Segmentation of Stochastic Images using ... - Jacobs University

Contents
Acknowledgement
Abstract
Notation
iii
v
ix
1 Introduction 1
2 Image Segmentation and Limitations 7
2.1 Mathematical Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Random Walker Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Mumford-Shah and Ambrosio-Tortorelli Segmentation . . . . . . . . . . . . . . . . 12
2.4 Level Sets for Image Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 Why is Classical Image Processing not Enough? . . . . . . . . . . . . . . . . . . . . 21
2.6 Work Related to the Stochastic Framework . . . . . . . . . . . . . . . . . . . . . . . 23
3 SPDEs and Polynomial Chaos Expansions 25
3.1 Basics from Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Stochastic Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Polynomial Chaos Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4 Relation to Interval Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4 Discretization of SPDEs 37
4.1 Sampling Based Discretization of SPDEs . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Stochastic Finite Difference Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.3 Stochastic Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.4 Generalized Spectral Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5 Adaptive Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5 Stochastic Images 47
5.1 Polynomial Chaos for Stochastic Images . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2 Generation of Stochastic Images from Samples . . . . . . . . . . . . . . . . . . . . 48
5.3 Comparison of the Space from [130] and the Space Used in this Thesis . . . . . . . . 52
5.4 Visualization of Stochastic Images . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6 Segmentation of Stochastic Images Using Elliptic SPDEs 57
6.1 Random Walker Segmentation on Stochastic Images . . . . . . . . . . . . . . . . . 57
6.2 Ambrosio-Tortorelli Segmentation on Stochastic Images . . . . . . . . . . . . . . . 67
7 Stochastic Level Sets 79
7.1 Derivation of a Stochastic Level Set Equation . . . . . . . . . . . . . . . . . . . . . 79
7.2 Discretization of the Stochastic Level Set Equation . . . . . . . . . . . . . . . . . . 83
7.3 Reinitialization of Stochastic Level Sets . . . . . . . . . . . . . . . . . . . . . . . . 84
7.4 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
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