Diffusion-driven Wavelet Design for Shape Analysis

220 Pages·2014·3.472 MB·English

by  Hou, Tingbo, Qin, Hong

DIFFUSION-DRIVEN
WAVELET DESIGN
SHAPE ANALYSIS
for
TINGBO HOU • HONG QIN
TINGBO HOU • HONG QIN
Diffusion-Driven Wavelet Design for Shape Analysis presents recent research results in
wavelet designs on 3D shapes and their applications in shape analysis. It explains how DIFFUSION-DRIVEN
to apply the design methods to various types of 3D data, such as polygonal meshes, point
clouds, manifolds, and volumetric images.

WAVELET DESIGN
The first part of the book introduces design methods of wavelets on manifold data,
incorporating interdisciplinary knowledge from differential geometry, functional analysis,

Fourier transform, spectral graph theory, and stochastic processes. The authors show
SHAPE ANALYSIS
how wavelets are purely determined by the shape geometry and how wavelet transforms for
are computed as inner products of wavelet kernels and input functions.
The second part presents applications in shape analysis/representation. The book
looks at wavelets as spectral tools for geometry processing with filters in a joint space-
frequency domain and examines wavelets as detail extractors for shape feature defini-
tion and detection. Going beyond these fundamental applications, the book also covers
middle- and high-level applications, including shape matching, shape registration, and
shape retrieval.
Unlike many other wavelet books, this one does not involve complicated mathematics.
Instead, the book uses simplified formulations and illustrative examples to explain deep
theories.
Features
• Collects the latest research results in wavelet design and applications, including the
space-frequency analysis of wavelet transforms on manifold geometry
• Describes design methods based on the heat diffusion theory
• Explains profound concepts through clear language and simple examples
• Presents the necessary implementation details and application algorithms
• Explores emerging research directions that take you to the frontier of the field
• Provides figures, videos, and sample codes on a supplementary website, enabling
you to implement the algorithms
K22193
Computer Game Development AN A K PETERS BOOK
Diffusion-Driven
Wavelet design

for Shape analySiS
Tingbo hou • Hong Qin
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Diffusion-Driven
Wavelet design

Shape analySiS
for
Tingbo hou • Hong Qin
Boca Raton London New York
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Version Date: 20140908
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To my family.
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Contents
List of Tables xv
List of Figures xvii
List of Algorithms xxiii
1 Introduction 1
1.1 Wavelets on 3D Shapes . . . . . . . . . . . . . . . . . . . . 1
1.2 Book Contents . . . . . . . . . . . . . . . . . . . . . . . . . 3
I Theories 7
2 Wavelet Theory 9
2.1 Classical Wavelet . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Subdivision Wavelet . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Diffusion Wavelet . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Spectral Graph Wavelet . . . . . . . . . . . . . . . . . . . . 19
3 Heat Diffusion Theory 25
3.1 Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Heat Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Applications in Shape Analysis . . . . . . . . . . . . . . . . 34
4 Admissible Diffusion Wavelets 39
4.1 Diffusion Operator . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Wavelet Construction . . . . . . . . . . . . . . . . . . . . . 41
4.3 Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . 44
4.4 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.5 Space-Frequency Processing Framework . . . . . . . . . . . 50
vii
viii CONTENTS
5 Mexican Hat Wavelet 53
5.1 Manifold Harmonics . . . . . . . . . . . . . . . . . . . . . . 53
5.2 Bivariate Kernels and Convolutions . . . . . . . . . . . . . . 55
5.3 Mexican Hat Wavelet . . . . . . . . . . . . . . . . . . . . . 57
5.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.5 Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . 65
6 Anisotropic Wavelet 69
6.1 Normal-ControlledCoordinates . . . . . . . . . . . . . . . . 69
6.2 Anisotropic Heat Kernel . . . . . . . . . . . . . . . . . . . . 73
6.3 Anisotropic Diffusion . . . . . . . . . . . . . . . . . . . . . . 75
6.4 Anisotropic Wavelet . . . . . . . . . . . . . . . . . . . . . . 77
7 Wavelet Generation 81
7.1 Volume Wavelets . . . . . . . . . . . . . . . . . . . . . . . . 81
7.2 Manifold Wavelet Generalization . . . . . . . . . . . . . . . 86
II Applications 91
8 Implementation 93
8.1 Discrete Laplace-Beltrami Operator . . . . . . . . . . . . . 93
8.2 Generalized Eigenvalue Problem . . . . . . . . . . . . . . . 99
8.3 Matrix Power . . . . . . . . . . . . . . . . . . . . . . . . . . 106
9 Shape Representation 111
9.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . 111
9.2 Heat Kernel Signature . . . . . . . . . . . . . . . . . . . . . 114
9.3 Wave Kernel Signature . . . . . . . . . . . . . . . . . . . . . 118
9.4 Wavelet Signature . . . . . . . . . . . . . . . . . . . . . . . 120
10 Geometry Processing 125
10.1 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . 125
10.2 Admissible Diffusion Wavelets . . . . . . . . . . . . . . . . . 128
10.3 Mexican Hat Wavelet . . . . . . . . . . . . . . . . . . . . . 131
11 Feature Definition and Detection 139
11.1 Saliency Visualization . . . . . . . . . . . . . . . . . . . . . 139
11.2 Feature Definition . . . . . . . . . . . . . . . . . . . . . . . 142
11.3 Feature Detection. . . . . . . . . . . . . . . . . . . . . . . . 146
CONTENTS ix
12 Shape Matching, Registration, and Retrieval 155
12.1 Shape Matching. . . . . . . . . . . . . . . . . . . . . . . . . 155
12.2 Shape Registration . . . . . . . . . . . . . . . . . . . . . . . 162
12.3 Shape Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . 170
Bibliography 181
Index 195