This book provides an introduction to the use of geometric partial differential equations in image processing and computer vision. It brings a number of new concepts into the field, providing a very fundamental and formal approach to image processing. State-of-the-art practical results in a large number of real problems are achieved with the techniques described. Applications covered include image segmentation, shape analysis, image enhancement, and tracking. The volume provides information for people investigating new solutions to image processing problems as well as for people searching for existent advanced solutions.
Author(s): Guillermo Sapiro
Publisher: CUP
Year: 2001
Language: English
Pages: 412
Cover......Page 1
About......Page 2
Geometric Partial Differential Equations and Image Analysis......Page 4
9780521790758......Page 5
Contents......Page 8
List of Figures......Page 12
Preface......Page 16
Acknowledgments......Page 18
Introduction......Page 22
1 Basic Mathematical Background ......Page 28
1.1. Planar Differential Geometry ......Page 29
1.2. Affine Differential Geometry ......Page 34
1.3. Cartan Moving Frames ......Page 39
1.4. Space Curves ......Page 42
1.5. Three-Dimensional Differential Geometry ......Page 44
1.6. Discrete Differential Geometry ......Page 47
1.7. Differential Invariants and Lie Group Theory ......Page 49
1.8. Basic Concepts of Partial Differential Equations ......Page 72
1.9. Calculus of Variations and Gradient Descent Flows ......Page 84
1.10. Numerical Analysis ......Page 88
Exercises ......Page 96
2.1. Basic Concepts ......Page 98
2.2. Level Sets and Implicit Representations ......Page 101
2.3. Variational Level Sets ......Page 118
2.4. Continuous Mathematical Morphology ......Page 119
2.5. Euclidean and Affine Curve Evolution and Shape Analysis ......Page 126
2.6. Euclidean and Affine Surface Evolution ......Page 156
2.7. Area- and Volume-Preserving 3D Flows ......Page 158
2.8. Classification of Invariant Geometric Flows ......Page 161
Exercises ......Page 169
3.1. Basic Two-Dimensional Derivation ......Page 170
3.2. Three-Dimensional Derivation ......Page 192
3.3. Geodesics in Vector-Valued Images ......Page 209
3.4. Finding the Minimal Geodesic ......Page 218
3.5. Affine Invariant Active Contours ......Page 224
3.6. Additional Extensions and Modifications ......Page 232
3.7. Tracking and Morphing Active Contours ......Page 234
3.8. Stereo ......Page 242
Appendix A ......Page 244
Appendix B ......Page 245
Exercises ......Page 247
4.1. Gaussian Filtering and Linear Scale Spaces ......Page 248
4.2. Edge-Stopping Diffusion ......Page 250
4.3. Directional Diffusion ......Page 268
4.4. Introducing Prior Knowledge ......Page 275
4.5. Some Order in the PDE Jungle ......Page 287
Exercises ......Page 292
5.1. Directional Diffusion of Multivalued Images ......Page 294
5.2. Vectorial Median Filter ......Page 296
5.3. Color Self-Snakes ......Page 308
Exercises ......Page 310
6 Diffusion on Nonflat Manifolds ......Page 311
6.1. The General Problem ......Page 314
6.2. Isotropic Diffusion ......Page 317
6.3. Anisotropic Diffusion ......Page 319
6.4. Examples ......Page 320
6.5. Vector Probability Diffusion ......Page 325
Appendix ......Page 331
Exercises ......Page 332
7 Contrast Enhancement ......Page 334
7.1. Global PDE-Based Approach ......Page 337
7.2. Shape-Preserving Contrast Enhancement ......Page 352
Exercises ......Page 364
8.1. Interpolation ......Page 365
8.2. Image Repair: Inpainting ......Page 370
8.3. Shape from Shading ......Page 382
8.4. Blind Deconvolution ......Page 384
Exercises ......Page 385
Bibliography ......Page 386
Index ......Page 408
