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9780387733005

Numerical Geometry of Non-rigid Shapes

by ; ;
  • ISBN13:

    9780387733005

  • ISBN10:

    0387733000

  • Format: Hardcover
  • Copyright: 2008-10-14
  • Publisher: Springer-Nature New York Inc
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Summary

Deformable objects are ubiquitous in the world surrounding us, on all levels from micro to macro. The need to study such shapes and model their behavior arises in a wide spectrum of applications, ranging from medicine to security. In recent years, non-rigid shapes have attracted growing interest, which has led to rapid development of the field, where state-of-the-art results from very different sciences - theoretical and numerical geometry, optimization, linear algebra, graph theory, machine learning and computer graphics, to mention several - are applied to find solutions.This book gives an overview of the current state of science in analysis and synthesis of non-rigid shapes. Everyday examples are used to explain concepts and to illustrate different techniques. The presentation unfolds systematically and numerous figures enrich the engaging exposition. Practice problems follow at the end of each chapter, with detailed solutions to selected problems in the appendix. A gallery of colored images enhances the text.This book will be of interest to graduate students, researchers and professionals in different fields of mathematics, computer science and engineering. It may be used for courses in computer vision, numerical geometry and geometric modeling and computer graphics or for self-study.

Table of Contents

Forewordp. VII
About the Authorsp. IX
Prefacep. XI
Introductionp. 1
Similarity of non-rigid shapesp. 3
Correspondence problemsp. 6
A landscape of problemsp. 7
Notesp. 9
A Taste of Geometryp. 11
Basic terms in metric geometry and topologyp. 11
Isometriesp. 13
Length spacesp. 17
Manifoldsp. 20
Embedded surfacesp. 21
Curvature and the second fundamental formp. 26
Intrinsic view on geometry of surfacesp. 29
Bending and rigidityp. 31
Intrinsic invariantsp. 34
Suggested readingp. 37
Problemsp. 37
Notesp. 39
Discrete Geometryp. 41
Point clouds and samplingp. 41
Farthest point samplingp. 43
Voronoi tessellationp. 46
Centroidal Voronoi sampling and the Lloyd-Max algorithmp. 48
Connectivityp. 52
Delaunay tessellationp. 53
Triangular meshesp. 54
Local feature size and curvature-dependent samplingp. 57
Approximation qualityp. 61
Suggested readingp. 63
Softwarep. 63
Problemsp. 63
Notesp. 64
Shortest Paths and Fast Marching Methodsp. 67
The shortest path problemp. 67
Dijkstra's shortest path algorithmp. 69
Fast marching methodsp. 71
Fast marching on parametric surfacesp. 81
Marching even fasterp. 83
Parallel distance computationp. 85
Minimal geodesicsp. 87
Suggested readingp. 89
Softwarep. 90
Problemsp. 90
Notesp. 91
Numerical Optimizationp. 93
Local versus global optimizationp. 93
Optimality conditionsp. 94
Unconstrained optimization algorithmsp. 97
The quest for a descent directionp. 100
Preconditioningp. 104
Let Newton be!p. 105
Truncated Newtonp. 106
Quasi-Newton algorithmsp. 107
Non-convex optimizationp. 108
Constrained optimizationp. 110
Penalty and barrier methodsp. 112
Augmented Lagrangian methodp. 114
Suggested readingp. 116
Softwarep. 116
Problemsp. 116
Notesp. 118
In the Rigid Kingdomp. 119
Moments of joy, moments of sorrowp. 120
Iterative closest point algorithmsp. 125
Enter numerical optimizationp. 128
Rigid correspondencep. 131
Suggested readingp. 133
Softwarep. 133
Problemsp. 133
Notesp. 134
Multidimensional Scalingp. 137
Isometric embedding problemp. 138
Multidimensional scalingp. 142
SMACOF algorithmp. 143
Second-order methodsp. 146
Variations on the stress themep. 148
Multiresolution methodsp. 153
Multigrid MDSp. 156
Vector extrapolationp. 160
A trouble with topologyp. 164
Suggested readingp. 165
Softwarep. 166
Problemsp. 167
Notesp. 167
Spectral Embeddingp. 169
Classic MDSp. 170
Local methodsp. 173
The Laplace-Beltrami operatorp. 176
To hear the shape of the drump. 178
Discrete Laplace-Beltrami operatorp. 180
Suggested readingp. 184
Softwarep. 184
Problemsp. 184
Notesp. 185
Non-Euclidean Embeddingp. 187
Spherical embeddingp. 187
Generalized multidimensional scalingp. 192
Representation issuesp. 194
Geodesic distance computationp. 197
Minimization of the generalized stressp. 198
Multiresolution encorep. 202
Suggested readingp. 203
Softwarep. 203
Problemsp. 204
Notesp. 204
Isometry-Invariant Similarityp. 205
Equivalence, similarity, and distancep. 205
Embedding distancep. 207
Gromov-Hausdorff distancep. 208
Intrinsic symmetryp. 211
Suggested readingp. 214
Problemsp. 214
Partial Similarityp. 217
Recognition by partsp. 218
Paretian approach to partial similarityp. 221
Scalar partial similarityp. 224
Fuzzy approximationp. 226
Extrinsic partial similarityp. 229
Intrinsic partial similarityp. 230
Not only size mattersp. 232
Suggested readingp. 236
Problemsp. 236
Notesp. 238
Non-rigid Correspondence and Calculus of Shapesp. 239
Intrinsic parameterizationp. 240
An image processing approachp. 241
Minimum distortion correspondencep. 244
Texture mapping and transferp. 246
Morphingp. 249
Guaranteed self-intersection free morphp. 254
Calculus of shapesp. 255
Suggested readingp. 258
Softwarep. 258
Problemsp. 259
Notesp. 259
Three-dimensional Face Recognitionp. 261
Some terminologyp. 263
A retrospectivep. 264
Isometric model of facial expressionsp. 268
Expression-invariant face recognitionp. 269
Comparison of photometric propertiesp. 273
Suggested readingp. 275
Notesp. 275
Epiloguep. 277
Solutions of Selected Problemsp. 279
Softwarep. 293
Notationp. 297
Acronymsp. 299
Glossaryp. 301
Referencesp. 307
Subject Indexp. 327
Author Indexp. 335
Table of Contents provided by Ingram. All Rights Reserved.

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