Mitiche Variational and Level Set Methods in Image Segmentation

1. Auflage 2010
ISBN: 978-3-642-15352-5
Verlag: Springer
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 192 Seiten

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.

Autoren/Hrsg.

Mitiche, Amar

Inhaltsverzeichnis

1;Contents;4
2;1 INTRODUCTION;8
2.1;References;17
3;2 INTRODUCTORY BACKGROUND ;21
3.1;2.1 Euler-Lagrange equations;21
3.1.1;2.1.1 Definite integrals;21
3.1.2;2.1.2 Variable domain of integration;23
3.2;2.2 Descent methods for unconstrained optimization;26
3.2.1;2.2.1 Real functions;26
3.2.2;2.2.2 Integral functionals;26
3.3;2.3 Level sets;28
3.4;2.4 Optical flow;31
3.4.1;2.4.1 The gradient equation;31
3.4.2;2.4.2 The Horn and Schunck formulation;32
3.4.3;2.4.3 The Aubert, Kornprobst, and Deriche formulation;34
3.4.4;2.4.4 Optical flow of rigid body motion;34
3.5;References;37
4;3 BASIC METHODS;38
4.1;3.1 The Mumford and Shah model;38
4.1.1;3.1.1 Bayesian interpretation;39
4.1.2;3.1.2 Graduated non convexity implementation;40
4.2;3.2 The minimum description length method of Leclerc;41
4.2.1;3.2.1 MDL and MAP;41
4.2.2;3.2.2 The piecewise constant image model;42
4.2.3;3.2.3 Numerical implementation;44
4.3;3.3 The region competition algorithm;45
4.3.1;3.3.1 Optimization;46
4.4;3.4 A level set formulation of the piecewise constant Mumford-Shah model;50
4.4.1;3.4.1 Curve evolution minimization of the Chan-Vese functional;51
4.4.2;3.4.2 Level set representation of curve evolution;53
4.4.3;3.4.3 Algorithm summary;54
4.4.4;3.4.4 Numerical implementation details of the level set evolution equation;55
4.5;3.5 Edge-based approaches;56
4.5.1;3.5.1 The Kass-Witkin-Terzopoulos Snakes model;56
4.5.2;3.5.2 The Geodesic active contour;57
4.5.3;3.5.3 Examples;59
4.6;References;62
5;4 MULTIREGION SEGMENTATION;64
5.1;4.1 Introduction;64
5.2;4.2 Multiregion segmentation using a partition constraint functional term;66
5.3;4.3 Multiphase level set image segmentation;67
5.4;4.4 Level set multiregion competition;71
5.4.1;4.4.1 Representation of a partition into a fixed but arbitrary number of regions;71
5.4.2;4.4.2 Curve evolution equations;72
5.4.3;4.4.3 Level set implementation;74
5.5;4.5 Multiregion level set segmentation as regularized clustering;75
5.5.1;4.5.1 Curve evolution equations;76
5.5.2;4.5.2 Level set implementation;78
5.6;4.6 Embedding a partition constraint directly in the minimization equations;79
5.6.1;4.6.1 Two-region segmentation: first order analysis;79
5.6.2;4.6.2 Extension to multiregion segmentation;81
5.6.3;4.6.3 Example;83
5.7;References;85
6;5 IMAGE MODELS;87
6.1;5.1 Introduction;87
6.2;5.2 Segmentation by maximizing the image likelihood;88
6.2.1;5.2.1 The Gaussian model;89
6.2.2;5.2.2 The Gamma image model;93
6.2.3;5.2.3 Generalization to distributions of the exponential family;95
6.2.4;5.2.4 The Weibull image Model;97
6.2.5;5.2.5 The Complex Wishart Model;99
6.2.6;5.2.6 MDL interpretation of the smoothness term coefficient;102
6.2.7;5.2.7 Generalization to multiregion segmentation;103
6.2.8;5.2.8 Examples;105
6.3;5.3 Maximization of the mutual information between the segmentation and the image;108
6.3.1;5.3.1 Curve evolution equation;110
6.3.2;5.3.2 Statistical interpretation;112
6.3.3;5.3.3 Algorithm summary;112
6.4;5.4 Segmentation by maximizing the discrepancy between the regions image distributions;113
6.4.1;5.4.1 Statistical interpretation;114
6.4.2;5.4.2 The kernel width;114
6.4.3;5.4.3 Algorithm summary;115
6.4.4;5.4.4 Example;115
6.5;5.5 Image segmentation using a region reference distribution;115
6.5.1;5.5.1 Statistical interpretation;117
6.5.2;5.5.2 Summary of the algorithms;118
6.5.3;5.5.3 Example;118
6.6;5.6 Segmentation with an overlap prior;118
6.6.1;5.6.1 Statistical interpretation;121
6.6.2;5.6.2 Example;121
6.7;References;124
7;6 REGION MERGING PRIORS;127
7.1;6.1 Introduction;127
7.2;6.2 Definition of a region merging prior;129
7.3;6.3 A minimum description length prior;130
7.4;6.4 An entropic region merging prior;130
7.4.1;6.4.1 Entropic interpretation;131
7.4.2;6.4.2 Segmentation functional;131
7.4.3;6.4.3 Minimization equations;132
7.4.4;6.4.4 A region merging interpretation of the level set evolution equations;134
7.4.5;6.4.5 The weight of the entropic prior;134
7.5;6.5 Example;136
7.5.1;6.5.1 Segmentation with the entropic region merging prior;136
7.5.2;6.5.2 Segmentation with the MDL region merging prior;137
7.5.3;6.5.3 Computation time;137
7.6;References;141
8;7 MOTION BASED IMAGE SEGMENTATION ;142
8.1;7.1 Introduction;142
8.2;7.2 Piecewise constant MDL estimation and segmentation of optical flow;144
8.2.1;7.2.1 Numerical implementation;146
8.2.2;7.2.2 Example;148
8.3;7.3 Joint segmentation and linear parametric estimation of optical flow;148
8.3.1;7.3.1 Formulation;150
8.3.2;7.3.2 Functional minimization;154
8.3.3;7.3.3 Level set implementation;158
8.3.4;7.3.4 Multiregion segmentation;158
8.3.5;7.3.5 Examples;158
8.4;References;161
9;8 IMAGE SEGMENTATION ACCORDING TO THE MOVEMENT OF REAL OBJECTS;164
9.1;8.1 Introduction;164
9.2;8.2 The functionals;167
9.3;8.3 Minimization of E1;169
9.3.1;8.3.1 Minimization with respect to the screws of motion;169
9.3.2;8.3.2 Minimization with respect to depth;170
9.3.3;8.3.3 Minimization with respect to the active curve;170
9.3.4;8.3.4 Algorithm;171
9.3.5;8.3.5 Uncertainty of scale in 3D interpretation;171
9.3.6;8.3.6 Multiregion segmentation;172
9.3.7;8.3.7 Example;172
9.4;8.4 Minimization of E2;172
9.4.1;8.4.1 Minimization with respect to the essential parameter vectors;172
9.4.2;8.4.2 Minimization with respect to optical flow;174
9.4.3;8.4.3 Minimization with respect to ;174
9.4.4;8.4.4 Recovery of regularized relative depth;174
9.4.5;8.4.5 Algorithm;175
9.4.6;8.4.6 Example;176
9.5;8.5 Minimization of E3;177
9.5.1;8.5.1 Example;178
9.6;References;181
10;9 APPENDIX ;184
10.1;9.1 The Horn and Schunck optical flow estimation algorithm;184
10.1.1;9.1.1 Iterative resolution by the Jacobi and Gauss-Seidel iterations;186
10.1.2;9.1.2 Evaluation of derivatives;187
10.2;9.2 The Aubert, Deriche, and Kornprobst algorithm;187
10.3;9.3 Construction of stereoscopic images of a computed 3D interpretation;189
10.4;References;191
11;Index;192