Geometric Partial Differential Equations and Image Analysis

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ISBN: 9780521685078 | 0521685079
Cover: Paperback
Copyright: 2/13/2006

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This book provides an introduction to the use of geometric partial differential equations in image processing and computer vision. This research area 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 in this book. Applications covered include image segmentation, shape analysis, image enhancement, and tracking. This book will be a useful resource for researchers and practitioners. It is intended to provide information for people investigating new solutions to image processing problems as well as for people searching for existent advanced solutions.

List of figures

Preface

Acknowledgments

xvii

Introduction

xxi

1 Basic Mathematical Background

(70)

1.1. Planar Differential Geometry

(5)

1.2. Affine Differential Geometry

(5)

1.3. Cartan Moving Frames

(3)

1.4. Space Curves

(2)

1.5. Three-Dimensional Differential Geometry

(3)

1.6. Discrete Differential Geometry

(2)

1.7. Differential Invariants and Lie Group Theory

(23)

1.8. Basic Concepts of Partial Differential Equations

(12)

1.9. Calculus of Variations and Gradient Descent Flows

(4)

1.10. Numerical Analysis

(8)

Exercises

(2)

2 Geometric Curve and Surface Evolution

(72)

2.1. Basic Concepts

(3)

2.2. Level Sets and Implicit Representations

(17)

2.3. Variational Level Sets

(1)

2.4. Continuous Mathematical Morphology

(7)

2.5. Euclidean and Affine Curve Evolution and Shape Analysis

(30)

2.6. Euclidean and Affine Surface Evolution

129

(2)

2.7. Area- and Volume-Preserving 3D Flows

131

(3)

2.8. Classification of Invariant Geometric Flows

134

(8)

Exercises

142

(1)

3 Geodesic Curves and Minimal Surfaces

143

(78)

3.1. Basic Two-Dimensional Derivation

143

(22)

3.2. Three-Dimensional Derivation

165

(17)

3.3. Geodesics in Vector-Valued Images

182

(9)

3.4. Finding the Minimal Geodesic

191

(6)

3.5. Affine Invariant Active Contours

197

(8)

3.6. Additional Extensions and Modifications

205

(2)

3.7. Tracking and Morphing Active Contours

207

(8)

3.8. Stereo

215

(2)

Appendix A

217

(1)

Appendix B

218

(2)

Exercises

220

(1)

4 Geometric Diffusion of Scalar Images

221

(46)

4.1. Gaussian Filtering and Linear Scale Spaces

221

(2)

4.2. Edge-Stopping Diffusion

223

(18)

4.3. Directional Diffusion

241

(7)

4.4. Introducing Prior Knowledge

248

(12)

4.5. Some Order in the PDE Jungle

260

(5)

Exercises

265

(2)

5 Geometric Diffusion of Vector-Valued Images

267

(17)

5.1. Directional Diffusion of Multivalued Images

267

(2)

5.2. Vectorial Median Filter

269

(12)

5.3. Color Self-Snakes

281

(2)

Exercises

283

(1)

6 Diffusion on Nonflat Manifolds

284

(23)

6.1. The General Problem

287

(3)

6.2. Isotropic Diffusion

290

(2)

6.3. Anisotropic Diffusion

292

(1)

6.4. Examples

293

(5)

6.5. Vector Probability Diffusion

298

(6)

Appendix

304

(1)

Exercises

305

(2)

7 Contrast Enhancement

307

(31)

7.1. Global PDE-Based Approach

310

(15)

7.2. Shape-Preserving Contrast Enhancement

325

(12)

Exercises

337

(1)

8 Additional Theories and Applications

338

(21)

8.1. Interpolation

338

(5)

8.2. Image Repair: Inpainting

343

(12)

8.3. Shape from Shading

355

(2)

8.4. Blind Deconvolution

357

(1)

Exercises

358

(1)

Bibliography

359

(22)

Index

381

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Geometric Partial Differential Equations and Image Analysis

Summary

Table of Contents

Supplemental Materials