Polygon Mesh Processing.pdf

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Contents

Preface

1. Surface Representations

1.1 Surface Definition and Properties

1.2 Approximation Power

1.3 Parametric Surface Representations

1.4 Implicit Surface Representations

1.5 Conversion Methods

1.6 Summary and Further Reading

2. Mesh Data Structures

2.1 Face-Based Data Structures

2.2 Edge-Based Data Structures

2.3 Halfedge-Based Data Structure

2.4 Directed-Edge Data Structure

2.5 Summary and Further Reading

3. Differential Geometry

3.1 Curves

3.2 Surfaces

3.3 Discrete Differential Operators

3.4 Summary and Further Reading

4. Smoothing

4.1 Fourier Transform and Manifold Harmonics

4.2 Diffusion Flow

4.3 Fairing

4.4 Summary and Further Reading

5. Parameterization

5.1 General Goals

5.2 Parameterization of a Triangulated Surface

5.3 Barycentric Mapping

5.4 Conformal Mapping

5.5 Methods Based on Distortion Analysis

5.6 Summary and Further Reading

6. Remeshing

6.1 Local Structure

6.2 Global Structure

6.3 Correspondences

6.4 Voronoi Diagrams and Delaunay Triangulations

6.5 Triangle-Based Remeshing

6.6 Quad-dominant Remeshing

6.7 Summary and Further Reading

7. Simplification & Approximation

7.1 Vertex Clustering

7.2 Incremental Decimation

7.3 Shape Approximation

7.4 Out-of-Core Methods

7.5 Summary and Further Reading

8. Model Repair

8.1 Types of Artifacts: The "Freak Show"

8.2 Types of Repair Algorithms

8.3 Types of Input

8.4 Surface-Oriented Algorithms

8.5 Volumetric Repair Algorithms

8.6 Summary and Further Reading

9. Deformation

9.1 Transformation Propagation

9.2 Shell-Based Deformation

9.3 Multi-Scale Deformation

9.4 Differential Coordinates

9.5 Freeform Deformation

9.6 Radial Basis Functions

9.7 Limitations of Linear Methods

9.8 Summary and Further Reading

A. Numerics

A.1 Discretizing Poisson and Laplace Equations

A.2 Data Structures for Sparse Matrices

A.3 Iterative Solvers

A.4 Sparse Direct Cholesky Solver

A.5 Non-Symmetric Indefinite Systems

A.6 Comparison

Bibliography

Index

Polygon Mesh Processing

Polygon Mesh Processing Mario Botsch Leif Kobbelt Mark Pauly Pierre Alliez Bruno L´evy A K Peters, Ltd. Natick, Massachusetts

Editorial, Sales, and Customer Service Oﬃce A K Peters, Ltd. 5 Commonwealth Road, Suite 2C Natick, MA 01760-1526 www.akpeters.com Copyright c 2010 by A K Peters, Ltd. All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form, electronic or mechani- cal, including photocopying, recording, or by any information storage and retrieval system, without written permission from the copyright owner. Library of Congress Cataloging-in-Publication Data Polygon mesh processing / Mario Botsch ... [et al.]. p. cm. Includes bibliographical references and index. ISBN 978-1-56881-426-1 (alk. paper) 1. Geometry–Data processing. 2. Mathematical models. 3. Computer graphics. 4. Polygons. I. Botsch, Mario. QA447.P62 2010 516.20285–dc22 2009024954 Printed in India 14 13 12 11 10 10 9 8 7 6 5 4 3 2 1

CONTENTS Preface 1 Surface Representations 1.1 Surface Deﬁnition and Properties . . . . . . . . . . . . . . 1.2 Approximation Power . . . . . . . . . . . . . . . . . . . . 1.3 Parametric Surface Representations . . . . . . . . . . . . . 1.4 . . . . . . . . . . . . . . 1.5 Conversion Methods . . . . . . . . . . . . . . . . . . . . . 1.6 Summary and Further Reading . . . . . . . . . . . . . . . Implicit Surface Representations 2 Mesh Data Structures 2.1 Face-Based Data Structures . . . . . . . . . . . . . . . . . 2.2 Edge-Based Data Structures . . . . . . . . . . . . . . . . . 2.3 Halfedge-Based Data Structure . . . . . . . . . . . . . . . 2.4 Directed-Edge Data Structure . . . . . . . . . . . . . . . . 2.5 Summary and Further Reading . . . . . . . . . . . . . . . 3 Differential Geometry 3.1 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Discrete Diﬀerential Operators . . . . . . . . . . . . . . . 3.4 Summary and Further Reading . . . . . . . . . . . . . . . ix 1 3 5 7 13 15 20 21 22 24 25 27 28 29 29 31 40 48 v

vi 4 Smoothing 4.1 Fourier Transform and Manifold Harmonics . . . . . . . . 4.2 Diﬀusion Flow . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Fairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Summary and Further Reading . . . . . . . . . . . . . . . 5 Parameterization 5.1 General Goals . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Parameterization of a Triangulated Surface . . . . . . . . 5.3 Barycentric Mapping . . . . . . . . . . . . . . . . . . . . . 5.4 Conformal Mapping . . . . . . . . . . . . . . . . . . . . . 5.5 Methods Based on Distortion Analysis . . . . . . . . . . . 5.6 Summary and Further Reading . . . . . . . . . . . . . . . 6 Remeshing 6.1 Local Structure . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Global Structure . . . . . . . . . . . . . . . . . . . . . . . 6.3 Correspondences . . . . . . . . . . . . . . . . . . . . . . . 6.4 Voronoi Diagrams and Delaunay Triangulations . . . . . . 6.5 Triangle-Based Remeshing . . . . . . . . . . . . . . . . . . 6.6 Quad-dominant Remeshing . . . . . . . . . . . . . . . . . 6.7 Summary and Further Reading . . . . . . . . . . . . . . . 7 Simpliﬁcation & Approximation 7.1 Vertex Clustering . . . . . . . . . . . . . . . . . . . . . . . 7.2 Incremental Decimation . . . . . . . . . . . . . . . . . . . 7.3 Shape Approximation . . . . . . . . . . . . . . . . . . . . 7.4 Out-of-Core Methods . . . . . . . . . . . . . . . . . . . . . 7.5 Summary and Further Reading . . . . . . . . . . . . . . . 8 Model Repair 8.1 Types of Artifacts: The “Freak Show” . . . . . . . . . . . 8.2 Types of Repair Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Types of Input 8.4 Surface-Oriented Algorithms . . . . . . . . . . . . . . . . 8.5 Volumetric Repair Algorithms . . . . . . . . . . . . . . . . 8.6 Summary and Further Reading . . . . . . . . . . . . . . . 9 Deformation 9.1 Transformation Propagation . . . . . . . . . . . . . . . . . 9.2 Shell-Based Deformation . . . . . . . . . . . . . . . . . . . 9.3 Multi-Scale Deformation . . . . . . . . . . . . . . . . . . . 9.4 Diﬀerential Coordinates . . . . . . . . . . . . . . . . . . . Contents 49 50 54 57 61 63 64 66 67 71 78 82 85 86 87 89 89 92 104 110 111 113 115 122 127 130 131 132 132 135 139 144 150 151 153 155 157 164

Contents 9.5 Freeform Deformation . . . . . . . . . . . . . . . . . . . . 9.6 Radial Basis Functions . . . . . . . . . . . . . . . . . . . . 9.7 Limitations of Linear Methods . . . . . . . . . . . . . . . 9.8 Summary and Further Reading . . . . . . . . . . . . . . . A Numerics A.1 Discretizing Poisson and Laplace Equations . . . . . . . . A.2 Data Structures for Sparse Matrices . . . . . . . . . . . . A.3 Iterative Solvers . . . . . . . . . . . . . . . . . . . . . . . A.4 Sparse Direct Cholesky Solver . . . . . . . . . . . . . . . . A.5 Non-Symmetric Indeﬁnite Systems . . . . . . . . . . . . . A.6 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography Index vii 169 173 175 177 181 181 184 187 193 196 197 203 226

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