Conformal geometry. Computational algorithms and engineering applications

Preface......Page 6
Contents......Page 8
 1.1 Riemann Mapping Theorem......Page 13
 1.2 Holomorphic Differential......Page 14
 1.3 Uniformization Theorem......Page 15
 1.4 Shape Space......Page 16
 1.5 Geometric Structures......Page 17
Part I Computational Algorithms......Page 20
 2.1 Halfedge Data Structure......Page 21
 2.2 Cut Graph and Fundamental Domain......Page 22
 2.3 Homotopy and Homology Group Basis......Page 23
 2.4 Canonical Homotopy Group Generators......Page 26
 3.1 Discrete Harmonic Energy......Page 29
 3.2 Topological Disk Surfaces......Page 31
 3.3 Topological Sphere Surfaces......Page 33
 3.4 Riemann Mapping......Page 35
 4.1 Characteristic Forms......Page 38
 4.2 Cohomology Basis......Page 40
 4.3 Harmonic 1-Form......Page 41
 4.4 Hodge Star Operator......Page 42
 4.5 Holomorphic 1-Form......Page 43
 5.1 Discrete Background......Page 48
 5.2 Discrete Surface Ricci Flow......Page 52
 5.3 Isometric Planar Embedding......Page 58
 5.4 Hyperbolic Ricci Flow......Page 60
 5.5 Isometric Hyperbolic Embedding......Page 61
Part II Engineering Applications......Page 66
 6.1 Introduction......Page 67
 6.2 Global Surface Parametrization......Page 68
  6.2.1 Optimal Global Conformal Parametrization Using Discrete One-Form......Page 69
  6.2.2 Global Conformal Parameterization Using Discrete Euclidean Ricci Flow......Page 74
 6.3 Uniform Remeshing......Page 76
  6.3.1 Voronoi Diagram in Different Spaces......Page 77
  6.3.2 Centroid Voronoi Tessellation in Different Spaces......Page 78
  6.3.3 CVT Energy in Different Spaces......Page 82
  6.3.4 Computing Centroidal Voronoi Tessellations......Page 84
  6.3.5 Uniform Remeshing and Partition......Page 86
 6.4 Metric-Driven RoSy Fields Design......Page 87
  6.4.1 Theory of Compatibility......Page 89
  6.4.2 Computing N-RoSy Fields......Page 92
  6.4.3 Remeshing......Page 95
 6.5 Computing Shortest Homotopic Cycles on Polyhedral Surfaces......Page 96
  6.5.1 Geodesic Uniqueness......Page 98
  6.5.2 Fuchsian Group Method......Page 99
  6.5.3 Birkhoff Curve Shortening Method......Page 101
  6.5.4 Extremal Quasiconformal Mapping......Page 105
 6.6 Summary and Further Reading......Page 109
 7.1 Introduction......Page 110
 7.2 Teichmuller Shape Space......Page 111
  7.2.1 Luo Coordinates......Page 113
  7.2.2 Fenchel-Nielsen Coordinates......Page 117
  7.2.3 Robustness of Teichmuller Space Coordinates......Page 123
  7.2.4 Surface Indexing and Classification......Page 124
  7.3.1 Generalized Discrete Ricci Flow......Page 125
  7.3.2 Doubly Connected Domain......Page 128
  7.3.3 Multiply Connected Domain......Page 129
 7.4 Shape Signature......Page 131
  7.4.1 Beltrami Equation......Page 132
  7.4.3 Holomorphic Differentials......Page 134
  7.4.4 Conformal Welding......Page 135
  7.4.5 Computing Shape Signatures of Planar Domains......Page 136
  7.4.6 Shape Clustering......Page 139
 7.5 Summary and Further Reading......Page 140
 8.1 Introduction......Page 141
  8.2.1 Definition......Page 143
  8.2.2 Properties......Page 144
  8.2.3 Surface Reconstruction......Page 146
  8.3.1 Definitions and Concepts......Page 151
  8.3.2 Equivalence to Affine Atlas......Page 154
  8.3.3 Existence......Page 156
 8.4 Constructing Affine Atlas......Page 157
  8.4.1 Computing Affine Structure Using Holomorphic 1-form......Page 158
  8.4.2 Computing Affine Structure Using Discrete Ricci Flow......Page 159
  8.4.3 Computing Affine Structure Using Polycube Maps......Page 164
  8.5.1 Manifold Triangular B-spline......Page 167
  8.5.2 Manifold Powell–Sabin Spline......Page 168
  8.5.3 Manifold T-spline......Page 169
 8.6 Handling Extraordinary Points......Page 170
  8.7.1 Domain Manifold......Page 174
  8.7.4 Manifold Splines Versus Subdivision Surfaces......Page 175
  8.7.5 General Manifold Spline Program......Page 176
 8.8 Summary......Page 179
 9.1 Introduction......Page 180
 9.2 Brain Surface Conformal Parameterization......Page 182
  9.2.1 Spherical Brain Conformal Parameterization......Page 183
  9.2.2 Planar Brain Conformal Parameterization  with Holomorphic Functions......Page 184
  9.2.3 Hyperbolic Brain Conformal Parameterization  with Ricci Flow Method......Page 187
 9.3 Brain Surface Registration......Page 190
  9.3.1 Optimization of Brain Conformal Mapping  with Landmarks......Page 191
  9.3.2 Constrained Harmonic Map......Page 195
  9.3.3 Hyperbolic Harmonic Map......Page 198
  9.3.4 Surface Fluid Registration......Page 211
 9.4 Global Transformation-Invariant Shape Descriptors......Page 217
  9.4.1 Spherical Harmonic Analysis Based Rotation Invariant Shape Descriptor......Page 218
  9.4.2 Conformal Welding Based Teichmüller  Shape Descriptor......Page 220
  9.4.3 Teichmüller Space Coordinates for Landmark Curve-Based Brain Morphometry Analysis......Page 226
  9.4.4 Riemannian Optimal Mass Transport Map and Riemannian Wasserstein Distance......Page 231
 9.5 Point-to-Point Local Surface Deformation Measurements......Page 245
  9.5.1 Genetic Influence of APOE4 Genotype on Hippocampal Morphometry......Page 246
  9.5.2 Ventricular Abnormalities in Mild Cognitive Impairment......Page 249
  9.5.3 Computer-Assisted Diagnosis with Computational Conformal Geometry......Page 253
 9.6 Summary......Page 256
 10.1 Introduction......Page 257
 10.2 Localization......Page 258
  10.2.1 Planar Sensor Network Localization......Page 259
  10.2.2 Surface Sensor Network Localization......Page 264
 10.3 Greedy Routing......Page 272
  10.3.1 3D Wireless Sensor Networks Without Hole......Page 273
  10.3.2 3D Wireless Sensor Networks with Internal Holes......Page 275
 10.4 Deployment......Page 276
  10.4.1 Optimal Surface Network Deployment......Page 277
  10.4.2 Optimal Solution......Page 278
  10.4.3 Decreased Sensing Unreliability......Page 281
  10.4.5 Special Scenarios......Page 282
 10.5 In-Network Data-Centric Processing......Page 285
  10.5.1 Double-Ruling in 3D Network......Page 286
  10.5.2 Data Replication......Page 288
  10.5.4 Delivery of Data and Query......Page 289
 10.6 Marching of Autonomous Networked Robots......Page 290
  10.6.1 Optimal Marching Problem......Page 291
  10.6.2 An Approximated Solution......Page 294
  10.6.3 Modified Harmonic Map......Page 295
  10.6.4 Minor Local Adjustment......Page 297
  10.6.5 Global Connectivity......Page 298
  10.6.7 Holes......Page 299
 10.7 Summary and Further Reading......Page 300
 BookmarkTitle:......Page 301
Index......Page 317