Variational and Level Set Methods in Image Segmentation

by Mitiche, Amar; Ayed, Ismail Ben

• ISBN13: 9783642153518
• ISBN10: 3642153518
• Format: Hardcover
• Copyright: 2010-09-29
• Publisher: Springer Verlag

Summary

Image segmentation consists of dividing an image domain into disjoint regions according to a characterization of the image within or in-between the regions. Therefore, segmenting an image is to divide its domain into relevant components. The efficient solution of the key problems in image segmentation promises to enable a rich array of useful applications. The current major application areas include robotics, medical image analysis, remote sensing, scene understanding, and image database retrieval. The subject of this book is image segmentation by variational methods with a focus on formulations which use closed regular plane curves to define the segmentation regions and on a level set implementation of the corresponding active curve evolution algorithms. Each method is developed from an objective functional which embeds constraints on both the image domain partition of the segmentation and the image data within or in-between the partition regions. The necessary conditions to optimize the objective functional are then derived and solved numerically. The book covers, within the active curve and level set formalism, the basic two-region segmentation methods, multiregion extensions, region merging, image modeling, and motion based segmentation. To treat various important classes of images, modeling investigates several parametric distributions such as the Gaussian, Gamma, Weibull, and Wishart. It also investigates non-parametric models. In motion segmentation, both optical flow and the movement of real three-dimensional objects are studied.

Table of Contents

Introduction	p. 1
References	p. 10
Introductory Background	p. 15
Euler-Lagrange equations	p. 15
Definite integrals	p. 15
Variable domain of integration	p. 17
Descent methods for unconstrained optimization	p. 20
Real functions	p. 20
Integral functional	p. 20
Level sets	p. 22
Optical flow	p. 25
The gradient equation	p. 25
The Horn and Schunck formulation	p. 26
The Aubert, Kornprobst, and Deriche formulation	p. 28
Optical flow of rigid body motion	p. 28
References	p. 31
Basic Methods	p. 33
The Mumford and Shah model	p. 33
Bayesian interpretation	p. 34
Graduated non convexity implementation	p. 35
The minimum description length method of Leclerc	p. 36
MDL and MAP	p. 36
The piecewise constant image model	p. 37
Numerical implementation	p. 39
The region competition algorithm	p. 40
Optimization	p. 41
A level set formulation of the piecewise constant Mumford-Shah model	p. 45
Curve evolution minimization of the Chan-Vese functional	p. 46
Level set representation of curve evolution	p. 48
Algorithm summary	p. 49
Numerical implementation details of the level set evolution equation	p. 50
Edge-based approaches	p. 51
The Kass-Witkin-Terzopoulos Snakes model	p. 51
The Geodesic active contour	p. 52
Examples	p. 54
References	p. 57
Multiregion Segmentation	p. 59
Introduction	p. 59
Multiregion segmentation using a partition constraint functional term	p. 61
Multiphase level set image segmentation	p. 62
Level set multiregion competition	p. 66
Representation of a partition into a fixed but arbitrary number of regions	p. 66
Curve evolution equations	p. 67
Level set implementation	p. 69
Multiregion level set segmentation as regularized clustering	p. 70
Curve evolution equations	p. 71
Level set implementation	p. 73
Embedding a partition constraint directly in the minimization equations	p. 74
Two-region segmentation: first order analysis	p. 74
Extension to multiregion segmentation	p. 76
Example	p. 78
References	p. 80
Image Models	p. 83
Introduction	p. 83
Segmentation by maximizing the image likelihood	p. 84
The Gaussian model	p. 85
The Gamma image model	p. 89
Generalization to distributions of the exponential family	p. 91
The Weibull image Model	p. 93
The Complex Wishart Model	p. 95
MDL interpretation of the smoothness term coefficient	p. 98
Generalization to multiregion segmentation	p. 99
Examples	p. 101
Maximization of the mutual information between the segmentation and the image	p. 104
Curve evolution equation	p. 106
Statistical interpretation	p. 108
Algorithm summary	p. 108
Segmentation by maximizing the discrepancy between the regions image distributions	p. 109
Statistical interpretation	p. 110
The kernel width	p. 110
Algorithm summary	p. 111
Example	p. 111
Image segmentation using a region reference distribution	p. 111
Statistical interpretation	p. 113
Summary of the algorithms	p. 114
Example	p. 114
Segmentation with an overlap prior	p. 114
Statistical interpretation	p. 117
Example	p. 117
References	p. 120
Region Merging Priors	p. 123
Introduction	p. 123
Definition of a region merging prior	p. 125
A minimum description length prior	p. 126
An entropic region merging prior	p. 126
Entropic interpretation	p. 127
Segmentation functional	p. 127
Minimization equations	p. 128
A region merging interpretation of the level set evolution equations	p. 130
The weight of the entropic prior	p. 130
Example	p. 132
Segmentation with the entropic region merging prior	p. 132
Segmentation with the MDL region merging prior	p. 133
Computation time	p. 133
References	p. 137
Motion Based Image Segmentation	p. 139
Introduction	p. 139
Piecewise constant MDL estimation and segmentation of optical flow	p. 141
Numerical implementation	p. 143
Example	p. 145
Joint segmentation and linear parametric estimation of optical flow	p. 145
Formulation	p. 147
Functional minimization	p. 151
Level set implementation	p. 155
Multiregion segmentation	p. 155
Examples	p. 155
References	p. 158
Image Segmentation According to the Movement of Real Objects	p. 161
Introduction	p. 161
The functionals	p. 164
Minimization of E1	p. 166
Minimization with respect to the screws of motion	p. 166
Minimization with respect to depth	p. 167
Minimization with respect to the active curve	p. 167
Algorithm	p. 168
Uncertainty of scale in 3D interpretation	p. 168
Multiregion segmentation	p. 169
Example	p. 169
Minimization of E2	p. 169
Minimization with respect to the essential parameter vectors	p. 169
Minimization with respect to optical flow	p. 171
Minimization with respect to¿	p. 171
Recovery of regularized relative depth	p. 171
Algorithm	p. 172
Example	p. 173
Minimization of E3	p. 174
Example	p. 175
References	p. 178
Appendix	p. 181
The Horn and Schunck optical flow estimation algorithm	p. 181
Iterative resolution by the Jacobi and Gauss-Seidel iterations	p. 183
Evaluation of derivatives	p. 184
The Aubert, Deriche, and Kornprobst algorithm	p. 184
Construction of stereoscopic images of a computed 3D interpretation	p. 186
References	p. 188
Index	p. 189
Table of Contents provided by Ingram. All Rights Reserved.