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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．Geodesics 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 B 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