Mathematical problems in image processing : partial differential equations and the calculus of variations

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Title:

Personal Author:

Aubert, Gilles

Series:

Applied mathematical sciences ; 147

Edition:

2nd ed.

Publication Information:

New York, NY : Springer, 2006

ISBN:

9780387322001

Subject Term:

Image processing -- Mathematics

Added Author:

Kornprobst, Pierre

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010119046	TA1637 A924 2006	Open Access Book	Book	Searching... Unknown

Partial differential equations (PDEs) and variational methods were introduced into image processing about fifteen years ago. Since then, intensive research has been carried out. The goals of this book are to present a variety of image analysis applications, the precise mathematics involved and how to discretize them.

Thus, this book is intended for two audiences. The first is the mathematical community by showing the contribution of mathematics to this domain. It is also the occasion to highlight some unsolved theoretical questions. The second is the computer vision community by presenting a clear, self-contained and global overview of the mathematics involved in image procesing problems. This work will serve as a useful source of reference and inspiration for fellow researchers in Applied Mathematics and Computer Vision, as well as being a basis for advanced courses within these fields.

During the four years since the publication of the first edition, there has been substantial progress in the range of image processing applications covered by the PDE framework. The main goals of the second edition are to update the first edition by giving a coherent account of some of the recent challenging applications, and to update the existing material. In addition, this book provides the reader with the opportunity to make his own simulations with a minimal effort. To this end, programming tools are made available, which will allow the reader to implement and test easily some classical approaches.

Foreword	p. iii
Preface	p. vii
1 Introduction	p. 1
1.1 The image society	p. 1
1.2 What is a digital image?	p. 3
1.3 About Partial Differential Equations (PDEs)	p. 5
1.4 Detailed plan	p. 5
Guide to relevant mathematical proofs	p. 23
Notations and symbols	p. 25
2 Mathematical preliminaries	p. 31
How to read this chapter?	p. 31
2.1 The direct method in the calculus of variations	p. 32
2.1.1 Topologies on Banach spaces	p. 32
2.1.2 Convexity and lower semi-continuity	p. 34
2.1.3 Relaxation	p. 39
2.1.4 About ¿-Convergence	p. 42
2.2 The space of functions of bounded variation	p. 44
2.2.1 Basic definitions on measures	p. 44
2.2.2 Definition of BV(¿)	p. 46
2.2.3 Properties of BV (¿)	p. 48
2.2.4 Convex functions of measures	p. 51
2.3 Viscosity solutions in PDEs	p. 52
2.3.1 Around the eikonal equation	p. 52
2.3.2 Definition of viscosity solutions	p. 54
2.3.3 About the existence	p. 55
2.3.4 About the uniqueness	p. 56
2.4 Elements of differential geometry: the curvature	p. 58
2.4.1 Parametrized curves	p. 59
2.4.2 Curves as isolevel of a function u	p. 60
2.4.3 Images as surfaces	p. 60
2.5 Other classical results used in this book	p. 61
2.5.1 Inequalities	p. 61
2.5.2 Calculus facts	p. 63
2.5.3 About convolution and smoothing	p. 64
2.5.4 Uniform convergence	p. 65
2.5.5 Dominated convergence theorem	p. 65
2.5.6 Well-posed problem	p. 65
3 Image Restoration	p. 67
How to read this chapter?	p. 67
3.1 Image degradation	p. 68
3.2 The energy method	p. 70
3.2.1 An inverse problem	p. 70
3.2.2 Regularization of the problem	p. 71
3.2.3 Existence and uniqueness of a solution for the minimization problem	p. 74
3.2.4 Toward the numerical approximation	p. 77
3.2.5 Some invariances and the role of ¿	p. 85
3.2.6 Some remarks in the nonconvex case	p. 88
3.3 PDE-based methods	p. 92
3.3.1 Smoothing PDEs	p. 93
The heat equation	p. 93
Nonlinear diffusion	p. 96
The Alvarez-Guichard-Lions-Morel scale space theory	p. 105
Weickert's approach	p. 112
Surface based approaches	p. 115
3.3.2 Smoothing-Enhancing PDEs	p. 120
The Perona and Malik model [209]	p. 120
Regularization of the Perona and Malik model: Catté et al [59]	p. 122
3.3.3 Enhancing PDEs	p. 127
The Osher and Rudin's shock-filters [199]	p. 127
A case study: construction of a solution by the method of characteristics	p. 128
Comments on the shock-filter equation	p. 133
4 The Segmentation Problem	p. 137
How to read this chapter?	p. 137
4.1 Definition and objectives	p. 138
4.2 The Mumford and Shah functional	p. 141
4.2.1 A minimization problem	p. 141
4.2.2 The mathematical framework for the existence of a solution	p. 141
4.2.3 Regularity of the edge set	p. 150
4.2.4 Approximations of the Mumford and Shah functional	p. 154
4.2.5 Experimental results	p. 159
4.3 Geodesic active contours and the level sets method	p. 161
4.3.1 The Kass-Witkin-Terzopoulos model [142]	p. 161
4.3.2 The Caselles-Kimmel-Sapiro geodesic active contours model [58]	p. 164
4.3.3 The level sets method	p. 170
4.3.4 Experimental results	p. 182
4.3.5 About some recent advances	p. 184
Global stopping criterion	p. 184
Toward more general shape representation	p. 186
5 Other Challenging Applications	p. 189
How to read this chapter?	p. 189
5.1 Sequence analysis	p. 190
5.1.1 Introduction	p. 190
5.1.2 The optical flow: an apparent motion	p. 192
The Optical Flow Constraint (OFC)	p. 193
Solving the aperture problem	p. 194
Overview of a discontinuity preserving variational approach	p. 198
Alternatives of the OFC	p. 201
5.1.3 Sequence segmentation	p. 203
Introduction	p. 203
A variational formulation (the time-continuous case)	p. 204
Mathematical study of the time sampled energy	p. 207
Experiments	p. 210
5.1.4 Sequence restoration	p. 212
5.2 Image classification	p. 218
5.2.1 Introduction	p. 218
5.2.2 A level sets approach for image classification [221]	p. 219
5.2.3 A variational model for image classification and restoration [222]	p. 224
A Introduction to Finite Difference	p. 237
How to read this chapter?	p. 237
A.1 Definitions and theoretical considerations illustrated by the 1-D parabolic heat equation	p. 238
A.1.1 Getting started	p. 238
A.1.2 Convergence	p. 241
A.1.3 The Lax Theorem	p. 243
A.1.4 Consistency	p. 243
A.1.5 Stability	p. 245
A.2 Hyperbolic equations	p. 250
A.3 Difference schemes in image analysis	p. 259
A.3.1 Getting started	p. 259
A.3.2 Image restoration by energy minimization	p. 263
A.3.3 Image enhancement by the Osher and Rudin's shock-filters	p. 266
A.3.4 Curves evolution with the level sets method	p. 267
Mean curvature motion	p. 268
Constant speed evolution	p. 270
The pure advection equation	p. 271
Image segmentation by the geodesic active contour model	p. 271
References	p. 273
Index	p. 291

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