This book is an introduction to the use of geometric partial differential equations (PDEs) in image processing and computer vision. This relatively new research area brings a number of new concepts into the field, providing, among other things, a very fundamental and formal approach to image processing. State-of-the-art practical results in problems such as image segmentation, stereo, image enhancement, distance computations, and object tracking have been obtained with algorithms based on PDE's formulations.

List of figures

Preface

Acknowledgments

Introduction

1 Basic Mathematical Background

　1.1 Planar Differential Geometry

　1.2 Affine Differential Geometry

　1.3 Cartan Moving Frames

　1.4 Space Curves

　1.5 Three-Dimensional Differential Geometry

　1.6 Discrete Differential Geometry

　1.7 Differential Invariants and Lie Group Theory

　1.8 Basic Concepts of Partial Differential Equations

　1.9 Calculus of Variations and Gradient Descent Flows

　1.10 Numerical Analysis

　Exercises

2 Geometric Curve and Surface Evolution

　2.1 Basic Concepts

　2.2 Level Sets and Implicit Representations

　2.3 Variational Level Sets

　2.4 Continuous Mathematical Morphology

　2.5 Euclidean and affine Curve Evolution and Shape Analysis

　2.6 Euclidean and Affine Surface Evolution

　2.7 Area-and Volume-Preserving 3D Flows

　2.8 Calssification of Invariant Geometric Flows

　Exercises

3 Geodesic Curves and Minimal Surfaces

 3.1 Basic Two-Dimensional Derivation

 3.2 Three-Dimensional Derivation

 3.3 Geodesics in Vector-Valued Images

 3.4 Finding the Minimal Geodesic

 3.5 Affine Invariant Active Contours

 3.6 Additional Extensions and Modifications

 3.7 Tracking and Morphing Active Contours

 3.8 Stereo

 Appendix A

 Appendix B

 Exercises

4 Geometric Diffusion of Scalar Images

 4.1 Gaussian Filtering and Linear Scale Spaces

 4.2 Edge-Stopping Diffusion

 4.3 Directional Diffusion

 4.4 Introducing Prior Knowledge

 4.5 Some Order in the PDE Jungle

 Exercises

5 Geometric Diffusion of Vector-Valued Images

 5.1 Directional Diffusion of Multivalued Images

 5.2 Vectorial Median Filter

 5.3 Color Self-Snakes

 Exercises

6 Diffusion on Nonflat Manifolds

 6.1 The General Problem

 6.2 Isotropic Diffusion

 6.3 Anisotropic Diffusion

 6.4 Examples

 6.5 Vector Probability Diffusion

 Appendix

 Exercises

7 Contrast Enhancement

 7.1 Global PDE-Based Approach

 7.2 Shape-Preserving Contrast Enhancement

 Exercises

8 Additional Theories and Applications

 8.1 Interpolation

 8.2 Image Repair: Inpainting

 8.3 Shape from Shading

 8.4 Blind Deconvolution

 Exercises

Bibliography

Index