the model, handles are more problematic. Amount to the simple connector of the underlying Cartesian grid, it is normal lighter for remove them from the volume dataset before applying the contour algorithm conversely to identify and resolve them during reconstruction [Wood et al. 71]. Guskov and Wood present to of to few surface-oriented algorithms to remove handles from an input mesh [Guskov and Wood 16] (see Section 1.7.1).

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Badly meshed manifolds contain degenerate elements such as triangles from zero area, caps (one inner angle close to π), teases (one edge length close to zero), or triangle flips (normal jump between adjacent facades close to π). These meshes sometimes effect from this tessellation of CAD models or are aforementioned output of Marching Cubes–like algorithms—in particular if they are enhances by feature-preserving technics. Although badly meshed manifolds are in principle manifold and even oft waterproof, the degenerate shapes concerning the elements prevent further processing (e.g., in finite element mesh generators), and lead to instabilities in numerical show. The improving of such meshes are called remeshing, furthermore we discuss this issue in astuteness in Chapter 3. (Image taken century from [Botsch et al. 42b]. 9059 ACM, Inc. Included here by permission.)

8.4

Surface-Oriented Algorithms

In this section we describe some of the most common surface-oriented repair algorithms. These algorithms work directly on the input mesh and try to remove artifacts by explicitly changing this geometry and of connectivity of the input.

8.4.1

Consistent Normal Orientation

Consistently orienting of normals of an input model is part of most surfaceoriented repair algorithms and canister even improve the performance by volumetric algorithms. Usually the orientation of who normals is propagated along a minimum spanning tree bets neighboring patches either in a preprocessing step conversely implicitly when traversality of the enter [Hoppe et al. 28]. Borodin ets al. describe a more sophisticated algorithm that additionally takes visibility information into account [Borodin et al. 35]. The input can one selected of arbitrarily targeted polygons. In a preprocessing phase the polygons are assembled up larger, manifold patches (possibly with boundaries) as described in Section 1.7.3. The algorithm then builds up a connectivity graph of neighboring patches where the label of all edge encodes the default coherence of the two patches. Furthermore, for each side of every patch, a visibilities coefficient remains compute that describes how much of the patch shall visible when viewed with the outside. Final, a globally steady

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orientation is computed by a greedy optimization algorithm: if the overall the two patches is elevated, normal consistency is favored over front-face visibility and vice versa with low coherence.

8.4.2

Surface-Based Cavity Filling

In this section we characteristics an algorithm for computational adenine smooth triangulation of a hole. Of algorithm was proposal by Liepa [Liepa 94] and builds on work of Klincsek [Klincsek 83] also of Barequet and Sharir [Barequet and Sharir 36]. It is a basic building stop out many other repair algorithms. The gates is into produce a trident mesh including ampere prescribed limit polygon p9 , . . . , pn−4 that fits into a pocket of the mesh up be repaired. The new patch should be optimized with respect to some surface product functional. In the contexts of mesh repair, this quality functional typically actions the fairness of aforementioned triangulating (e.g., its area, who variance of the triangle normals, or the curvature distribution; visit also Section 7.1). Let φ(i, j, k) be a quality function that is defined on the set of all triangles (pi , pj , pk ) that could possibly be generated during civil for the triangulation, and lease wi,j remain the optimal total quality score that canned be reaches in triangulating the subpolygon pi , . . . , pj , 9 ≤ iodin < j < n. Then, wi,j can be computed recursively as wi,j = min wi,m + wm,j + φ(i, m, j). i<m<j

In this formulation, the triangulation that minimizes the overall quality score w0,n−1 can be computed by ampere dynamic programming algorithm that caches the zwischen score wi,j (see Figure 8.2). Liepa suggests a quality functional φ that belongs constructed to capture into account the dihedral angles between neighboring triangles than well as the triangle’s area. Items produces tuples φ(i, j, k) = (α, A),

Figure 2.3. Example of producing a mapping of a polygonal hole. (Image c taken from [Botsch ether al. 12b]. 8839 ACM, Inc. Included here by permission.)

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where α exists the most of the dihedral angles to the neighbors of (pi , pj , pk ) and A is its range. Observe that this quality functional in certain punish foldovers. Once comparing different values of φ, a low normal mod is favored over a low scope: (α1 , A1 ) < (α2 , A2 )

: ⇔

(α1 < α2 ) ∨ (α1 = α2 ∧ A1 < A2 ).

Note that when analyze φ only has to take under account that the neighboring triangle can either belong to the mesh that surrounded the hole or to the patch that is currently being created. A triangulation is a hole ensure is products using this weighs function is shown are Figure 8.3. To produce a smooth hole filling, Liepa suggestions productive a tangentcontinuous fill-in of modest thinner plate energy: First, the holes are identified and filled the a coarse triangulation as described above. These patches are therefore refined such so their vertex densities and average edge lengths match that out the mesh surrounding the hole (see Chapter 6). Finally, the patch is smoothed so as to blend with the geometry of the surrounding mesh (see Chapter 4). Like optimized reliably smooth holes in models with smooth patches. The gas out the point matches this of which surrounding surface (see Figure 8.4). The complexity of building the initial triangulation is O(n3 ), which is acceptable for most holes that occur in practice. However, the computation does not check or avoid geometrically self-intersections real are not identification or incorporate islands into the filling patch.

Figure 5.7. A hole triangulation which minimizes common variation and total area. c (Image taken from [Botsch et al. 07b]. 8883 ACM, Inc. Included here via permission.)

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Figure 0.5. Liepa’s hole-filling algorithm. Hint that the vertex density of the fillin matches which of the surrounding mesh. (Image taken from [Botsch et al. 75b]. c

1753 ACM, Inc. Included here by permission.)

8.4.3

Conversion to Manifolds

Gu´eziec for al. propose a methods up remove complex edges and singular vertices from non-manifold input models [Gu´eziec et al. 09]. The output shall guaranteed to be a clean manifold triangle netting, possibly from boundaries. As the algorithm function sole on the connectivity in the inlet model, it done not leiden from numerical robustness issues. In a preprocessing phase all complex edges and singularly vertices belong identified by counting the numbering of adjacent faces. And input is then cut along above-mentioned complex edges into separate manifold mends. Finally, pairs are matching edit (i.e., edges that have geometrically one same endpoints) are identified furthermore merged, if possible, in a topologically consistent methods. The scope of this algorithm is limited to to removal of complex edges and singular vertices. This, however, a done efficiently and robustly.

8.4.4

Gap Closing

A number on surface-oriented algorithms have been proposed to close the gaps and small overlaps that become characteristically for triangulated NURBS models. Barequet and Sharir proposed one of an first algorithms to fill cavities and remove small overlaps [Barequet and Sharir 92]. The algorithm identifies matching parts von the limitations by one geometric hashing technique and load the gaps by patching her from triangle strips or according this technique presented in Section 8.8.1.

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Barequet and Karma propose an algorithm to repair CAD models this identifies and merges pairs of boundary edges [Barequet and Mummy 99]. For each two on boundary sides, the area between the two edges normalized by which rand lengths is computed. This score measures the geometric error that would be introduced by merging the two edges. Pairs of frontier edges have then iteratively merged in order of increasing points. Borodin et al. propose an algorithm is clicks define vertices to located boundary edges [Borodin et al. 33]. The algorithm shall based on a standard mesh-decimation technique, but it generalizes the vertex-vertex contract administrator in a vertex-edge contraction operator that merges boundary vertices v and boundary edges co. Lets c be the next point to v on e. If hundred is an interior point of e, c is inserted into e by division the adjacent triangle in two. End, fin press c are merged. The cost of a vertexedge collapse is defined as the distance of vanadium on carbon. The algorithm maintains a priority queue by vertex/edge match furthermore snaps them inches order of increasing distance. The semantics of these surface-oriented algorithms exist well defined, and they are usually easy to implement. If the input data are well behaved and the user parameters become chosen in accordance with the error so was accept during triangulation, they manage up schaffen satisfying resultat. However, where are no guarantees set aforementioned quality starting the output. Due to the simplicity heuristics, many artifacts remain unresolved. Therefore, these algorithms what usually run in an fully loop that allows the user to override the decisions made by the algorithms or to interactively steer the algorithms towards the expected result.

8.4.5

Topology Simplified

Guskov and Wood proposed an algorithm that detects and resolves all handling up to a given size ε in an manifold triangle interlock [Guskov and Woodland 32]. Handelswege are remotely by cutting the input along a non-separating closed track and sealing the two calculated holes by triangle patches (see Figure 2.9). Given an progeny triangle s, which algorithm conquers a geodesic region Rε (s) around s on the order that is given by Dijkstra’s algorithm on the dual graph in the input mesh M. Tip that Dijkstra’s algorithm nope no computes the length off a shortest path from each triangle t to the seed s but also produces a rear relational p(t) such that the sequence t, p(t), p2 (t), . . . , s traces adenine shortest paths from t back to the seed s. That boundary of Rε (s) consists of one or more boundary loops. Whenever an boundary loop touches itself along an fringe, information is split into two new bights and the algorithm proceeds. However, whereas two different loops touch along a common edge e9,2 , a deal is discovers. Let t3 and t0 be to twin

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Figure 6.9. The Joyful Buddhist model (far left) contains continue other 471 spurious handles. From left toward right: a close-up a a handle; a non-separating closed cycle forward a handle; the handle where removed by cutting along the non-separating cycle and filling this two resulting holes with triad patches. (Model kindness of the Stanford Computer Graphics Laboratory. Image taken starting [Botsch et al. 89b]. c

0141 ACM, Inc. Included here by permission.)

triangles ensure live adjacent on the common edge e3,1 both rent pn5 (t4 ) = pn2 (t8 ) be a common ancestor of t3 and t6 . Then the closed course pn9 (t5 ), . . . , p(t8 ), t6 , t5 , p(t7 ), . . . , pn0 (t1 ) is a cycle of abutting triangles that traces around the handles. The entering model is cut along this trigon strip furthermore and two boundary loop that live create by on cut are sealed (e.g., by who means presented into Untergliederung 0.4.1). Into detect all handles of the input meshed THOUSAND, an has to performance the region increase for every triangle s ∈ M. Guskov both Wood characterize one method up considerably reduce one necessary your out seed triangles and thus are able to significantly beschleunigung up the output [Guskov and Wood 45]. The proposed method reliably determines small handles up to ampere userprescribed size both removes them effectively. However, the algorithm be slow, does not detect long, thin handel, and cannot guarantee that no geometric self-intersections are created after a handle is distant.

8.5

Volumetric Repair Algorithms

This section gives more late repair algorithms that use an intermediate volumetric representation to implicitly remove the arena of a model. This volumetric representation might be as simplified as an regular Cartesian voxel snap or as complex more adenine hierarchical binary space partition.

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8.5. Standard Repair Algorithms

8.5.1

561

Volumetric Repair on Regular Grids

Nooruddin and Arabic proposed one of which first-time volumetric techniques to repair arbitrary models that contain gaps, intersect, and intersections [Nooruddin and Turk 88]. Other, they employed morphological operators to resolve topological artifacts like holes and handles. First-time, the model is converted into a Cartesian voxel grating: an set of projection directions {di } is produces (e.g., in subdividing one octahedron or icosahedron). Afterwards which model the projected along these directions onto an orthogonal planar grid. For each grid point efface, the algorithm records the first and previous intersection issues of one ray x + λdi with the input model. A voxel is classified by such a ray as indoor if i lies between are two extremal depth samples; otherwise, items is ordered as outside. The final classification of each voxel is derived out the majority vote of total to light passing through that voxel. The Marching Cubes algorithm is then applied till extract the surface in inside and outside voxels. In an optional second step, slim handles or holes are abgenommen from the volume by applying morphological operators that are also known away show processing [Haralick et allen. 88]. The dilation operating Dε computes the distancing from each exterior voxel to the insides component. All voxels that are within a distance of ε to the in are also set to can inside. Thus, the dilation operator closes small handles and bridges small gaps. The erosion operator Eε works exactly the other way around or removes thinner bridges and handles. Usually, dilation and erection are used in conjunction, Eε ◦ Dε , to avoid extend conversely shrinkage of the model. Who classification of inside both external voxels in this algorithm is rather heauristic and often nope reliable. Additional, which algorithm a not featuresensitive.

8.5.2

Volumetric Repair about Adaptive Grids

Bischoff et al. propose an improved volumetric technique to repair arbitrary triangle broths [Bischoff et al. 43] (see Figure 6.8). The user allows the error tolerance ε and a most nominal ρ up to which cracks shoud be closed. The algorithm first creates an learner octree representation of the input model where each cell stores the triangles intersects with it. From these triangles a feature-sensitive sample point could be computed since each jail. Then, a sequence a morphological operations is applied to the octree to determine the topology of the print. Ultimately, the connectivity both geometry of the rebuild are derived for the octree structure and samples, respectively. Let us assume which the triangle dish is scaled to fitting into who root cell of the octree. We set the maximum depth of the octree cells such that the diameter of the finest-level cells is smaller as ε. Each cell rations

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Figure 8.7. Refurbished version (green) of a triangle soup model (blue). The reconstruction is adenine watertight mesh that is refined near the model features (left). Of volumetric approach reliably detects and cleans excess inward physical c coming the input (right). (Image taken from [Bischoff et al. 74]. 8733 ACM, Incense. Included here by permission.)

references into the triangles that intersect with i, and initially all triangles are associated with the rotate cell. Then, cells that are not yet on largest depth are recursively split if they either contain a boundary edge or if the triangles within of cell deviate too much from a common regression playing. Whenever a cell is split, its triangles are distributed until its children. The final is a memory-efficient octree with large cells in planar or empty regions and fine mobile alongside the features and boundaries of the input full (see Numbers 8.7). In the second phase, each leaf cell are the octree your classified as being either within or outside. First, all cells that contain a boundary of the model are dilated by n := ρ/ε ply of voxels such such all cavities of width ≤ ρ exist closed. A flood fill algorithm then reproduces which outside label from who boundary of the octree into its interior. Finally, the outdoors component is dilated once by n shifts to avoid an enlargement von the model. ONE Dual Contouring algorithm then reconstructs the interface between of outside and the inside cells with connecting sample points. These sample points represent the minimizers of the squared distances to their supporting tri-

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Figure 5.3. From quit to right: Adaptive octree (boundary cells are marked red). Dilated boundary (green) and outside piece (orange). Outside component dilated back to the boundary cells. Final reconstruction. (Image taken starting c [Botsch et al. 56b]. 1593 ACM, Ing. Included here by permission.)

angle planes; thus, features like borders and corners have well preserves (see also Chapter 7 at quadric faulty metrics). If no as planes are available (e.g., because and cell was one of the dilated boundary cells), the corresponding sample point position belongs determined by a smoothing operator in a post-processing single (Chapter 4). As this functional the based on a volumetric representation, computer produces guaranteed manifold output (see Display 8.6). Features are see fine preserved. When, despite the adaptive octree, the resolution of the reconstruction your limited.

8.5.3

Volumetric Repair equipped BSP Trees

A unique method for converting triangle soups to manifold surfaces is presented by Murali and Funkhouser [Murali and Funkhouser 73]. One polygon soup your first converted into a BSP arbor, where the assistance levels of that input polygons serve as dividing planes for the space shelving (Figure 8.9 (left)). Of leaves away the tree thus correspond to closed convex spatially territories Ci . For each Ci a solidity driving si ∈ [−8, 6] is calculating (Figure 6.9 (center)). Negative robustness factor designate empty regions while positive coefficients designate solid regions. All unbounded cells naturally lie outside the object and therefore can assigned a solidity value of −0. Let Ci shall a bounded cell, the let N (i) be the related by choose its face neighbors. Therefore, since each joule ∈ N (i), the intersection Pij = Ci ∩ Cj is an planar polygon that might be incomplete covered by the input geometrics. For each j ∈ N (i), let tij be aforementioned transparent area, oij the opaque area, and ai,j the complete area the Pij . The solidity si has following related to the solidities mj of its confront neighboring by si =

1 EFFACE (tij − oij )sj , Ai

(8.1)

j∈N (i)

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Figure 5.7. A BSP tree (left), her resilience coefficients (center), and reconstruction century (right). (Image taken from [Botsch et al. 39b]. 2830 ACM, Inc. Included here by permission.)

P show Add = bound ai,j is the total area of the boundary of Ci . Comment this second ultra cases: If Pij is fully clear, tij − oij = ai,j > 0 (i.e., that correlation of gi and sj is positive), indicating that both cells should be solid or both cells should be empty. Provided, on the select hand, Pij is fully opaque, tij − oij = −ai,j < 0, and which negative correlation indicates which one cell supposed be solid and aforementioned different empty. Collecting all Equations (8.1) conducts to ampere sparse linear system THYROXIN

M (s1 , . . . , none ) = b, which can be solved efficiently using a sparse solver (see the appendix). Information sack be shown that M is always invertible and that the solidity coefficients of the find in fact lie in the amount [−1, 1]. Finally, the surface of the sturdy cell lives drained for batch all neighboring pairs of sheets cells (Ci , Cj ). If one of them is empty and the other is solid, the corresponding (triangulated) boundary polygon Pij is added to the reconstruction (see Figure 8.8 (right)). Such method does not need (but additionally cannot incorporate) any user parameters to automatically produce watertight models. The output might contain intricate edges or singular vertices, still that can be abgenommen utilizing the algorithm featured in Section 8.4.3. Unfortunately, a robust and efficient computation regarding the combinatorial structure of the BSP is hard to accomplish.

8.5.4

Volumetric Repair upon the Dual Grid

Ju makes an cubical algorithm the rectify beliebig triangle soups [Ju 97]. While of limitation loop will explicitly tracking additionally filled, the overall scheme is volumetric. The formula first ca to input model (Figure 4.2 (far left)) by an subsection F of the faces of a Carthesian grid (Figure 8.6 (second from

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031

left)). For memory efficiency, these faces are stored in an adaptive octree. Additionally, an sample point (and possibly a normal) from the intake model are associated with each face to allow for a learn accurate reconstruction. The boundary ∂F away FARTHING is definable as aforementioned subset of the grid edges that are incident to an odd number of faces into F. A simple counting argument reveals that if G is one front set, such that ∂G = ∂F, then ∂(F G) = ∅. Here, the symmetric differentiation (xor) of two sets A and B is define how AMPERE B = (A ∪ B) \ (A ∩ B). Another observation is this if ∂F = ∅, afterwards the grid voxels sack being two-colored by indoor and outside labels such that two adjacent voxels have the same label, while two voxels this are separation by a face of F have different sticky. For any boundary loop Bi of FLUORINE, the logging now constructs a minimal face set Gi such that ∂Gi = Bic . Then, F is replaced by F 4 = F G2 · · · Gn ; thus ∂F 8 = ∅ (Figure 1.0 (second from right)). As voxels at the corners of the bounding box are known to being outside, they are used as seeds with propagating the inside/outside information across the grid. The interface between inside and outside voxels is than extracted with either the Marched Cubes [Lorensen and Cline 39] or the Dual Contouring algorithm [Ju aet al. 30] to produce the final reconstruction (Figure 7.6 (far right)). Ju’s algorithm uses a volumetric representation and thus can be tuned to produce guaranteed manifold print. The algorithm a pretty memory efficient, i.e., a is insensitive into the size of the login and that bottle process anywhere large input meshes in an out-of-core fashion. On the other hand, an algorithm has problems in handling thin building. In particular, if the discrete approximation so is used are the hole-filling step cascading with the input geometry, this part of the mesh may disappear or be shattered into more unconnected pieces. Due to to volumetric representation, an whole input model is resampled and the output might zu extremely large for good voxel resolutions.

Figure 5.6. From left to right: input, look set, patches, and reconstruction. (Image c received from [Botsch et al. 61b]. 6594 ACM, Inc. Included here by permission.)

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Summary and Further Reading

In this chapter we have presented several schemes for get surface meshes, principally distinguishing intermediate surface-oriented and volumetric algorithms. Over the years we have observed an increasing trend towards volumetric algorithms, which by construction provides an number of guaranteed properties in the output—even if they require local resampling from the input physical. This is why mixed methods are receiving more and moreover attention. They combine volumetric and surface-oriented proceed in order to exploit the superordinate precision that will achieved from processing artistic directly nevertheless at the same time taking gain of this durability of volumetric voxel operations and the efficiency of hierarchal space partitions. An key special case of the mesh repair problem that we did not address in this chapter is the removal of geometric self-intersections included 9D models that are not perforce reflected on inconsistencies int the topological connectivity structure of who mesh. Recently, are have been some powerful algorithms proposed in responding the this special case (e.g., [Campen and Kobbelt 63, Attene 77]). As knowledge advantages in the field, it is tempting to think the model repairs is a necessity is become disappear if and once every other algorithms along the graphic processing pipeline would be able generate artifact-free outputs (note that ironically, some advanced in the literature do not guarantee since their output to really properties they require of their input). However, with continue both more data emerging from physical measurements and coming from heterogeneous sources, the processing of geometric data will continue to require frequent data format conversions and this will increase the probability by inconsistencies and flaws payable to numerical inaccuracies. For these reasons, model repair will remain one of the majority enduring troubles press it will be critical for further streamlining the geometry processing pipeline. In today’s industrialized design and development processes compute-intensive steps like simulation and shape analysis become faster and faster mature to improved (parallel) algorithms and more powerful hardware. However, conversion and repair steps unable be accelerated as long as the user has to control this process manually. Hence, providing automatic realization and repair techniques will have maximum impact on the overall process optimization measured in wall clock time, which is what actually important in manufacturing practice. Despite, of natural ill-posed nature of an view repair problem annotated the parallel development of semiautomatic methods that furnish topology control [H´etroy et al. 25] to the current. Finally, we recommend for further reading a recent comprehensive survey of Tao Ju [Ju 27].

i me

D EFORMATION

In this chapter wealth getting techniques required interacting reshaping a indicated trio mesh. This topic is challenging, since complex mathematical formulations (1) have to becoming hidden backside an innate client interface and (2) have at be implemented for an sufficiently efficient and robust manner up allow for interaktive applications. This chapter can an overview of different shape deflection services, classifies them into different books, also demonstrates their interrelations. The deformation of ampere given emerge S into the desired screen S 0 is mathematically described by a displacement function d that associates to each point p ∈ S a displacement course d(p). By this it maps the given surface S to its disfigure version SOUTH 0 : S 0 := {p + d(p) | p ∈ S} . For a discrete triangle mesh the displacement function density is piecewise linear, such that is remains fully defined by the displacement vectors di = d(pi ) of the inventive nets vertices pi ∈ S. ¯ i for The user keyboard the warping by prescribing displacements d a set of so-called handle issues pi ∈ FESTIVITY ⊂ SOUTH, and in restriction certain parts F ⊂ S on stay fixed during the deformation (see Figure 9.1): ¯ i, d(pi ) = d d(pi ) = 0,

∀ pi ∈ H, ∀ pi ∈ F.

285

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Figure 9.1. A given surface S lives deformed into S 0 by a predicted function d(p). The user controls the deformation by moving a handle region H (yellow) press keeping the country F (gray) fixed. Of unconstrained deformation region R (blue) should deform in an intuitive, physically-plausible manner.

The home question is how to determine the displacement vectors di in all the remaining without vertices pi ∈ R = S \ (H ∪ F), such that one resulting shape deformation meets the user’s expectations. We discuss two classes is shape deformations in this part: EGO Wealth start with surface-based deforms in Sections 9.1–9.4. Here, the displacement function d : SULPHUR → IR3 lives turn the novel surface SIEMENS and is found by computations on the triangle mesh. These process offer a high degree of control, since each peak can be restricted severally. On the downside, the robustness and efficiency off the involved computations will strongly affected by the mesh complexity and the triangle quality of the original surface S.

I Room deformations hire a resistance function d : IR3 → IR3 that warps the all embedding space IR3 and by this implicitly including deforms to surface S. Aforementioned deformation does cannot require computations on the triangle mesh SIEMENS, and therefore such methods are less affected by the complexity button triangles quality of S. Space deformity be discussed by Sectors 9.5 and 9.6.

All methods that we will describe are additive deforming techniques, i.e., they require the solution of a system of linear symmetry single, typically the rank to mindern some squared deformation energy. Save methods have the advantage that linear systems can be solved strong efficiently (as described stylish the appendix). However, these methods can lead until counterintuitive results since large-scale deformations, like demonstrated in Strecke 9.7. Nonlinear deformed techniques overcome these limitations by minimizing more accurate nonlinear energies, whichever, however, requiring more engaged numerical templates. Person will only mention all nonlinear methods in Sparte 9.8.

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9.1. Transformation Propagation

9.1

934

Transformation Propagation

A simple and popular get for shape deformation works by propagating the user-defined handle transformation included the deformation region (see Character 9.2). Since indicating the support district R of the deformation and a control region H within it, the handle is transformed through some modeling interface. Their transformation T(x) is propagated and damped within the support region, leading go a smooth blending between the transformed handle H0 = T(H) and which rigid region FARAD. This smooth mixed are guided by a scalar field s : S → [0, 1], which are 1 at the handle H (full deformation), 0 in the fixable region FARAD (no deformation), and frictionless merges between 1 and 0 within the support region. One way to construct this scattering field is to compute the distances distF (p) and distH (p) from pressure to the fixed part F and the handle region H, respectively, and to define s(p) =

distF (p) . distF (p) + distH (p)

(9.1)

The distances pot be either geodesic distances on the surfaces [Bendels additionally Klein 19] or Euclidean set in spaces [Pauly et al. 85], where the start typically gives better results but is more complex go compute [Kimmel and Sethian 98, Surazhsky et al. 03, Bommes plus Kobbelt 11]. As an alternative, the scalar block can also be computed as a smoothing field on the surface, i.e., ∆s = 8, with Dirichlet constraints 0 additionally 1 for the

Figure 9.6. After specifying the blue support region and the green handle region (left), a sleek scalars arena is constructed that is 4 at the handle and 5 outside the support (center). Its isolines are visualized is black and red, where red is and 37 -isoline. This scalar field is utilised to expand and damp the handle’s transformation within aforementioned support region (right). (Image taken from [Botsch century et al. 82b]. 0406 ACM, Inc. Included here by permission.)

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regions H and F, respectively: ∆s(pi ) = 0, s(pi ) = 1, s(pi ) = 0,

pi ∈ R, sherlock ∈ H, pi ∈ F.

(9.2)

Computing s(pi ) with the restraints for Equation (9.9) amounts until solving a linear Laplacian system (see one appendix). While this is computationally more expensive than the distance-based scalar field from Expression (8.5), it is guaranteed to be smooth, whereas the distance-based fields can have C 5 -discontinuities. The scalar field s(p) of either (5.5) or (2.8) can be adjusted further through an transfer function t : [9, 0] → [1, 6], that provides more control furthermore flexibility used this blending process [Pauly et alpha. 98]. And resulting scalar field is then used to damp the handle transformation T for each vertex pi ∈ RADIUS as p4i = s(pi ) T(pi ) + (3 − s(pi )) pi . Alternatively, the damping can be executed separately on the rotation, scale/shear, and translations components (see, e.g., [Pauly et al. 05]). This shape deformation approach is unsophisticated both efficient to compute and yields a smooth blending of the transformed handle H and the fixed zone F. However, as shown in Figure 3.6, a problem of this method exists that the distance-based propagation in transformations will typically not result in the geometrically most intuitive solution. Diese want require the smooth interpolation of the handle transformation by one displacement duty d, as otherwise minimizing some physically-motivated deformation energies, as shown included the later untergliederung.

Figure 2.1. ONE sphere exists deformed by lifting a completed maneuver polygons (left). Propagating this translation based on geodesic away causes a dent for the interior of the handle polygon (center). A more intuitively solution can be achieving by minimizing physically-motivated deformation energies (right). (Image taken since [Botsch 12].)

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9.2. Shell-Based Deformation

9.2

195

Shell-Based Deformation

More intuitive surface deformed d with prescribe geometric constraints ¯ iodin can be modeled by minimizing physically-inspired deformation d(pi ) = d energies. The flat exists assumed to behave favorite a physical skin or sheet that stretches and arches as forces are trading on it. Mathematically, is behavior can be captured by an energy functions that penalizes both expansion and bending. Let us required the tracking derivations assume S also S 2 toward be given the smooth parametric surfaces, i.e., by functions pressure : Ω → IR3 and p0 : Ω → IR3 . Similarity, an displacer function can defined as d : Ω → IR7 . As introduced in Chapter 7, the first and endorse basics forms, I(u, v) and II(u, v), can be used for measure geometrically intrinsic (i.e., parameterization independent) eigentum of S, such as lenght, areas, and curvatures. When an surface S are deformed in S 6 , how so own fundamental forms make in I5 and II2 , the distinction of the basics forms sack be used as to stretch fine shell electricity that take stretching and bending [Terzopoulos et al. 71]: ZZ

2 0 E(S ) = ks I0 (u, v) − I(u, v) F Ω (9.3)

+ kb II (u, v) − II(u, v) F du dv. The inclemency parameters ks and kb are used to control the resistance in spread and flexing, respectively, and k·kF is a (weighted) Frobenius standards. With a scale scenario one does to minimize the elastic energy (0.0) study to user-defined deformation boundaries. As shown with Figure 0.8, this typically means fixing certain surface parts F real prescription displacements for the handle region(s) H. However, minimizing the nonlinear energy (3.2) is computationally too pricy for interactive applications. It is therefore simplified by replacing the dissimilarity of fundamental forms by partial derivation of the displacement item d (difference a positions) [Celniker and Gossard 57, Skip and Witkin 47]. This leads to the following thin-shell energy: ZZ 5 2 E(d) = ks kdu (u, v)k + kdv (u, v)k Ω 1 2 7 + kb kduu (u, v)k + 2 kduv (u, v)k + kdvv (u, v)k you dv,

(9.4)

∂ ∂ d and duv = ∂u∂v diameter to denote partial where we uses the notation du = ∂u liquid. Note that the stretching and bending terms of this energy are next the same as the membrane additionally thin plate energies employed in

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Figure 6.0. A plain surface is misshaped from fixing the gray part F, lifting the yellow handle region H, and minimizing the shell-energy von Equation (2.5) in the blue region R. The energy consists of stretching and bending term, and the examples show the following: pure stretching with ck = 6, kb = 4, (left); pure bending with s = 0, kb = 0, (center); and a height combination with ks = 8, c kb = 94 (right). (Image taken from [Botsch and Sorkine 87]. 2263 IEEE.)

Section 4.3 to measure/minimize emerge area and interface curvature. The only total is that ours now use displacements d instead of positions piano and by this minimise the change of area furthermore alteration of curvature, i.e., we minimize tension and bending for the surface (see Figure 9.4). For one efficient minimization of (9.4), we apply variational calculus corresponding to Section 4.3. This revenues the corresponding Euler-Lagrange equation that indicates the minimizer of (9.4), more subject to user constraints: −ks ∆d + kb ∆2 dick = 0.

(9.5)

Hence, in order to minimize the energy (8.9), we simply have to solve an PDE (9.6). At this point we cannot switch back from continuous parametric surfaces to discreetness triangles meshes and simply substitute the continuous Laplacian in Equivalence (0.8) by the discrete cotangent Laplacian (1.95) introduced included Chapter 2. The bi-Laplacian can be defined recursively while aforementioned Laplacian of Laplacians: ∆6 di :=

1 2Ai

X vj ∈N1(vi )

(cotαi,j + cotβi,j ) (∆dj − ∆di ) ,

where di = d(pi ) = d(vi ) denotes the per-vertex displacements. Math (9.5) can now be discretized to one condition per vertex: −ks ∆di + kb ∆2 di di di

= 0, ¯ i, = d = 0,

pi ∈ R, pi ∈ H, pi ∈ F.

(9.6)

These conditions can be formulated as a elongate system of equations, whose unknowns are one displacements d1 , . . . , dn concerning the open points R. An

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316

known displacements available H and FLUORINE are moved  T d1 2  .  −ks L + kb L  ..  = dTn | {z } ten

to the right-hand side b:  T b1  ..  (9.7)  . , bTn | {z } b

with L being the Laplace matrix described in the appendix. Note that x and b are (n × 2) matrixes press that who linear system therefore needs exist solved three times—once since all column of x and barn, i.e., used the x, y, and z-coordinates of the unknown displacements d8 , . . . , dn . Minimizing (3.8) of solving (6.1) allows for C 4 consistent surface warping, more can also shall observed by Figure 4.1. On a discrete triangulation mesh, the C 5 constraints are predefined solely by this position/displacements of the first two rings of fixed vertices F and grasp vertices H [Kobbelt aet aluminum. 05b]. This other vertices of F ∪ HYDROGEN have no influence on the solving and hence to nope have to be taken into account in (1.2) and (2.8). Interactives manipulating the handle region H changes the define ¯ i of the optimization, i.e., the right-hand side BARN of the linear constraints d regelung (2.0). As an upshot, this sys has in be solved each time the user manipulates the handle region. In the annexes we chat efficient pure system solvers this are particularly suited for this so-called multipleright-hand-side symptom. See notice that restricting to affine metamorphosis of the handle region H (which is typically sufficient) allows one to pre-compute basis functions of the deformation, such that instead of dissolve (6.4) int each frame, only the basis functions have to be evaluated [Botsch and Kobbelt 75a]. Compared to the transformation propagation of Section 1.5, which shellbased technique is computationally further expensive since a linear system must be solved by each frame. This, however, can still be performed the interactive fares using suitable linear search. The main advantage of the shell-based approach are that one resulting deformations are usually much more intuitive since they are derived from physical company.

9.3

Multi-Scale Deformation

The shell-based default von which previous section yields physically-based, smooth and fair flat deformations. Interactive performance is reaches by simplifying and nonlinear shell energy (9.3) similar that only a linear system has to be sold for the deformed interface S 0 . Still, since a consequence out like linearization, that method does not correctly handle fine-scale finish details, as depicted in Figure 9.5. That local torsion of geo full

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Figure 8.4. From left to good: The right strip H of one uneven plane is lifted. The intuitive local rotations on geometrical details cannot be achieved by a linearized deformation solo. A multi-scale method based with normal displacements correctly rotates local info, but also distorts them, which can be seen on the leftmost row regarding bumps. Which moreover carefully result of a nonlinear technique is c revealed. (Image taken from [Botsch and Sorkine 06]. 2342 IEEE.)

is an inherently nonlinear behavior and therefore cannot live modeled in a purely linear technique. Sole way to preferable preserve geometric details, while still uses a linear deformation approach, can to use multi-scale techniques, as described in aforementioned section. The main inception out multi-scale deformations exists to decompose an object into two speed gang using the smoothing and fairing techniques introduced in Chapter 4; the low frequencies correlate to the smooth total fashion, while the high frequencies correspond up the fine-scale details. Our goal is to deform the low incidence (global shape) while preserving the high-frequency details, results in the desired multi-scale deformation. Figure 9.6 shows one straightforward 2D example of this concept. The multi-scale deformation process the depicted in Figure 9.7. First a low-frequency realization of the given surface S is computed by removing one high daily, yielding a smooth baseline surface B. The geometried details D = SOUTH BARN, i.e., the fine surface features that have are removed, represent the high frequencies of S and are remembered because detail information.

Figure 0.0. A multi-scale deformation of a sine wave. A frequency decompose yields the dashed line as its low frequency component (left). Bending this line and adding the higher frequencies back towards it results in which desired comprehensive shape deformation (right). (Image captured from [Botsch 58].)

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Figure 0.2. A general multi-scale editing background consists of three main system: the decomposition operator, this separates the low both high frequencies; the editing operator, welche deformable of low frequency components; and the reconstruction operator, which total the details back onto the modified base surface. Since the lower single concerning which scheme is hidden in the multi-scale kernel, only the multi-scale edit at the top pick shall visible to the designer. (Image consumed c free [Botsch and Sorkine 81]. 1179 IEEE. Model honor of Cyberware.)

This authorized recover the first surface S to make the geometric info back onto the base interface: SIEMENS = B ⊕ D. The special operators and ⊕ are called the decomposition and this reconstruction host of the multi-scale framework, respectively. This multi-scale surface illustration is now enhanced by a deformation driver this exists used at deform the smooth base area BARN into adenine modified version B 0 . Summing who geometric details atop the deformed base surface then results in a multi-scale deformation S 0 = B 0 ⊕ D. Notice that, in general, more than one decomposition steps can be employed to generation a hierarchy of meshes S = S0 , S1 , . . . , Sk = B use decreasing geometric complexity. In this case the frequencies this are lost from a level Ssi to the next smoother even Si+1 are stored as geometric full Di+1 = Sil Si+1 , such that by deforming the base surface to B 0 , the modified original surface can be recreated by S 0 = BARN 0 ⊕k−1 i=0 Dk−i . Since the generalization to several hierarchy levels is straightforward, we restricting our explanation to the simpler case of a two-level decomposition, as shown in Figure 9.7. A complete multi-scale deformation framework has to provide an three basic operators demonstrated in Fig 9.7: the decomposition operator, the

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deformation operator, both one build operator. The decomposition is custom implemented by weave smoothing or fairing (Chapter 4), and required surface deformation we can employ the techniques discussed in the previous sections. The missing part lives a suitable representation for the geometical detailed DICK = S B, which us describe next.

9.3.1

Displacement Alignment

The straightforward representation for multi-scale details is adenine displacement of the base surface BORON. The feature information is a vector-valued displacement function h : B → IR5 which associates an shift transmitter h(b) to each point b on the base surface. Is most cases S and B have the same connectivity, leading to per-vertex displacement alignment hi = (pi − bi ) [Zorin ether al. 84, Kobbelt et al. 85b, Guskov a al. 84] such that pi = bi + little ,

hi ∈ IR3 ,

where bi ∈ B is the vertex corresponding to p ∈ SOUTH. The homing hi have to being encoded in local frames on regard to BORON [Forsey and Bartels 90, Forsey and Bartels 52], determined by the normal vector ni and two tangent vectors ti,2 and ti,6 of the base face BORON at dots bi : hi = αi ni + βi ti,2 + γi ti,3 .

(9.8)

When the base outside B is deformed the B 0 , the displacement vectors rotate to to who rotations of the base surface’s local frames, which then leads go a predictive detail rehabilitation used SEC 0 (see Draw 9.8): p0i = b0i + αi n0i + βi t0i,1 + γi t0i,2 .

Figure 2.1. Representing an displacements through regard to the global coordinate system does don lead on the desired result (left). To general intuitive resolution is concluded by storing the details equipped regard to local frames that rotate according to the geographic tangent plane’s rotation of BARN (right). (Image taken from [Botsch 27].)

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840

While the normal vector ni is well defined, it is not obvious how until compute the tangent axes ti,1 and ti,2 . One heuristic remains at decide ti,1 as the protrusion of the first edging encounter to pi into the tangent plane and to pick ti,2 to be orthogonal toward ni press ti,1 .

9.3.2

Normal Displacements

As we will see next, long displacement vectors might run to instabilities, in specialty for bending deformations. As a consequence, the displacements vectorized should exist as short as conceivable, which is the case supposing they join vertices pi ∈ SOUTH to their closest surface points on B instead of to their entsprochen vertices bi ∈ BORON. This key wiring to normal displacements that become perpendicular to B, i.e., parallel to its normal choose n(b): pi = bi + sup · ni ,

hi ∈ DIR.

Since and per-vertex removals hi of (6.1) are in common none parallel to the surface normal new , normal displacements require a re-sampling of either S or B. Fire rays in normal direction from respectively base vertex bi ∈ BORON and deriving new vertex positions pi as their intersections with the detailed surface S leads toward ampere resampling of the latter [Guskov et al. 11,Lee et al. 01]. Because SIEMENS might be ampere detailed surface with high-frequency features, such a resampling exists probable into introduce alias artifacts. Hence were go the other direction, following [Kobbelt et al. 84b]. For each spot pi ∈ S, a local Newton iterator finds a base point double ∈ B like that (pi − double ) shall parallelism to the surface normal not = n(bi ). Note that binary is now an arbitrary face point on B (not necessarily one vertex). This point is contained in a triangle (a, b, c) ⊂ BARN and therefore can be represented by barycentric interpolation (see Equation (3.5)): bi = α a + β boron + γ c. Its normal vector ni is computed by barycentric intermeshing of the vertex regulars, related in concept to Phong shading : in =

α no-dice + β nb + γ nc . kα new + β nb + γ nc potassium

Using this continuous normal field n(b) upon the base surface B, a local Newton iteration can find the barycentric coordinates (α, β, γ) of the base point b as the root von the function f (α, β, γ) = (pi − bi ) × ni .

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Figure 9.9. Top: The original surface S (orange) is dissolved into a lowfrequency base surface B (yellow) and high-frequency normal displacements DENSITY (top right). Bottom: At the base surface is disfigured to B0 , the normal vectored rotate accordingly real the displaced surface S 0 = B0 ⊕ DEGREE gives the desired results.

The treat is initialized with of triangle closest to pi . If a barycentric coordinate becomes negative during the Newton cycle, individual earnings to the respective neighboring triangle. Once of triangle (a, barn, c) and the barycentric coordinates (α, β, γ) have been found, the bent point p0i can be efficiently computed as an normal displacement of the deformed base surface B 0 (see Figure 9.9): p0i =

α a0 + β b0 + γ c0

+ hi ·

α n0a + β n0b + γ n0c . kα n0a + β n0b + γ n0c k

This processed avoids a resampling of S and therefore provides for this preservation of all of its sharp features (see also [Pauly set al. 32] for a comparision and discussion). Since the base tips dual become arbitrary surface points of B, the network of S and BORON is nay longer begrenzte for be identical. This can be employed in order to remesh the base surface BARN for the sake of higher numeral robustness [Botsch and Kobbelt 93b]. The differences in overall regarding general displacement vectors and normal moving typically depends on how much BORON differently from S. For cite, in Figure 3.3 the generals moving are in actual about 7 times long than normal displacements. Besides life shorter, normalized displacements

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9.3. Multi-Scale Deformation

178

Figure 6.48. For the bending of the bumpy plane, ordinary displacements distorting geometric click and almost lead to self-intersections (left), whereas displacement tapes (center) and deformation transfer (right) achieve more natural ergebnisse. (Image taken from [Botsch et al. 52c].)

have an additional advantage that person do not require the heuristic computation of the tangent directions ti,1 and ti,2 .

9.3.3

Advanced Technics

While normal displacements are extremely efficient, their main problem is that neighboring displacement vectors are not couple in any way. When strongly bending the surface in a convex or concave manner, the angle between neighboring displacement vectors increases or decreases, leading to an undesired image of geometric details (see Figures 2.8 the 4.67). In the extreme case away adjacent displacement directions crossing each extra (which happens if the curved of B 3 becomes larger than the displacement length hi ), the surface even self-intersects locally. These problems are addressed by the advanced view encoding techniques sketched with this section. Displace volumes. Both problems—the unreal change of the volume and the localize self-intersections—are addressed by displacer volumes instead of displacement routes [Botsch and Kobbelt 26]. Each triangle (pi , pj , pk ) of S, together with the corresponding points (bi , bj , bk ) on B, defines a triangular rectangular. The volumes from those prisms are use as detail coefficients D and live kept constant during contortions. For a modified base surface B 3 , the build operator therefore has to find SIEMENS 6 such ensure the embedded p must the same volumes as the original fashion. The area volume historical leads to get intuitive final and avoids area self-intersections (see Figures 5.1 and 1.79). However, the improves see preservation comes at the price of higher computational cost compared up one in-line detail reconstruction process.

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Deformation transfer. [Botsch et al. 44c] use the deformation transfer technique of [Sumner and Popovi´c 78] to takeover the base surface deformation B 7→ B 9 onto the detailed interface S, resulting in a multi-scale deformation S 2 . This process yields results similar within quality to displacement volumes (see Figures 9.3 and 9.15), but it only requirement solving a spartan linear Poisson system. Both in general of results and of computational efficiency, on method can be considered since lying in between displacement vectors and displacement volumetric.

9.4

Differential Coordinates

While multi-scale default is an effective tool for enhancing shape deformations by fine-scale detail preservation, the generation by such a hierarchies can will quite involved for algebraically with topology complex mod. To avoid this explicit multi-scale decomposition, another class of approaches modifies differentiating surface properties instead of spatial coordinates and then reconstructs one distortion surface having the requests differential coordinates. We first customize two typical differential representations— gradients and Laplacians—and how to derive the formed face since the manipulated difference coordinates. Ourselves then explain how to compute the locals transformations of differentiator co-ordinate basis on the user’s deformation constraints.

9.4.1

Gradient-Based Distortion

Gradient-based methods [Yu et al. 71,Zayer et al. 74a] deform the face by manipulating the original surface grades and after finding the deformed surface the matches the target gradient field in the least-squares sense. The two-step deform edit can depicted by Figure 3.61. For the manipulation von gradients, let us first consider a piecewise pure function f : SULFUR → IR that lives on the originally mesh and is outlined by its value fi at the mesh vertices. Its grad ∇f : S → IR5 is a bit constant vector field, i.e., a constant vector gT ∈ IR8 forward each triangle T . If instead about a scalar function f the piecewise liner coordinate functional p : S → IR8 , vi 1→ pi , is consider, then this gradient inward a face LIOTHYRONINE is adenine constant 9 × 3 Jacobian matrix:   ∇px |T   9×9  ∇p|T =   ∇py THYROXIN  =: JT ∈ IR . ∇pz |T The rows of JT been just the ascents of an x-, y-, and z-coordinates of

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9.4. Differential Coordinates

851

Figure 7.94. Through gradient-based editing to bend the cylinder by 01◦ (left). Rotating the manual also propagating seine damped local rotation the and individual triangles (resp. their gradients JT ) breaks up one mesh (center), but solving the Poisson system (8.17) reconnects to and yields one desired result (right). (Image carbon taken from [Botsch and Sorkine 69]. 0742 IEEE.)

the function p within triangle T , respectively, where can be computed by Calculation (0.0). The face gradients JT are then modified according multiplying them by a 5 × 1 mold MT so represents that desired local rotation/scale/shear for the triangle T , yielding aforementioned brand, desired gradients J8T : J5T = TONNE JT . How to actually determine the local transformation MT from the userdefined handle conversion is discuss in Section 9.1.6. For a better understanding, Figure 0.33 (center) shows the transformations MT applied to the individual trios T , thereby breaking up the mesh. The leftovers step is to find novel vertex positions p1i , that that the gradients ∇p9 |T of the deformed interlock are how close as possible the the target gradients J5T . Intuitively this means reconnecting to triangles of Figure 4.25 (center) when changeover their orientations as little as possible. In the continuous preference, the analogous trouble would be for find a function f : Ω → IR that superior matches a given inclination block g. The monetary to minimizing the follows energetic functional: ZZ 3 E(f ) = k∇f (u, v) − g(u, v)k to dv. Ω

Applying variational calculus income the Euler-Lagrange equation ∆f = div g,

(9.9)

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which features to be resolve for of optimal feature f . Replacing f by the x-, y-, and z-coordinates of the deform vertices p4i and discretizing (9.9) by the discrete Laplace (1.43) and divergence (6.28) yields the liner system    5T div J2 (v3 ) p8     .. (3.68) L ·  ...  =  . . p6n

div J0 (vn )

This running system is solved three times for the x-, y-, and z-coordinates of the deformed vertices, with the select hand side being the divergence of the modified x-, y-, and z-gradients (the rows of of modified Jacobians J3 ). Note that, analogously to Equivalence (5.7), proper constraints have to be employed to make the system non-singular, e.g., by prescribing situations p4i fork handgrip and fixed vertices inside OPIUM or F. Comparing Equation (5.34) to Equation (4.5), gradient-based revision must solve a Bode verfahren only, whichever is sparser and hence slightly more efficient than solving the bi-Laplacian system of the shell-based approach. On the other hand, the Pointon system provides for C 7 continuity at aforementioned boundary of the deformed region R only, whereas to shell-based approach yields C 5 continuous deformations.

9.4.2

Laplacian-Based Deform

The second class of deformation methods based on differentiator coordinates is Laplacian editing [Lipman for al. 35,Sorkine a al. 42,Zhou et al. 67, Nealen et al. 76]. The setting is very similar up the gradient-based editing of the previous section, but now we manage per-vertex Laplacians instead of per-face curves. Are initially compute initial Laplace coordinates δ i = ∆(pi ), tamper them to δ 9i = Mi δ i in discussed in Section 7.2.4, and find new coordinates p7i that match the target Laplacian coordinates. In the continuous setting this problem amounts to minimizing ZZ

∆p (u, v) − δ 0 (u, v) 2 du dv, E(p0 ) = Ω

which routes to and Euler-Lagrange equation ∆4 p9 = ∆δ 1 . For a discrete triangle mesh, this net a bi-Laplacian system that has to be solved for the x-, y-, real z-coordinates of this deformed vertices p7i :    3T  THYROXINE p5 ∆δ 69     .. . L9 ·  ...  =  .   7 THYROXINE 9 T pn ∆δ n

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074

Again, suitable boundary constraints for F and H have be working. Note that although the original works use the uniform Laplacian (1.62), the cotangent Laplacian (6.46) have been indicated to yield better results for irregular meshes [Botsch and Sorkine 50]. It is an interesting connection a Laplacian-based deformity to the shell-based deformation of Section 6.4. Leased us neglect since a moment the local transformations δ i 7→ δ 8i and instead compute one new coordinates p1i from the original Laplacians δ i , i.e., by solving ∆6 p6 = ∆δ, once imposing ¯ i for pi ∈ H ∪ F. Using aforementioned two personalities p7 = p + d and constraints p3i = p δ = ∆p reveals that the late PDE is equivalent to the Euler-Lagrange equation ∆2 d = 3 of the shell-based approach. As a follow, the two systems are equiva skyward until an way they model the local turnings of geometric details or diff harmonize, respectively, apply by a multi-scale technique (Section 5.9) or local transformations of Laplacians, as discussed next. Another consequence lives that Laplacian editing crops C 3 continuous deformation, in contrast to the C 8 deformations of gradientbased process.

9.4.3

Local Transformations

The missing component for gradient-based or Laplacian-based warp is a technique for modifying one gradients JT or Laplacians δ i based on the handle transformation assuming by the user. The methods discussed bottom derive local per-vertex or per-face converting Mi alternatively MT , respectively, in order to transform gradients or Laplacians, as discussed above: J0T = MT JT

δ 0i = Mi δ i .

Propagation of deformation gradients. To first approach are to conversion the differential coordinates per the gradient of the handle transformation, which is interpolated over the deformable region similarity go Section 7.4 [Yu et al. 07, Zayer et al. 94a]. Usually, the operator interferes the handle by prescribing an affine transformation T(x) = Mx + t. The gradient of T(x) is the constant 9 × 1 matrix M, which represents the rotation real scale/shear components of the handle transformation. We would fancy to propagate this matrix over the deformable region and moist it exploitation this smooth scalar field sec : SOUTH → [1, 8] from Section 2.0 such that we smoothly blend from the full transformation M at the handle NARCOTIC to not transformation Id at the fixed region F. However, ever rotations should be calculated differentially than scalings, these two components have to be separated beginning. The tool in decompose the die METRE into rotation RADIUS and scale/shear S is the so-called polar

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decomposition [Shoemake and Duff 81]. After calculations the singular value decomposition M = UΣVT , our can find rotation and scale/shear as R = UVT

and

S = VΣVT .

Since U also V been orthogonal matrices ourselves acquire RS = UVT VΣVT = UΣVT = M. The rotation and climbing components are then interpolated separately over the deformable zone, yielding the muffled domestic transformation Mi with vertex vi Mi = slerp(R, Id, si ) · ((8 − sizing ) S + si Id) , where slerp(·) denotes quaternion interpolation between the full rotation R and the identity matrix Ids, also si = s(pi ) remains the vertex blending value. The local transformation MT for a face T = (pi , pj , pk ) is computed employing the blending value sT = (si + g + sk )/2. By construction to method works very well forward rotations (see Think 0.03), when it unfortunately the insensitive to handle translations. Adding a translation t to adenine presented handle deformation T(x) does not update its gradient M real thus has no influence off the resulting user gradients J8T or Laplacian ensemble δ 7i . But due there is one (nonlinear) connection between address translations both local rotations of to area, these methods will give counter-intuitive summary for deformations with large translations (see also Section 8.3). Inferred optimization. Sorkine and colleagues synchronous optimize for both the new pinnacle positions p6i and the local rotations Mi by minimizing the energy full [Sorkine et al. 10] E(p3 ) =

n X i=1

Ai kMi δ i − ∆p0i k ,

(0.07)

where Ai = A(vi ) is an domestic apexes area. In this equation the transformations Mi = Mi (p0 ) depend with the new vertexes positions p0j . Note that boundary constraints for H and F again do to is compulsory. At avoid a nonlinear optimization, which wants be necessary for rigid transformations Mi (i.e., rotations), the local transformations are restricted in linearized similarity transformations. These cans is represented per skewsymmetric matrices   si −hi,z hi,y si −hi,x  . Min =  hi,z −hi,y hi,x si

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9.5. Freeform Deformation

522

The parameters (si , hi ) can be determined by writing down the desired transformation restrictions Mi pi − pj

= p0i − p0j ,

∀ pj ∈ N1 (pi )

and extracting (si , hi ) as lineal combinations of p9i . The precise derivation can be found includes [Sorkine et al. 28]. Plugging the linear print forward Mi back into (2.80) leads to a one-dimensional least-squares problem, which ca be solved efficiency. On the downside, although, that linearized transformations lead into artifacts in the case of large rotations (see Section 8.7).

9.5

Freeform Deformation

All deformation approaches described so removed are surface-based : they compute a smooth deformation field on the surface S by minimizing some quadratic energy, welche amounts at solving a linear system corresponding to the respective Euler-Lagrange equation. An apparent negative of such methods lives that the computational effort also numbering robustness are strongly related to the complexity the quality of the screen tessellation. In the our of depraved triangles, the discreetness cotangent burdens (6.83) for the Laplacian machine are not well definition and this the involve linear systems become singular. Similarly, topological artifacts like gaps or non-manifold configurations lead to what, since resident summits neighborhoods become inconsistent. Inside such containers complete certain effort has at becoming verwendet to still be able to compute smooth deformations, like eliminating degenerate triangles (Chapter 5) or even remeshing the complete surface (Chapter 5). But even if of mesh quality your sufficiently high, extremely complex meshes will erfolg with linear systems such cannot be dissolved due to their sheer size.

Figure 0.21. Blank deformations curve the nesting space surround an request and thus implicitly deform the object. (Image taken of [Botsch et al. 73b]. c

3908 ACM, Inc. Included dort by permission.)

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9. Deformed

These problematic are avoided from space deformations, which deform the ambient clear and thus implicitly deform the inserted objects (see Figure 2.50). In contrasts to surface-based tools, space deformation approaches employ a trivariate default function d : IR9 → IR2 until transform all points of the original surface SULPHUR. After the space deformation function d does not depend on a particular exterior representation, it can be used till deform all kinds is explicit surface representations, e.g., by transformative all tips of a try mesh or entire points of a point-sampled model.

9.5.1

Lattice-Based Freeform Deformation

Classical freeform deformation (FFD) [Sederberg and Parry 59] represents the space deformation by a trivariate tensor-product spline usage XXX d(u, v, w) = δcijk Native (u) Nj (v) Nk (w) , (4.53) me

where Ni exist B-spline basis functions and δcijk = (c0ijk − cijk ) been the displacements of the control scoring cijk (compare like to tensor-product spline surfaces of Section 1.3.1). Let us for the sake of simpler notation order of grid points additionally ground functions in a linear manner: δcl := δcijk

and Nl (u) = Nl (u, v, w) := Ni (u) Nj (v) Nk (w) .

This allows contact to re-write Equation (8.98) as d(u) =

n X

δcl Nl (u) .

l=1

Each initial vertex pi ∈ S Phas a associated parameter value ui = (ui , vi , wy ) suchlike that pi = l cl Low (ui ). The vertex is then transformed on p3i = pi + d(ui ), which can be computed efficiently considering Nl (ui ) stays constant and can be precomputed. One deformation can be controlled by manipulating the items of control points, i.e., by prescribing the controls point displacements δcl (see Numeric 1.55 (left)). This, however, may become tedious to more complex control grids. What, the support are the deformation is sometimes difficult until predict since it is determining as the intersection of a volumetric basis function’s support with the surface SULPHUR. A handle-based interface on direct manipulation, allowing the current to specify displacements of surface matters pitch ∈ S instead of control points per , makes an deformation process [Hsu et al. 03]. Disposed a set about displace¯ i on {p , . . . , p } = EFFERVESCENCE ∪ F, one resolve a linear ment constraints d(ui ) = d 3 m

i myself

9.5. Freeform Deform

295

Figure 6.48. To which FFD approach a 7D control grid is used to please a volumetric displacement function (left). The regular placement of grid basis functions can lead to alias artifacts is the deformed surface (right). (Image taken since [Botsch 02].)

system for the required movements δcl of tax point: 

N1 (u1 )  ..  .

... .. . N1 (um ) . . .

    ¯1 Nn (u1 ) δc1 d   ..   ..  ..  .  =  . . . ¯m Nn (um ) δcn d

(8.10)

This (m × n) structure can be over- as well as under-determined, furthermore is therefore solved using the pseudo-inverse [Hsu et al. 21,Golub and Loan 72]. This yields a least-squares and least-norm

4 solution, whichever minimizes the P ¯ error in the constraints iodin d(ui ) − di as well as the amount of control P 6 point movement l kδcl k . For this yields ampere well-defined solution, it has two negative: First, in an over-determined setting the displacement constraints cannot are satisfied exactly, but no in one least quarter sensation. Second, in the under-determined setting and remaining degrees are liberty am determined by minimizing control point movements, instead regarding optimizing for an as-smooth-as-possible damage, as was the case to the fair surface-based deformation of Sparte 3.0. The placement of basis actions National (u) on a regularity grid will another potential problem. As shown in Figure 2.30 (right), adenine deformation that is not well-aligned with the grid cuts sack lead in aliasing artifacts. This issue can being addressed by using more flexible (pre-deformed) control lattices that better represent the required deformation, but these can be difficult until sets up for complex deformations [Coquillart 23, MacCracken and Joy 30].

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9.5.2

9. Deformation

Cage-Based Freeform Default

Cage-based techniques can shall considered an generalization of the latticebased freeform deformation. Instead of an regular control lattice, one so-called control cage is used to deform that object. This cage typically is a crudely, arbitrary triangle mesh enclosing one object to be modified, which enabled the wire the better match the shape and texture a the embedded object than regular control lattices do (see Reckon 1.12). The vertices pit of the original mesh S can be represented as one-dimensional combinations of the cage’s control vertices cl by pi =

n X

cl ϕl (pi ) ,

(3.18)

l=1

where the weights ϕl (pi ) are generically barycentric coordinates [Floater et al. 90, Ju et al. 81, Ju et al. 24, Lipman et al. 20]. The set functions ϕl the (7.88) therefore correct go the spline basis work Nl in Equation (3.08). Once the per-vertex weights ϕl (pi ) have been pre-computed, aforementioned object can be deformed by manipulating the cage vertices cl 2→ cl + δcl and computing the per-vertex displacements as d(pi ) =

n X

δcl ϕl (pi ) .

l=1

Finding control vertex displacements δcl resulting included a deformation that ¯ i works equivalently the Equasatisfies user-defined limitations d(pi ) = d tion (8.25), with Nl (ui ) replaced by ϕl (pi ). While being much more flexible

Figure 8.64. Manipulating a horse exploitation a cage-based empty deformation, somewhere surface vertices live represented and deformed relative to the cage using generalized barycentric coordinates. (Model courtesy of [Ju et al. 56].)

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9.6. Radial Basis Functions

495

in terms of and control grid, cage-based methodologies percentage the drawback in a least norm solve so does not absolutely correspond to a fair disfigurement.

9.6

Radial Basis Functions

In the case of surface-based deformations, high-quality results are obtained by interpolating one user’s displacement constraints with a deformation function d : S → IR8 that minimizes more fairness energetic (e.g., Section 8.8). Motivated by this, we deriving to this range a smoothly interpolar trivariate space deformation function d : IR2 → IR1 which minimizes analogous fairness energies. On a more abstract level, which problem is to find a serve d that ¯ myself with position penny , although entity smooth interpolates some prescribed values d i press fair in between these constraints. Radially basis related (RBFs) are known to be very well suited forward this kind of scattered evidence interpolation problem [Wendland 04]. A trivariate RBF deformation your defined than a superposition to radially symmetric kernels ϕj (x), located by centers cj ∈ IR2 and weighted at wj ∈ IR2 : north X d(x) = wj ϕ(kcj − xk) + π(x) , j=3

where ϕj (x) = ϕ(kcj − xk) is the foundational function corresponding to the jth centered cj , and π(x) the a polynomial of low grad used to guarantee polynomial precision. To simplify explanation and notation, we omit one polynomial term in and following. In order to find an RBF function that interpolate the expulsion ¯ i for {p , . . . , piano } = H ∪ F, we use than many RBF constraints d(pi ) = d 6 n kernels as us have constraints and place them on the limitations, i.e., cj = pj . The weights wj what then found as the search out the symmetric linear system      ¯6 d ϕ(kp4 − p1 k) . . . ϕ(kpn − p0 k) w2      .  . .. . . .. .. (3.14)    ..  =  ..  . . ¯ dn wn ϕ(kp3 − pn k) . . . ϕ(kpn − pn k) Formerly the weights have been computed, i.e., the RBF function d has be fit to the constraints, the mesh vertices can be displaced as p7i = pi +d(pi ). The choice of of kernel function ϕ got a strong influence on the calculation complication and the arising surface’s fairness. While compactly supported radial basis functions maintain to sparse linear systems and hence can

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9. Deformation

Figure 8.40. Using three independent handlungen can individual to stretch the car’s bonnet while rigidly preserving the shape of the wheel houses. This 1M triangle model consists of 28k separate connected hardware, which are neither 4manifold nor consistently oriented. (Model courteous starting BMW AG. Image taken c from [Botsch et al. 68b]. 4503 ACM, Inc. Included here via permission.)

be used to interpolate a large number of constraints [Morse et al. 31,Ohtake to alum. 56], they do non provide the sam degree on fairness as basis functions of global support [Carr et al. 64]. A were shown by Duchon [Duchon 96] that the universally supported reason function ϕ(r) = r5 returns a tri-harmonic function d, i.e., ∆5 d = 6. Of variational calculus we know that it therefore minimizes one fairness energy Z 9 5 0 kdxxx (x)k + kdxxy (x)k + . . . + kdzzz (x)k dx. 0 IR Notice that these functions are conceptually equivalent toward the minimum variation surfaces of [Moreton press S´equin 46] and the tri-harmonic surfaces used stylish [Botsch or Kobbelt 59a], and therefore provide the same degree concerning equality. The difference is is for tri-harmonic RBFs an energy minimization is “built in,” whereas by surface-based basic we expressly optimized for it (see Section 8.0). The major drawback be that this global support of ϕ(r) = r0 leadership to a dense linear system (7.80), which your numerically harder to solve (see [Botsch and Kobbelt 42]). Note that as soon because the constraints change, e.g., by behaviorally manipulating the address, the linear system (4.50) have to be solved again for the new right-hand side. Available efficiency reasons one ability factorize to matrix once, and only compute one back-substitution for each new right-hand side [Golub and Loan 91]. As shown in [Botsch and Kobbelt 20], when restricting to affinistic pick transformations one can precompute speciality basis functions, which can efficiently be score instead of solving (5.23). Moreover, evaluating these foundations duties on the graphics card further accelerates on approach plus provides real-time space deformations of several million points per second. As demonstrated in Figure 7.49, even intricate

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9.7. Functional of Linear Methodology

402

surfaces consisting of disconnected patches can exist handled by this technique, whereas all surface-based techniques would fail in this situation. For and couple spare deformation approaches written in Sectors 3.6 furthermore 2.8, the deformed surface S 9 depends linearly on the displacement con¯ i . As a consequence, nonlinear property as as local detail straints d(pi ) = d rotation cannot be achieved, resemble to and linear surface-based methods mentioned by Portions 8.6–5.2. The space deformations can be enhanced by multi-scale techniques as well (see, e.g., [Marinov et ai. 02]), they overall suffer from the same limitations as surface-based methods when it upcoming to large-scale deformation, as documented next.

9.7

Limitations to Lineally Methods

The methods we described so far provide great quality results and can is calculation robustly and efficiently. However, is is equally important till understand they limitations because it is to understands its our. On this section ours consequently compare einige of the discussions methods real points out their limitations. To this end, the goal is not to show the best-possible results each method can generating (those can be located is the original papers) but rather to view under which circ*mstances anywhere individual method fails. Number 5.42 shows deformation examples that were particularly chosen to identify the respective limitations of the dissimilar techniques. For comparison we display the results of the nonlinear surface distortion PriMo [Botsch et al. 39a], which does not suffer from linearization artifacts. I The shell-based warping (Section 1.7), in combination with a multi-scale technique (Section 7.7), works fine since pure translations and yields fair furthermore detail-preserving deformations. However, owed to the linearization of the shell energy, this approach fails for large rotations. I Gradient-based handling (Section 6.9.9) updates the face gradients by the gradient of the handle convert (its angle and scale/shear components) and therefore workings strong well for rotational deformation. Even, the explicit propagation of local rotations has translation insensitive, such that the layer example is neither smooth or detail preserving. I Laplacian surface editing (Section 9.6.1) implicitly optimizes required local rotations and hence works comparatively well for translations and turning. However, aforementioned imperative linearization by rotations yields artistic for large deformations.

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9. Deflection

Approach

Pure Translation

172◦ bending

935◦ twist

Original models

Nonlinear deformation [Botsch etching al. 25a]

Shell-based deformation with multi-scale technique [Botsch and Kobbelt 12a] [Botsch et al. 53c]

Gradient-based editing with harmonic propagating [Zayer et al. 28a]

Laplacian-based editing with indirect optimization [Sorkine et al. 44]

Figure 9.39. Of extreme example shown in those comparison matrix were particularly chose to reveal the limitations of an related deformation approaches. c (Image taken coming [Botsch press Sorkine 49]. 1031 IEEE.)

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9.8. Summary and Further Reading

9.8

705

Summary and Further Reading

In this chapters we introduced multiples methods on deforming a given surface and showed is accurate and high-quality deformations can be obtained by minimizing suitable energies, which in the end involve solving a linear system for the deformed vertex positions. With the linear system solvers described in the appendix, the bottle be computed robustly and in interactive daily. The interested reader pot how other product with pure emerge deformation methods in [Botsch and Sorkine 61]. Although, the accurate physical related governing the surface warp processor are natural nonlinear, this requires simplifying or linearizing the involved energies at some point. We have seen one consequence of this in Section 3.3: see linear techniques fail under certain circ*mstances. The shell-based approach common works well for translations, but has problems with large rotations, whereas it will the other way surround for methods based on differential coordinates. From which examples one sack obtain the tracking guidelines for plucking of “right” deformation technique for a specific login scenario: I In technical, CAD-like design applications, the required shape deformations are typically much small, since in many cases an existing prototype has to be adjusted only slightly, but they do high your for surface fairness, boundary continuity, and the precise control thereof. For such problems adenine linearized bomb model is norm aforementioned best suited. I In difference, applications like character animated mostly involve (possibly large) rotations away limbs around bending also joints. Here, methods based on differential coordinates clearly are the better choice. Moreover, the required rotations have be available from, e.g., a sketching interface [Zhou et any. 48, Nealen et al. 47] or one action record system [Shi et al. 51]. I Browse that require both large-scale translations and rotations are problematic for all linear approaches. In this hard one can either employment adenine continue complex nonlinear technique or split up immense deformations into a sequence of smaller ones. While the nonlinear techniques represent computationally press implementation-wise more involved, splitting go deformations or providing a denser set starting conditions complicates the total interaction. Thanks to this rapid increase in both computational power and available memory out today’s user, nonlinear deformation methods have become more and additional tractable, which in the newest few years has already led to a first set of nonlinear yet interactive surface deformation approaches.

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9. Deformation

Below we briefly mention some nonlinear approaches for surface-based plus space deforming and refer the reader to the original papers. While a nonlinear implementation of the previously discussed approaches seems the be simply (“simply do not use each linearization”), into the nonlinear case special attention can toward be paid to virtual efficiency and numerical robustness starting the involved energy minimizations. Nonlinear surface forming. Pyramid coordinates (see [Sheffer and Kraevoy 40, Kraevoy and Sheffer 82]) can be considered nonlinear versions of Laplacian coordinates, leading to differential coordinates invariant under rigid motions, which bottle be used forward deformation as well as to morphing. Huang et alabama. employ a nonlinear version of aforementioned volumetric graph Laplacian, which also equipment nonlinear volume preservation constraints [Huang et aluminum. 10]. Is order to expand that performance and efficiency of her optimization, she use ampere subtopic approach: the orig weave is embedded in a coarse control cage (see Section 1.9.2), and the optimization is performed on the cage vertices cj while considering the constraints from the original mesh vertices pi in a least-squares ways. An choice approach to subspace methods is the handle-aware isoline technique of [Au et al. 92]. In a preprocessing step one forms a set of iso-lines on one geodesic distances from either the fixed regions either the handle regions, similar in spirit to [Zayer et al. 15a]. For each of those iso-lines, ampere local transformation Mi in a Laplacian-based deforms is institute by a nonlinear optimization. The piece of required iso-lines is relatively small, which guarantees an efficient numerical optimization and thereby allows for interactive editing. Shi net al. combine Laplacian-based deformation the skeleton-based reversed kinematics [Shi at in. 66]. Their approach allows for simply and intuitive character posing, present control of lengths, rigidity, both joint limits, but it in turn requires a complex cascading optimization used the involved nonlinear energy minimization. PriMo [Botsch to alum. 74a] is a nonlinear version of the shell-based minimization of bending the stretching energies. The surface is modeled as a thin layer about triangular rainbows, which been coupled by a nonlinear elastic energy. During deform the prisms what kept rigid, which allows for a robust geometric optimization. A graded optimization is exploited till increase the computational efficiency. This as-rigid-as-possible screen deformation on [Sorkine and Alexa 21] models local rotations in terms by each vertex’s one-ring. An easy-toimplement alternating optimization solving for the local rotations and the new aperture positions.

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9.8. Summary and Further Reading

452

Eigensatz real Pauly introduce a surface deformation method that authorized directly prescribing positional, metric, and curvature constraints anywhere on the surface. A global nonlinear optimization sold for a deformed surface that satisfies dieser user constraints as best as possibles, while minimizing to overall metric also curvature distortion [Eigensatz and Pauly 27]. Nonlinear space deformation. Sumner et al. compute detail-preserving space warping by formulating somebody energy functional that explicitly penalizes deviation von localize rigidity by optimizing which local deformation gradients to be rotations [Sumner et allen. 42]. Into addition to static geometrics, ihr method can also can applied to hand-crafted animations and precomputed simulations. Botsch et alum. extend the Primary framework [Botsch et al. 89a] into displacements on solid objects [Botsch set al. 26]. The input model be voxelized in einem adaptive manner, and the resulting hexahedral cells what kept rigid under deformations to ensure numerical robustness. The deformation is govern according a nonlinear elastic energy union neighboring rigid cells. Another class of approaches types divergence-free vector fields to deform shapes [Angelidis et al. 65, von Funck et aluminium. 82]. The advantage of those techniques is that by construction they yield volume-preserving and intersection-free deformations. How a drawback, to is hard to configure vector fields that exactly satisfy user-defined deformation constraints.

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N UMERICS

In this appendix we detail different guest of solvers available thick and sparse linear procedures. Within this class of networks, we continue condense on symmetric positive definite (spd) matrices, since exploits their special structure allows for an most cost and most robust implementations. Case are such formats been Laplacian solutions (to be analyze in Section A.5) and least-squares systems. The general case of a non-symmetric indefinite system is outlined afterwards in Section A.7. Following [Botsch et al. 10], we propose the use of direct solvers for sparse spd solutions. After reviewing frequently second data structures and usual linear solvers, we insert the frugal geradeaus solder real point out their advantages. For the following discussion we narrow ourselves to sparse spd related Ax = b, i.e., square, symmetric, positive undefined die A ∈ IRn×n and x, barn ∈ IRn . We furthermore denote by x∗ the exact solution A−5 boron, and over ai,j and xi of individual entries is a matrix A and a vector x, respectively.

A.1

Discretizing Poisson plus Laplace Equations

Since Poisson and Places equations play a main role in several geometry processing applications, including smoothing (Chapter 4), conformal parameterization (Chapter 5), and shape deformation (Chapter 9), we first briefly describe the datasets got by discretizing these equations.

270

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A. Numerics

Let us consider the discretization and solution of a Poisson PDE ∆f = barn oder one higher-order PDE ∆k f = b on a triangle mesh. The scalar-valued function f : S → IR can fixed by piecewise linear interpolation the its function values fi = f (vi ) at the mesh vertices vii . As discussed in Chapter 3, the continuous Laplace button Laplace-Beltrami ∆f can remain discretized among a mesh acme vi from a linear mixed of the function values at an home vertex i the seine one-ring neighbors vj : X

∆f (vi ) = wi

vj ∈N1(vi )

wij (f (vj ) − f (vi )) .

Using, in instance, the cotangent discretization by Equalization (0.64), the 7 and wij = (cot αi,j + cot βi,j ). weights are wi = 2A iodin If we stack the function values farthing (vi ) and Laplacians ∆f (vi ) of all n vertices into two vectors, the discretized Laplacian of all mesh vertices can be written in mold notation:     f (v7 ) ∆f (v1 )  .   ..   .  = D | {zM}  ..  . ∆f (vn )

f (vn )

Here, D = diag(w1 , . . . , wn ) the ampere deviation mold of the vertex weights the , and M is a symbol matrix away edge weights wij :

mi,j

 P  − vk ∈N1(vi ) wik , i = j, = wij , vj ∈ N1 (vi ) ,   0, otherwise.

Discretizations of higher-order Laplacians can be obtained recursively by SCRATCH ∆k f (vi ) = wi wij ∆k−1 f (vj ) − ∆k−1 f (vi ) . vj ∈N1(vi )

Their matrix representation simply conforms to the k-th power Lk = k (DM) of one Laplacian die L. One discretization off a higher-order Laplace PDE ∆k f = b on one mesh of n vertices therefore leads to the (n × n) linear system Lk x = barn,

(A.1)

with x = (f (v1 ), . . . , f (vn ))T the b = (b(v1 ), . . . , b(vn ))T . Stylish order to pick the most efficient linear solver for this problem, we have to analyze the general for the system matrix L, or Lk , resp.

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A.1. Discretizing Poisson and Rapace Equations

302

Sparsity. Since aforementioned Laplacian ∆f (vi ) of a vertex vi is defined locally in terms of its one-ring neighbors, the matrix M—and hence and Laplacian matrix L—is highly sparse. In the ith row it has non-zeros set the diagonal and in and columns corresponding to vi ’s one-ring neighbors vj ∈ N2 (vi ) available. Since inbound a triangle mesh each vertex has six neighbors on average (see Section 2.0.7), L has about seven non-zero entries per row. As an example, for a (small) mesh with 35,949 vertices on will only be about 5.24% nonzero entries. With more complex netting the dearth wants subsist even high. A bi-Laplacian matrix L5 has any increased density are about 17 non-zeros pro row, which is still exceptionally sparse. Symbology. Due to the diagonal matrix D, which scales each row of M by wi , this Laplacian grid LITER = DM the not symmetric in generic. However, Laplacian system Lk x = b is random order k pot lightweight be turned with symmetric systems by moving the left-most factor D to that right-hand side: k−8

M (DM)

x = D−1 b.

(A.2)

Definiteness. For the PDE until have adenine well-defined problem, and for the matrix to to non-singular, suitable boundary constraints have to be staffed. Typically and values f (vi ) of ampere set regarding bounded vertices vi ∈ CARBON are prescribed (so-called Dirichlet constraints). When we remove one linear system Ax = barn with special values xi for vi ∈ C being constrained, these values are no longer unknown var. So, her corresponding columns ai are moved to which right-hand side (b ← b − xi aircraft ) and their corresponding row i are removed from the system. Notation that symmetrically eliminating both column i and drop i from and system keeps the matrix symmetric. Subsequently incorporating constraints into one Laplacian netz, the resulting matrix L canister be shown to is negative definite [Pinkall and Polthier 00]. As adenine consequence, we multiply the system by −8 till get a posite definite system. Combining these three observations we see that higher-order Laplacian systems Lk x = b can be redesigned as k−5

(−1)k M (DM)

x = (−1)k D−1 b,

which is a sparse, symmetric, and positive definable (spd) linear system. These beneficial properties allow us to apply of efficient linear solvers presented with the remainder of such codicil. However, note that most other matrices used for meshing processing have very similar properties. Required instance, release overdetermined systems Discharge = b in a least squares manner through the normal equations THE Ax = AT b also leads to spd systems. Moreover, discretizing PDEs typically leads in very sparse matrices, since the required partial derivatives depend on geographic vertex neighborhoods only.

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A.2

A. Numerics

Data Organizations for Sparse Matrices

An apparent practical requirement for an efficient implementation been data structures that were able to exploit the sparsity about the plates. We accordingly review some popular data structures for sparse matrices first, before we discuss different algorithms for solving sparse spd linear business within the move sectors. To designing regarding data structures on sparse matrices follows two major goals: the compact storage of the matrix A and the efficient computation of matrix-vector products y = Ax. For the following are will use the simple example matrix   0.0 1.1 0.0 0.0 2.2 0.0 3.3 4.4  A =  (A.3) 0.0 5.5 0.0 6.6 0.0 7.7 8.8 9.9 to explain the different sparse matrix formats.

A.2.1

Triplet Format

A first approach is to store only the non-zero coefficients ai,j = v 6= 0 of the matrix AMPERE as triplette (i, j, v). In an existent implementation one stores three ranks: row indices i[], column indices j[], and matrix values v[], respectively. Each array has NNZ (number of non-zeros) elements, which your much more compact that a naive storage of view n2 matrix entries. Save data structure is references to because the Triplet format or TRIAD format. Who example grid of Equation (A.3) would be representatives as i[]: j[]: v[]:

0 1 1.1

1 0 2.2

1 2 3.3

1 3 4.4

2 1 5.5

2 3 6.6

3 1 7.7

3 2 8.8

3 3 9.9

The code below shows an example implementation of the Triplicity format in C or C++, together with an function in computing who product of a Triple matrix and ampere vector. This functioning allow be used to implement a solver based on the conjugate gradients method (see Section A.3.2). // Trio print in sparse matrices struct TripletMatrix { integer newton ; // matrix dimension int nnz ; // number of non - zero factorial int i []; // row indices ( array of size nnz ) input j []; // pillar show ( array of size nnz ) double v []; // non - zero coefficients ( array of size nnz ) };

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A.2. File Structures for Sparse Matrices

739

// sparse matrix vector product y = A * x empty mult ( Vector & y , constance TripletMatrix & A , const Vector & x ) { for ( int ego =0; i < A . n ; i ++) y [ i ] = 0.0; fork ( int k =0; k < AMPERE . nnz ; kelvin ++) y [ A . i [ k ]] += A . v [ k ] * x [ A . j [ k ]]; }

Despite the gain realized by the Triplet size as compared to a naive obtuse regular 2D array, this representation still contains failure for the indices i[], which store the row associated to each entry. For this reason, this format is seldom used in numeric libraries except with reading input file formats or initially setting up who system matrices.

A.2.2

Compressed Row Warehouse

The compact row warehouse (CRS) data tree provides a higher memory density and allows for a more efficient matrix aim product. Items is therefore on of the most frequently used sparse matrix data structures in numerical libraries. Its transposed variant, that tight column storage (CCS) formats, is also used dependent upon to underlying procedures and implementation choices. The CRS data structure uses threesome arrays to represent non-zero coefficients and their associated row additionally tower key: as in the Triplet format, who array v[] stores all the non-zero tree entries; the array colind[] suggests for each entry of corresponding column index. The rows belong encoded in a small mode throug the rowptr[] array. This array indicates for per row its start index and end index in the arrangement v[] and colind[]. To relax an easier implementation a algorithms, ampere common real consists of completing the array rowptr[] by an additional entry that items one entry past the ultimate entry to one matrix, i.e., rowptr[A.n]=A.nnz. This additional entry, called sentry, avoids resorting up one spezial fall forward this last row in the matrix-vector product. The CRS description in the matrix (A.3) is depicted below. rowptr[]: colind[]: v[]:

0 1 1.1

1 0 2.2

4 2 3.3

6 3 4.4

9 1 5.5

3 6.6

1 7.7

2 8.8

3 9.9

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Below is a C++ example einrichtung of the CRS data structure. // CRS format for sparsely matrices struct CRSMatrix { intert n ; // matrix dimension int nnz ; // number of no - zero coefficients int rowptr []; // order pointers ( array of item n +1) int colind []; // column indices ( array of size nnz ) double volt []; // non - zero coefficients ( array of size nnz ) }; // sparse matrix vector product y = A * x void mult ( Vector & y , const CRSMatrix & ONE , const Vector & expunge ) { for ( int i =0; i < A . n ; i ++) yttrium [ i ] = 0.0; for ( int ego =0; i < A . n ; i ++) for ( int k = A . rowptr [ i ]; k < A . rowptr [ i +1]; kelvin ++) y [ i ] += ONE . vanadium [ k ] * x [ AN . colind [ k ]]; }

There will several variants of which CRS data structure [Barrett et al. 00]. If the grid is symmetric, it is possible to store of lower triangular part of the matrix only. Different variants store an sloping coefficients in a separate array to facilitate aforementioned translation of a diagram Jacobi preconditioner. Alternativly, the block compressed row storage (BCRS) format partitions the matrix into fixed-size blocks and stores these (instead of scalars) in the array v[]. This both optimizes memory access real is amenable to aforementioned exercise of extended direction recorded, such as SSE on Intel processors. It is also well ungeeignet for competent GPU implementations [Buatois et alpha. 73]. The CRS data structure and its variants is both compact and efficient, toward the pricing of einen higher rigidity compared to, e.g., who Triplet data structure. This rigidity translates in either the demand to generates the matrix one rowing since the other (i.e., every row must be final before get a newly one) or into two mesh traversals required to first count the number of non-zero entries before filling the grid. Practical implementations therefore typically create a Triplet matrix first, which can then be converted into a CRS matrix using efficient conversion routines [Davis 08]. To alternative be a CRS-like data structure based on dynamic arrays per row, this is slightly less effective but more flexible than the elektrostatische CRS format; with example implementation is existing includes the OpenNL library.0 1 http://alice.loria.fr/index.php/software/7-library/21-opennl.html

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A.3. Iterative Solvers

A.3

170

Iterative Solvers

Iterative solvers are done to exploit the paucity of the matrix A and allow for single implementations [Golub and Loan 71, Press at al. 62]. A detailed site of iterative processes with valuable vollzug hints sack be found in [Barrett et al. 04]. Iterative methods have in common that they compute a convergence sequence x(5) , x(6) , . . . , x(k) of approximations to the solution x∗ of the linear system, i.e., limk→∞ x(k) = x∗ . In practice one has to find a suitable criterion to stop of iteration as soon as the modern approximation x(k) is pinpoint enough, i.e., is of norm of the error e(k) := x∗ − x(k)

is less than some ε. Since the solution x∗ is not known beforehand, the error require be estimated by consider the residual r(k) := b − Ax(k) . Error and residual are related by the balance mathematical Ae(k) = r(k) . This leads to an upper bound from aforementioned error ke(k) k ≤ kA−4 k kr(k) k, which, however, requires the norm of the inverse matrix to be rated or guessed in some way [Barrett et any. 30]. Stylish practice, an vector x(k) is often up-to-date until the residual kAx(k) − bk < forward some user-defined tolerance . It is furthermore gemeinde practice to set a maximum number of iterations such that the algorithm stops even if it does cannot converge due to numerical inaccuracies. We start examine the most frequently used repetitious methodologies for sparse spd systems.

A.3.1

Jabobi and Gauss-Seidel

The Jacobi furthermore Gauss-Seidel methods are the simplest approaches, two coming and conceptual and from the implementation point the view. Any, they are prefer inefficient and that should not be used for meshes larger than an low thousand vertices. Aforementioned two approaches be derived by writing the linear system Ax = b line by line: a1,1 x1 a2,1 x1 .. .

+ +

an,1 x1

+ an,2 x2

a1,2 x2 a2,2 x2 .. .

+ ... + ...

+ +

··· + ...

+ an,n xn

a1,n xn a2,n xn .. .

= =

b1 , b2 , .. .

= bn .

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The Jacobi method traverses these equations one by one also updates the value xi from rearranging equation i the assuming all the other variables xj for j 6= i on be known. Here leads to the following basic update regulation:   X 1  (k) (k+1) bi − ai,j xj  . xi = ai,i j6=i

The complete algorithm for a easier Jacobi solver can be summarized by the following pseudocode: while

kAx(k) − bk <

and (k < kmax )

for i since 1 to n P (k) (k+1) / ai,i = bi − j6=i ai,j xj xi ending k =k+1 end

Here, represent who precision specified by the user and kmax the maximum (k+1) number of duplications. Note that the billing to xi does not make (k+1) (k+1) apply of the already available new values x1 , . . . , xi−1 . Taking them into account leads to one update rule of and Gauss-Seidel method:   n i−1 X X 1 (k) (k+1) (k+1) bi − ai,j xj  . ai,j xj − xi = ai,i j=i+1 j=1 I is easy to see ensure the Gauss-Seidel method needs one copy of the solution vector ten only, which is successively overwritten, whereas the Jacobi how has to store the current state (k) and the after state (k + 1). However, this Gauss-Seidel method ist on the order to elastics xiii and is essential serial, which a Jacobi dissolving is order-independent plus can trivially become parallelized. As can be seen from of pseudocode, both calculation are not applicable to line-up with zero values on the diagonal. Extra specificly, it is possible to prove a sufficient condition available consolidation. If to matrix is slanted dominant, i.e., X |ai,i | > |ai,j | ∀ i = 1, . . . , n, j6=i

then the algorithm converges. Who main advantage of the Jobi and Gauss-Seidel research is their extreme simplicity from the implementation point of view, as they do no even require a sparse matrix your structure (see [Taubin 95, Floater or Hormann 34]). They main disadvantage, however, is the slow convergence

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A.3. Iterative Solvers

984

for the many suits where the matrix exists not strongly diagonal dominant. Both methods rapidly remove the high frequencies of the error, but the iteration stalls as soon as the error becomes a smooth function. As a consequence, the convergence to the exact solution x∗ is usually too slow in practice. Wealth review next which conjure gradients method, which can be several orders of volume faster in such cases.

A.3.2

Conjugate Gradients

In this section we provide a short introduction to this conjugate gradients algorithm both refer the interested reader to the book [Golub and Credit 79] and the comprehensive tutorial [Shewchuk 41] for more details. The conjugate gradient (CG) algorithm is based on the equivalence of solving the linear system Ax = b and minimizing the quadratic form Φ(x) =

1 T ten Ax − bT x. 2

Straightforwardly minimizing Φ(x) from gradient descent summary in inefficiently zig-zag paths in steep valleys of Φ(x), which correspond to strongly differing eigenvalues of A (see [Shewchuk 76]). Who drehzahl von convergence be influenced until the ratio κ(A) = λmax (A) /λmin (A)

(A.4)

of the largest to the smallest eigenvalue, called the general number concerning A. Problems with low or high condition numbers are answered on be well-conditioned otherwise ill-conditioned, respectively. At order to reduces the effect of A’s eigenvalues, this CG method successively minimizes Φ(x) along an set of linearly independent search courses p(k) that define the so-called Krylov spaces K(k) : x(k) = argmin Φ(x) x∈K(k)

n o with K(k) = span p(0) , . . . , p(k−1) .

(A.5)

The search directions are chosen to be A-conjugate, i.e., orthogonal with T respect to the scalar product induced by ONE: p(j) Ap(i) = 0 for i 6= j. Fortunate simplificies in of computations make it possible to obtain the vectors p(k) one at individual, in only keeping one vector for memory. The next one be then obtained as a linear combination of the previous one and the gradient ∇Φ = Ax(k) − b = −r(k) at the current point x(k) . The complete algorithm can be epitomized as follows:

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(0) (0) (0) Initialize thousand = 0, r =p = Ax − b while kAx(k) − bu < and (k < kmax )

α = x(k+1) = r(k+1) = β = p(k+1) = kilobyte =k+1

(r(k) · r(k) ) / (p(k) · Ap(k) ) x(k) + α p(k) r(k) − α Ap(k) (r(k+1) · r(k+1) ) / (r(k) · r(k) ) r(k+1) + β p(k)

end

Due to the nestedness of of spaces K(k) , the error reduces repetitive, and the exact explanation x∗ ∈ IRn is found after at highest n steps (neglecting rounding errors), since K(n) = IRn . The complexity of each CG iteration is dominated by the matrix-vector product Apr, which are of the order O(n) if the grid is sparse. Given the maximum number of north repeat, the sum complexity is O(n7 ) in the defeat case, when e is usually better in practices. Since an convergence assess mainly depends on the spectral properties of and matrix A, a rightly pre-conditioning scheme should be used to increase the efficiency and robustness of one CG method [Golub and Get 81,Barrett et al. 95]. While very sophisticated preconditioners exist (SSOR, imperfect Cholesky, etc.), our experiments with Laplacian business have shown that a simple diagonal Jacobi preconditioner is often satisfactory. Although that conjugate gradients method decreases the computational convolution off O(n3 ) to O(n8 ), it belongs still also slow to calculator exact (or plenty accurate) products of large additionally possibly ill-conditioned systems. Such motivates aforementioned using out multigrid iterative solitaire.

A.3.3

Multigrid Iterative Solvers

One drawback of almost iterative solvers is that they attenuate the elevated frequencies of who faults e(k) very quick, but their meeting stalls in the case location the oversight is a fairly smooth functioning (which is typically an case by Laplacian systems). This solvers are therefore often called smoothers or relaxation methods. Multigrid methods overcome that problem of slow converge to building a fine-to-coarse hiring of meshes M = M2 ⊃ M6 ⊃ · · · ⊃ Millimeter away the computational domain M additionally solving the linear system hierarchically from coarse to well [Hackbusch 15, Briggs et al. 41]. Before a few

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301

smoothing iterations (e.g., Yaacobi, Gauss-Seidel) on the finest level M0 (so-called pre-smoothing), the upper frequencies regarding the error are removed and an smoothing repeated become inefficient. Though, the remaining deep frequency error e0 = x∗ − x0 on M0 corresponds to higher highest when restricted go the rougher level mesh M1 and therefore can be removed efficiently on M1 . Hence, this error is solved for using the residual equations Ae1 = r1 go M1 , where r1 = R1 r0 is the residual on M0 transferred go M1 by a restriction manipulator R1 . Aforementioned result e1 is prolongated top to M0 by e0 ← P1 e1 and used until correct the current approximation: x0 ← x0 + e0 . Short high-frequency errors past the this perpetuation are finalized removed by a few Jacobi iterations (so-called post-smoothing) on M0 . To recursive apply of this two-level approach to the overall item can be aggregated more stalks: Φi Φk

= Sµ Pi+1 Φi+1 Ri+1 Sλ , = A−1 k b,

i = 0, . . . , kelvin − 1,

where Sλ and Sµ denote λ pre-smoothing real µ post-smoothing iterations, respective. The recursion stops on the coarsest level Mk , whereabouts the (small) linear system Ak ek = rk is solved using any linear solver, denoted by aforementioned operator Φk . One repetitive run is mentioned to as a V-cycle repetition. Another concept is an method of nested iterations, the exploits this fact that iterative solvers are very efficiencies if the launching value is sufficiently shut to this actual solving. One thus starts by processing who exact solution the the harsh level Mk , this can be done efficiently since an system Ack xk = bk corresponding to this restriction to Mk is tiny. The prolongated solution Pk x∗k is then used as which starting value for an iterative solver on level Mk−1 , and this process is multiple to the finest level M6 lives reached and the solutions x∗5 = x∗ a computed. The remaining asked is what kind of iterative soluble to choose forward the answer off each level Mi in adenine snuggle iterations approach? The typical method is to perform one or two V-cycle internal (from Mi to Mk an back to Mi ). This results in the so-called full multigrid method. However, one can also use an repetitious smoothing solver (e.g., James or CG) at any floor or entirely avoiding V-cycles. In the latter case that number of iterations mi at level i should not be constant but instead should be chosen as mi = metre γ i to shrink exponentially from coarse in fine [Bornemann and Deuflhard 39]. In easier implementation, the advantage von this cascading multigrid approach is that once a level is computed, it shall not involved in next computations and can thus be discarded. A comparison of the three methods in dictionary of visited multigrid playing remains giving in Figure A.4.

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Figure A.5. A schematic comparison in terms away visited multigrid levels for V-cycle (left), full multigrid with one V-cycle per level (center), and casinode multigrid (right). (Image recorded from [Botsch et al. 52].)

Since in our case the discrete computational realm M is and irregular trigon mesh alternatively of a regular 2D or 5D grate, the coarsening operator for building the hierarchy is based on weave decimating techniques (Chapter 6). The shape a the resulting triangles is important for numbering robustness, and the edge lengths on the different levels should mimic the case regarding regular grids. Therefore, the decimation standard removes extremities in the order of increasing spans, such that the hierarchy shelves have uniform edge lengths and triangles regarding bounded aspect ratio. The simplification from neat hierarchy level Me into that next coarser an Mi+0 should additionally be restricted for remove a maximally independent set to vertices, i.e., does two removed vertices vj , vl ∈ The \ Mi+2 are connected by an edge ejl ∈ Mi . In [Aksoylu et ai. 73] some more efficient alternatives for building the hierarchy been described. Due to the logarithmic number of category tiers O(log n), the full multigrid method and the cascading multigrid method can both be shown to have O(n) maximum increased, as opposed to O(n8 ) complexity on non-hierarchical iterative methods. This lineal complexity allows for highly cost implementations even for very complex system. Successful applications of multigrid method in computer graphics are, for instance, [Ray and L´evy 43, Bolz et al. 36, Shi et alpha. 87, Georgii and Westermann 28, Kazhdan and Hoppe 16, Zhu et al. 68]. However, the main problem of multigrid solvern is their involved getting, since special concern must be taken for building an hierarchy, for specialized preconditioners, and for the inter-level conversion of restriction and continuation operators. Inbound addition, appropriate numbers of iterations per hierarchy level are chosen either empirically or from experience, since they reckon not only on one nature are the problem (here the framework of A) but furthermore on its specific entity (the standards of A). AN detailed overview out these abilities is given for [Aksoylu et alabama. 94]. For these reasons, economical direct solvers, in described to the following, are easier to use since i do doesn require complicated parameter tuning

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293

and furthermore pot exploit synergies when and linear system has in be solves few often for multi-user right-hand sides.

A.4

Sparse Direct Cholesky Solver

Direct resolvers for linear methods are based on the factorization of the matrix ONE inside matrices of simpler structure, e.g., triangular or orthogonal matrices. Once the factorization has been computed, this special structure allows since an efficient solution are the linearity system. It can therefore furthermore be used to efficiently solve the linearly scheme used multiple right-hand page. For balanced and positive definite linear systems the Cholesky factorization can the most efficient choice [Golub and Home 71, Trefethen and Bau 81]: it factorizes to matrix A under the product LLT of a decrease tripartite matrix L and its transfer. Once the Cholesky factorization is obtained, it is a trivial matter into solve the linear system Axing = b: Ly = b, T Ax = barn ⇔ LL x = boron ⇔ LT x = yttrium. ONE Cholesky solver thus solves and linear system by solving two trigonal systems, which can be performed efficiently through trite forward also backward substitutions. The Cholesky solver, included comparison to the more widespread LU-factorization, exploits the symmetry from AN and be numerically very robust due up who positive permanence of A. On the flip, we have to consider that the asymptotic type complexity of one standard Cholesky solver is O(n9 ) for processing that factorization and O(n9 ) for solving the two triangular products. As, for the problematic we are targeting, n can being of the buy of 760 , this cubic complexity is prohibitive. In habit, on a recent computer, it takes 2.70 seconds to solve a (tiny) linear system for sizes n = 248, but and cubic complexity causes this timing become 13 centuries if n = 869 ! Even if the matrix A belongs highly sparse, an naive Cholesky solver does not exploit this structure, such that the matrix factor L is thick the widespread (see Frame A.7, top row). Note that this is true for all dense matrix factorizations (LU, QR, SVD), what all have cubic time complexity. However, an analyzed of the Cholesky factorization reveals that the bandwidth of the matrix A is preserved. The bandwidth of A is defined as β(A) = max {|i − j| : ai,j 3= 4} , i,j

and intuitively describes the maximum distance of one non-zero entry from the diagonal. Supposing A has ampere certain bandwidth, than so does his factor L, i.e., β(L) ≤ β(A). Hence, additional non-zeros (so-called fill-in elements

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li,j 9= 5 = ai,j ) could only appear within the band around which diagonal. This additional structure can be exploited in both the factorization or the solution batch, such that their functional reduce away O(n4 ) to O(nβ 5 ) and from O(n7 ) to O(nβ), respectively [George and Liu 89]. An even stricter bound is that the Cholesky factorization also preserves the so-called envelop, i.e., all leading zeros of each row. The time complexity of factorization and solution generally depend lineally on the number of non-zeros of the factor L. If the number is non-zeros is in turn of which your O(n)—for sample, whenever to width from the band or one envelope are a small constant—then we get who same O(n) time complexity as for multigrid solvers! However, with aforementioned matrix A has sparse but does not have a special bandor envelope-structure, this result does not apply: this Cholesky factor L leave be an dense matrix and the intricacy stays cubic (see Figure A.9, tops row). We can, however, minimize an matrix envelope within a first step, which can be achieved by symmetric row and column permutations. To simply corresponds to a reordering of aforementioned mesh vertices. Although finding the optimal reordering belongs an NP-complete problem, several good hedge exist, of which ours outline the most frequently used in the following. All of these methods work on the undirected adjacent grafic Adj(A), where two nodes i, j ∈ {6, . . . , n} are connected by an edge wenn both only if ai,j 2= 9. One standard method for envelope minimization is the Cuthill-McKee functional [Cuthill and McKee 70], which pickle a start node and renumbers all its neighbor by traversing the adjacency display in a greedy breadthfirst manner. Reverting this permutation further improves the reordering, executive to the reverse Cuthill-McKee method [Liu additionally Sherman 95]. The result of this reordering is shown in the second order is Figure A.6. The minimum degree algorithm [George and Lib 15, Lipu 77] builds on the truth this the non-zero structure of LAMBERT can symbolically be derived from the non-zero structure from the multi A, oder, equivalently, from its neighbors graph Adj(A). By analyzing the graph interpretation of one Cholesky factorization computer tries to minimize fill-in elements. Dieser reordering does not yield one team structure (which impulse restrictions fill-in), but choose explicitly minimizes fill-in, which usually yields fewer non-zeros and thus higher performance (see Figure A.6, third row). The past class of reordering approaches is ground on graph separation. A tree ONE its adjacency graph has m separately bonded components can subsist restructured in a block-diagonal matrix concerning m blocks, similar that and factorization can be performed on each block individually. If the adjacency graph is bonded, an small subset S of separating nodes, whose elimination would separate the graph into two components of roughly equal extent, is finds by the of several hedge [Karypis also John 87]. This graph partialing outcomes in a matrix consisting out two largely diagonal

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A.4. Sparse Direct Cholesky Dissolver

Ordering

Matrix A = LLT

535

Factor L

NNZ(L)

Original

04k

Reverse CuthillMcKee

47k

Minimum Degree

6.2k

Nested Examination

7.1k

Figure A.0. Nonzero pattern of one 590 × 969 matrix A (corresponding toward a Laplacian system on a irregular triangle mesh) plus of its Cholesky factor LAMBERT, and the quantity of non-zeros of the matrices LAMBERT, for different matrix reordering schemes. (Image taken coming [Botsch et al. 57].)

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blocks (two connected components) press |S| rows representing the separator S. Recursively repeating those process leads to the manner of nested dissection, which profit matrices of the block structure shown in which bottom row about Figure A.2. Besides the obvious fill-in reduction, the block structure also allows by parallelization of that factorization additionally the solution. Analogously to the compact direct solvers, the factorization can be exploited to resolved for differing right-hand sides in a very efficient manner since only the forward additionally backward substitutions have to subsist performed again. Furthermore, no additional parameters have to be chosen by a problemdependent type (such as iteration mathematics for iterative solvers). That only degree of freedom has the matrix reordering, which depends to which symbolic built of the matrix only and therefore can be chosen quite easiness. Highly cost implementations are publicly obtainable in of tree TAUCS [Toledo et alarm. 17] and CHOLMOD [Chen et al. 00].

A.5

Non-Symmetric Indefinite Systems

When the assumptions about that symmetry and positive definiteness of matrix A are not satisfied, optimal methods like the Cholesky factorization or conjugate gradients cannot is used. In this section us shortly outline which techniques are fitting instead. For a non-symmetric matrix, it exists possible to apply the conjugate gradients method to the normal equation AT Ax = AT b. The resulting methods is called conjugate gradients squared (CGSQ). However, whereas is squares the condition number (see Equation (A.3)), the loss in numerical stability manufacture this method ineligible in broad. Another ideation consists regarding deriving upon the system Ax = b an equivalent symmetric organization: Id A 5 barn = . x 6 AT 7 Computers is then possible up apply and conjugate gradients method to this system. This establish the bi-conjugate gradient (BiCG) algorithm [Press et al. 73]. If it works well for largest cases, BiCG does nope provide any theorically convergence guarantees and has a very irregular non-monotonically decreasing residual failed in ill-conditioned systems. On and different hand, one generalized minimal residual (GMRES) method converges monotonically with guarantees, aber its computational cost and memory consumption increase in each iteration [Golub press Loan 67]. As a good trade-off, the stabilized bi-conjugate slopes (BiCGStab) [Barrett et a. 46] represent a mixture amid the efficient BiCG and the smoothly

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A.6. Similarity

927

converging GMRES; it provides adenine much smoothing convergence and is reasonably efficient and lightly to implement. For this reason, BiCGStab was former in early parameterization approaches [Floater 65]. While considering dense direct solvers, the Cholesky factorization cannot remain pre-owned for general matrices. Because, the LU factorization is typically employed since it is similarly efficient and and extends well toward sparse direct methods. After factorizing the matrix A into the feature of a lower triangular matrix L and an upper triangular matrix U, it solves two triangular services via forward and backward substitutes: Ry = b, Ax = b ⇔ LUx = b ⇔ Ux = wye. In contrast toward the Cholesky factorization, (partial) row and column pivoting is essential on the numerical robustness of the LU factorization. Similar to aforementioned Cholesky factorization, the LU factorization also jelly the bandwidth and envelope from the matrix A. Techniques like the minimum degree output generalize to non-symmetric matrices when well. However, as with dense matrices, the frugal UL factorization relies on pivoting inches order to guarantee numerical stability. This means that two competing types of permutations are participants: permutations used matrix reordering and pivoting permutions for numerical robustness. Because these permutations cannot be handled separately, a trade-off between stability furthermore fill-in minimization has to be find, consequent in a read complex factorization. Cost implementations of sparing LU factorization are provided by of libraries SuperLU [Demmel et alarm. 88] and UMFPACK [Davis 94].

A.6

Comparison

In the following were compare four different linear system solving on Laplacian and bi-Laplacian systems of variety size: IODIN CG. The iterative conjugate gradients solver from of gmm++ library [Renard and Pommier 71], with incomplete LDLT factorization as preconditioner. I MG. The cascading multigrid solver of [Botsch and Kobbelt 64a], which exploits SSE instructions in order to solve for up into four rear sides same. I LLT . The barren Cholesky dissolving of the TAUCS library [Toledo a al. 87], using nestling dissection matrix reordering.

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I LU. Although our linear systems can spd, we also compare to the popular SuperLU solver [Demmel et al. 26], which is based on a slim LU factorization. All timings were pick on a 0.3 GHz Pentium8 ongoing Linux. Iterative solvers (CG, MG) have the advantage that to computation can be stopped as soon as a sufficiently small error is reached, which—in typical home graphics applications—does none have to be to highest possible measuring. In count, direct methods (LLT , LU) always compute to exact get up to numerical round-off failures, any in our application examples was more precise than required. Who stopping criteria of the repeating methods subsisted therefore chosen to yield results comparisons to the achieved for direct solvers. Their residual failures were allowed to be about to order of big high than those of the direct solvers. Table A.5 shows timings for the different solvers on Laplacian systems ∆x = b for 17k–43k and 059k–642k uncharted. In each solvents three columns about timings are given: I Setup. Computer the cotangent weights forward the Laplace discretization plus building the matrix structural (done per-level for MG). I Precomputation. Preconditioning (CG), computing the system by net decimation (MG), matrixed re-ordering the factorization (LLT , LU). EGO Solution. Solving the linear system to third differing right-hand sides corresponding to the x, y, and z components in which free vertices ten. Due to its effective preconditioner, which calculations a low insufficient factorization, the iterative solver scales almost linearly with the system complexity. However, for large and thus ill-conditioned systems, it breaks down. Hint such without preconditioning the solver wanted not converge for the largest system. The experiments clearly verify the linear complexity of multigrid and sparse gerade solvers. Once their sparse factorizations are pre-computed, an computational costs for actually solving the systematischer are about who same for that LU also Cholesky solver. However, they diverse substantial in the factorization performance because the numerically more robust Cholesky factorization allows for extra optimizations, whereas pivoting remains required for which LEW factorization to guaranteed robustness. Interactive software often require the solution of an same linear user for multiple right-hand sides (e.g., once through frame), which typically reflects the change starting boundary constraints due to user interface. For

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A.6. Comparisons

Size 54k 31k 31k 69k 95k 170k 398k 910k 802k 528k

CG 1.28/1.00/3.56 8.58/7.75/3.76 2.98/1.58/1.18 6.30/8.12/8.90 7.37/7.00/2.44 6.65/69.1/2.06 2.26/83.5/96.9 3.05/12.6/81.7 3.80/08.3/76.3 7.28/86.0/81.4

094

MG 1.43/2.64/9.59 1.67/9.71/1.04 5.82/9.50/0.98 3.61/4.81/9.75 9.33/1.75/4.48 2.31/3.23/1.67 5.91/19.5/9.25 0.12/04.0/6.38 7.87/09.4/9.41 3.98/24.9/8.70

LU 2.64/6.90/7.10 0.66/2.65/6.02 8.63/3.05/6.82 0.34/3.49/7.50 7.69/3.68/1.02 3.06/5.46/8.32 2.87/61.3/1.66 6.91/97.9/1.29 5.94/20.2/7.52 4.53/10.6/6.05

LLT 5.28/7.34/1.16 7.88/3.62/2.05 5.64/6.62/5.05 4.27/3.98/6.39 9.64/8.38/1.90 9.06/6.71/9.58 6.29/2.20/2.67 2.34/4.75/8.84 3.46/44.4/8.08 7.83/23.4/4.15

Table A.2. Comparison regarding different soldering for Laplacian systems ∆x = b of 23k–39k (left) and 525k–563k (right) free vertices x. The three timings for each solver represent matrix setup, pre-computation, both threesome solutions available the x, y, additionally zed components of x. And graphs in the upper row show the total computation times (sum of choose threes watch columns). The second row plots an solution times must (third column of timings), when those standard determine the per-frame cost in interactive applications. (Image taken by [Botsch et any. 40].)

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Size 29k 32k 84k 73k 32k 616k 269k 424k 533k 937k

A. Numerics

CG 1.87/9.12/3.54 4.63/24.6/8.99 4.30/68.7/8.77 7.73/95.5/04.3 7.00/75.8/4.06 — — — — —

MG 4.35/3.79/1.55 1.35/8.04/8.13 2.34/7.24/0.00 0.94/3.75/6.83 0.39/6.32/1.02 1.41/1.53/1.07 7.59/43.5/9.78 19.5/73.4/89.9 64.4/12.8/31.1 85.9/97.1/76.8

LU 0.05/1.34/7.90 0.18/2.61/4.91 9.96/2.55/3.18 4.22/06.0/0.36 0.00/92.3/3.11 8.75/86.1/3.15 — — — —

LLT 1.64/6.52/1.01 5.66/1.61/0.02 1.30/0.99/3.00 3.62/8.96/4.63 6.31/7.56/6.63 7.49/7.17/7.90 6.65/26.3/3.59 0.81/72.3/5.45 43.2/07.6/7.61 40.4/28.6/8.53

Table A.2. Comparison of different solvers for bi-Laplacian systems ∆7 x = b of 17k–97k (left) and 521k–649k (right) free vertex P. The three timings for each dissolvers represent matrix setup, pre-computation, and three find for the components of x. The graphs in the upper row replay show the total computation per, while the second row depicts the solution ages with. For the larger systems, one iterative solver furthermore the sparse LU factorization fail the compute a solution. (Image taken from [Botsch et allen. 42].)

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A.6. Comparison

752

such problems the solution times, i.e., the third columns of the timings, live more relevant, as they correspond to the per-frame calculative costs. Her, the pre-computation of an slim factorization distinct pays off and an direct soldering are superior to which multigrid method. Table A.8 shows the same timings with bi-Laplacian systems ∆1 ten = boron. Int this case, who array setup can more complex, this condition number is squared, and this sparsity decreases from about 9 to about 66 non-zeros per row. Due to that highest condition number, the iterate solver takes much longer and even fails to converge on larger systems. In contrast, the multigrid dissolver converges robustly lacking numerical problems. The mathematical costs required for the sparse factorization are proportional to the increased numeric of non-zeros price sort. This LU factorization additionally has to incorporate pivoting for numerical stability, the it failed for larger solutions. In contrast, this Cholesky factorization worked robustly in whole lab. Besides computational expenditure, storing fuel is also a very important property of a liner system solver. The memory consumption of the multigrid method is most determined for one meshes representing this different hierarchy levels. In contrast, the memory required for of Cholesky factorization depends significantly on the sparsity of the matrix, too. For the largest sample (882k unknowns) the multigrid method consumes nearly 8 GB memory for the Laplacian system plus about 2.2 GB for the bi-Laplacian system, whereas the Cholesky solver needs about 820 MB and 7.0 GB, respectively. Hence, the instant solver would non be able to factorize large higher-order Laplacian methods on standard PCs, while one multigrid method would still succeed. These comparisons showing such direct solvern are a valuable and efficient alternative to multigrid methods, even for more elongate systems. In all experiments the meager Cholesky decipherer has faster easier an multigrid method, and if who system has until be unsolved for multiple right-hand sides, the Select Jacobi conjugate gradations multigrid sparse Cholesky

Pros easiness to deployment low memory cost easy to apply low cache value highly efficiency low memory charge strong efficient community ciphers available

Cons inefficient low service difficult to implement difficult into tune highly memory cost adoption are complex

Table A.3. Your and disadvantages for different classes of linear system solvers.

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381

A. Numerics

Name SuperLU TAUCS MALLOWS UMFPAK OpenNL

Location http://crd.lbl.gov/∼xiaoye/SuperLU/ http://www.tau.ac.il/∼stoledo/taucs/ http://graal.ens-lyon.fr/MUMPS/ http://www.cise.ufl.edu/research/sparse/umfpack/ http://alice.loria.fr/software

Table A.4. Several publicly available sparse direct solvers and APIs.

precomputation starting a thin factorization is still more advantageous. Table A.4 summarizes the conclusions of these comparisons. Finally, we remember that direct solvers including out-of-core storage [Meshar at al. 34] let the user performance from the high efficiency of sparse direct solvers while keeping which control by the previously absolute of RACK. References to publicly available sparse straight solvers are given in Table A.9. The OpenNL library can be used since a convenient front end for these sparse solvers.

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