List of figures ................................................ xi
Preface ........................................................ xv
Acknowledgments .............................................. xvii
Introduction .................................................. xxi
1 Basic Mathematical Background ................................ 1
1.1 Planar Differential Geometry ............................ 2
1.2 Affine Differential Geometry ............................ 7
1.3 Cartan Moving Frames ................................... 12
1.4 Space Curves ........................................... 15
1.5 Three-Dimensional Differential Geometry ................ 17
1.6 Discrete Differential Geometry ......................... 20
1.7 Differential Invariants and Lie Group Theory ........... 22
1.8 Basic Concepts of Partial Differential Equations ....... 45
1.9 Calculus of Variations and Gradient Descent Flows ...... 57
1.10 Numerical Analysis ..................................... 61
Exercises ................................................... 69
2 Geometric Curve and Surface Evolution ....................... 71
2.1 Basic Concepts ......................................... 71
2.2 Level Sets and Implicit Representations ................ 74
2.3 Variational Level Sets ................................. 91
2.4 Continuous Mathematical Morphology ..................... 92
2.5 Euclidean and Affine Curve Evolution and Shape
Analysis ............................................... 99
2.6 Euclidean and Affine Surface Evolution ................ 129
2.7 Area- and Volume-Preserving 3D Flows .................. 131
2.8 Classification of Invariant Geometric Flows ........... 134
Exercises .................................................. 142
3 Geodesic Curves and Minimal Surfaces ....................... 143
3.1 Basic Two-Dimensional Derivation ...................... 143
3.2 Three-Dimensional Derivation .......................... 165
3.3 Geodesies in Vector-Valued Images ..................... 182
3.4 Finding the Minimal Geodesic .......................... 191
3.5 Affine Invariant Active Contours ...................... 197
3.6 Additional Extensions and Modifications ............... 205
3.7 Tracking and Morphing Active Contours ................. 207
3.8 Stereo ................................................. 215
Appendix A ................................................. 217
Appendix В ................................................. 218
Exercises .................................................. 220
4 Geometric Diffusion of Scalar Images ....................... 221
4.1 Gaussian Filtering and Linear Scale Spaces ............ 221
4.2 Edge-Stopping Diffusion ............................... 223
4.3 Directional Diffusion ................................. 241
4.4 Introducing Prior Knowledge ........................... 248
4.5 Some Order in the PDE Jungle .......................... 260
Exercises .................................................. 265
5 Geometric Diffusion of Vector-Valued Images ................ 267
5.1 Directional Diffusion of Multivalued Images ........... 267
5.2 Vectorial Median Filter ............................... 269
5.3 Color Self-Snakes ..................................... 281
Exercises .................................................. 283
6 Diffusion on Nonflat Manifolds ............................. 284
6.1 The General Problem ................................... 287
6.2 Isotropic Diffusion ................................... 290
6.3 Anisotropic Diffusion ................................. 292
6.4 Examples .............................................. 293
6.5 Vector Probability Diffusion .......................... 298
Appendix ................................................... 304
Exercises .................................................. 305
7 Contrast Enhancement ....................................... 307
7.1 Global PDE-Based Approach ............................. 310
7.2 Shape-Preserving Contrast Enhancement ................. 325
Exercises .................................................. 337
8 Additional Theories and Applications ....................... 338
8.1. Interpolation ......................................... 338
8.2 Image Repair: Inpainting .............................. 343
8.3 Shape from Shading .................................... 355
8.4 Blind Deconvolution ................................... 357
Exercises .................................................. 358
Bibliography .................................................. 359
Index ......................................................... 381
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