Curve and Surface Reconstruction

Algorithms with Mathematical Analysis

By: Tamal K. Dey

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Hardcover | 8 January 2007

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Hardcover
232 Pages

Dimensions(cm)
22.86 x 15.24 x 1.75

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Many applications in science and engineering require a digital model of a real physical object. Advanced scanning technology has made it possible to scan such objects and generate point samples on their boundaries. This book, first published in 2007, shows how to compute a digital model from this point sample. After developing the basics of sampling theory and its connections to various geometric and topological properties, the author describes a suite of algorithms that have been designed for the reconstruction problem, including algorithms for surface reconstruction from dense samples, from samples that are not adequately dense and from noisy samples. Voronoi- and Delaunay-based techniques, implicit surface-based methods and Morse theory-based methods are covered. Scientists and engineers working in drug design, medical imaging, CAD, GIS, and many other areas will benefit from this first book on the subject.

Industry Reviews

'The text is well written, and present the algorithms in a way that makes them quite understandable. Instead of presenting the algorithms as single, monolithic and complex methods, they are broken into parts that can be explained and mathematically analyzed in an order that makes clear how they are later composed into the larger task.' SIGACT News

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Preface	p. xi
Basics	p. 1
Shapes	p. 2
Spaces and Maps	p. 3
Manifolds	p. 6
Complexes	p. 7
Feature Size and Sampling	p. 8
Medial Axis	p. 10
Local Feature Size	p. 14
Sampling	p. 16
Voronoi Diagram and Delaunay Triangulation	p. 18
Two Dimensions	p. 19
Three Dimensions	p. 22
Notes and Exercises	p. 23
Exercises	p. 24
Curve Reconstruction	p. 26
Consequences of [epsilon]-Sampling	p. 27
Crust	p. 30
Algorithm	p. 30
Correctness	p. 32
NN-Crust	p. 35
Algorithm	p. 35
Correctness	p. 36
Notes and Exercises	p. 38
Exercises	p. 39
Surface Samples	p. 41
Normals	p. 43
Approximation of Normals	p. 43
Normal Variation	p. 45
Edge and Triangle Normals	p. 47
Topology	p. 50
Topological Ball Property	p. 50
Voronoi Faces	p. 53
Notes and Exercises	p. 57
Exercises	p. 57
Surface Reconstruction	p. 59
Algorithm	p. 59
Poles and Cocones	p. 59
Cocone Triangles	p. 62
Pruning	p. 64
Manifold Extraction	p. 66
Geometric Guarantees	p. 70
Additional Properties	p. 72
Topological Guarantee	p. 73
The Map [nu]	p. 73
Homeomorphism Proof	p. 75
Notes and Exercises	p. 76
Exercises	p. 78
Undersampling	p. 80
Samples and Boundaries	p. 80
Boundary Sample Points	p. 81
Flat Sample Points	p. 82
Flatness Analysis	p. 83
Boundary Detection	p. 87
Justification	p. 88
Reconstruction	p. 89
Notes and Exercises	p. 90
Exercises	p. 91
Watertight Reconstructions	p. 93
Power Crust	p. 93
Definition	p. 94
Proximity	p. 97
Homeomorphism and Isotopy	p. 99
Algorithm	p. 101
Tight Cocone	p. 104
Marking	p. 105
Peeling	p. 107
Experimental Results	p. 109
Notes and Exercises	p. 111
Exercises	p. 111
Noisy Samples	p. 113
Noise Model	p. 113
Empty Balls	p. 115
Normal Approximation	p. 119
Analysis	p. 119
Algorithm	p. 122
Feature Approximation	p. 124
Analysis	p. 126
Algorithm	p. 130
Notes and Exercises	p. 131
Exercises	p. 132
Noise and Reconstruction	p. 133
Preliminaries	p. 133
Union of Balls	p. 136
Proximity	p. 142
Topological Equivalence	p. 144
Labeling	p. 145
Algorithm	p. 147
Notes and Exercises	p. 149
Exercises	p. 150
Implicit Surface-Based Reconstructions	p. 152
Generic Approach	p. 152
Implicit Function Properties	p. 153
Homeomorphism Proof	p. 154
MLS Surfaces	p. 155
Adaptive MLS Surfaces	p. 156
Sampling Assumptions and Consequences	p. 159
Influence of Samples	p. 162
Surface Properties	p. 166
Hausdorff Property	p. 167
Gradient Property	p. 170
Algorithm and Implementation	p. 172
Normal and Feature Approximation	p. 172
Projection	p. 173
Other MLS Surfaces	p. 175
Projection MLS	p. 175
Variation	p. 176
Computational Issues	p. 176
Voronoi-Based Implicit Surface	p. 179
Notes and Exercises	p. 180
Exercises	p. 181
Morse Theoretic Reconstructions	p. 182
Morse Functions and Flows	p. 182
Discretization	p. 185
Vector Field	p. 185
Discrete Flow	p. 187
Relations to Voronoi/Delaunay Diagrams	p. 190
Reconstruction with Flow Complex	p. 192
Flow Complex Construction	p. 192
Merging	p. 193
Critical Point Separation	p. 194
Reconstruction with a Delaunay Subcomplex	p. 196
Distance from Delaunay Balls	p. 196
Classifying and Ordering Simplices	p. 198
Reconstruction	p. 201
Algorithm	p. 202
Notes and Exercises	p. 205
Exercises	p. 205
Bibliography	p. 207
Index	p. 213
Table of Contents provided by Ingram. All Rights Reserved.

Curve and Surface Reconstruction

Algorithms with Mathematical Analysis

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

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