Segmentation of Stochastic Images using ... - Jacobs University

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7.5 Segmentation of Stochastic Images Using Stochastic Level Sets . . . . . . . . . . . 88 8 Segmentation of Classical Images Using Stochastic Parameters 97 8.1 Random Walker Segmentation with Stochastic Parameter . . . . . . . . . . . . . . . 98 8.2 Ambrosio-Tortorelli Segmentation with Stochastic Parameters . . . . . . . . . . . . 101 8.3 Gradient-Based Segmentation with Stochastic Parameter . . . . . . . . . . . . . . . 104 8.4 Geodesic Active Contours with Stochastic Parameters . . . . . . . . . . . . . . . . . 105 9 Summary, Discussion, and Conclusion 107 9.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 9.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 9.3 Outlook and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 List of Figures 111 List of Tables 117 A Publications Written During the Course of the Thesis 119 A.1 Publications Related to Stochastic Images . . . . . . . . . . . . . . . . . . . . . . . 119 A.2 Publications Related to Radiofrequency Ablation . . . . . . . . . . . . . . . . . . . 119 Bibliography 121 viii
Notation u image function ⊗ tensor product D image domain ∂D boundary of the domain D R real numbers S ρ,k,m Kondratiev space P i finite element hat function H univariate polynomial I node set of a finite element grid Ψ multivariate polynomial D c ( Cantor measure a b) binomial coefficient BV functions with bounded variation ξ basic random variable SBV special BV space (D c = 0) δ i j Kronecker delta GSBV generalized SBV space ∂ f ∂x partial derivative sign sign function ∂ t partial temporal derivative H d d-dim. Hausdorff measure τ time step size K edge set (discontinuities) of an image h spatial grid spacing φ phase field or level set V finite element space H 1 (D) Sobolev space H 1 over D S stochastic space, ⊂ L 2 (Ω) | · | absolute value of real numbers ‖ · ‖ x x-norm ∆ Laplace operator F cumulative distribution function tanh hyperbolic tangent N normal vector κ curvature of level sets or phase fields T tangential vector (of level sets) ∗ convolution operator (·) ′ derivative of univariate function E expected value W width of the tangential profile of a phase field Ω probability (event) space L p Lebesgue spaces H m Sobolev spaces ix
Page 1: Segmentation of Stochastic Images u
Page 5: Abstract The task of segmentation,
Page 11 and 12: Chapter 1 Introduction The developm
Page 13 and 14: Figure 1.3: This thesis combines fi
Page 15: the presentation of the stochastic
Page 18 and 19: Chapter 2 Image Segmentation and Li
Page 36 and 37: Chapter 3 SPDEs and Polynomial Chao
Page 48 and 49: Chapter 4 Discretization of SPDEs F
Page 50 and 51: Chapter 4 Discretization of SPDEs 4
Page 52 and 53: Chapter 4 Discretization of SPDEs a
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Chapter 5 Stochastic Images
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Chapter 5 Stochastic Images Figure
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Chapter 5 Stochastic Images like qu
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Chapter 5 Stochastic Images Figure
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Chapter 6 Segmentation of Stochasti
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6.1 Random Walker Segmentation on S
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6.2 Ambrosio-Tortorelli Segmentatio
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Chapter 7 Stochastic Level Sets whe
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Chapter 7 Stochastic Level Sets Fig
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Chapter 7 Stochastic Level Sets and
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Chapter 7 Stochastic Level Sets rar
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Chapter 7 Stochastic Level Sets MC
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Chapter 7 Stochastic Level Sets act
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Chapter 8 Segmentation of Classical
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Chapter 9 Summary, Discussion, and
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List of Figures 1.1 Left: CT image
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List of Figures 6.2 Mean and varian
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List of Figures 7.12 Mean (left) of
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Appendix A Publications Written Dur
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Bibliography [14] L. Ambrosio and M
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Bibliography [46] B. Echebarria, R.
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Bibliography [81] B. N. Khoromskij
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Bibliography [113] A. Nouy. A gener
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Bibliography [147] J. Tryoen, O. Le
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