Segmentation of Stochastic Images using ... - Jacobs University

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Chapter 2 Image Segmentation and Limitations Figure 2.4: From left to right: Three steps of the interactive random walker segmentation. We show the seeds and the image to segment in the upper row and the segmentation corresponding to this particular choice of the seeds in the lower row. The addition of seed regions for the object and the background yield an iterative refinement of the segmentation. 2.3 Mumford-Shah and Ambrosio-Tortorelli Segmentation The minimization of a functional, as seen in the random walker segmentation, is a common technique for segmentation problems. The next method, the Mumford-Shah segmentation, bases on the minimization of a functional, too. The Mumford-Shah functional is not as easy as the random walker functional, because the Mumford-Shah functional involves two unknowns, the image and an additional edge set. This leads to a couple of mathematical problems for the theoretical proof of existence and uniqueness of minimizers and it is hard to discretize the Mumford-Shah functional directly. We avoid the numerical problems by introducing the Ambrosio-Tortorelli approximation that Γ-converges to the Mumford-Shah functional [14]. Mumford and Shah [107] proposed to minimize the functional ∫ ∫ E MS (u,K) := (u − u 0 ) 2 dx + µ |∇u| 2 dx + νH d−1 (K) , (2.14) D\K where u 0 : D → IR is the initial image, u : D → IR is an image that is smooth and differentiable in D\K, K ⊂ D a set of discontinuities, µ,ν are nonnegative constants, and H d−1 (K) is the d −1-dimensional Hausdorff measure of the edge set K. The aim is to find an image u and a set K such that the functional is minimal. Roughly speaking, the minimizer u must be an image, which is close to the initial u 0 away from the edges (then ∫ D\K (u − u 0) 2 dx is small) and smooth away from the edges (then ∫ D\K |∇u|2 dx is small). Moreover, the length of the edge set K must be small (then H d−1 (K), measuring the length of the edge set, is small). The direct minimization of the Mumford-Shah energy is difficult due to the different nature of u and K: u is a function and K is a set. In addition, the proof of existence of a D\K 12
2.3 Mumford-Shah and Ambrosio-Tortorelli Segmentation minimizer is a challenging problem, cf. [35]. Since the functional is not differentiable, the estimation of minimizers based on the Euler-Lagrange equations is impossible. Instead, researchers proposed regularized approximations (see [17]). The following paragraph summarizes one of these methods, proposed by Ambrosio and Tortorelli [14]. Remark 4. All components of the Mumford-Shah functional are essential to get a segmentation of the image u, i.e. it is impossible to omit one of the components to end up with a mathematically and numerically easier problem. If we omit the first component, we have no control over the difference between the image and the smooth approximation and u = 0, K = /0 minimize the remaining parts. We obtain another trivial solution if we omit the second component: Now u = u 0 and K = /0 minimize the functional. When omitting the last component, K = D minimizes the functional. Thus, the Mumford- Shah functional contains the minimal number of components necessary for segmentation, and it is essential to discretize them well to get meaningful numerical solutions. 2.3.1 Ambrosio-Tortorelli Segmentation As already mentioned, the Ambrosio-Tortorelli segmentation [14] is a kind of regularization of the Mumford-Shah functional. Ambrosio-Tortorelli segmentation uses a function φ : D → IR, the phase field, instead of the edge set K. The phase field is a smooth indicator function of the edge set K. It is zero on the edge set K and goes smoothly to one away from the edge set. An additional variable ε controls the width of the transition zone. When ε goes to zero, the phase field goes to the characteristic function of the edge set. In a following section, we will recap that the Ambrosio-Tortorelli energy converges in the Γ-sense to the Mumford-Shah energy [14]. The idea of the Ambrosio-Tortorelli segmentation for a given initial image u 0 is to find a phase field φ and a smooth image u minimizing the energy where E AT [u,φ] := Efid,u ε [u] + Eε reg,u[u,φ] + Ephase ε [φ] , (2.15) E ε fid,u [u] = ∫ D ∫ Ereg,u[u,φ] ε = ∫ Ephase ε [φ] = D D 1 2 (u − u 0) 2 dx µ ( φ 2 ) + k ε |∇u| 2 dx ( νε|∇φ| 2 + ν 4ε (1 − φ)2) dx . (2.16) The first energy, the fidelity energy, ensures closeness of the smoothed image to the original u 0 . The second energy, the regularization energy, measures smoothness of u apart from areas where φ is small (the edges), and enforces φ to be small close to edges. The parameter k ε ensures coerciveness of the differential operator and existence of solutions, because φ 2 may vanish. The third energy, the phase energy, drives the phase field towards one and ensures small edge sets via the term |∇φ| 2 . The parameter ε controls the scale of the detected edges, µ the amount of detected edges, and ν the behavior of the phase field. k ε is a small regularization parameter. The relation between the first two components of the Ambrosio-Tortorelli and the Mumford-Shah energy are obvious. The third component, the phase energy, is a combination of a term forcing φ to be one and the term ∫ D ε|∇φ|2 . In the limit ε → 0, it can be shown to be equal to H d−1 (K) by using the co-area formula [105]. A minimizer of this energy is an image that is flat away from edges and a phase field, which is close to zero at edges only. To obtain a minimizer of an energy, a widely used technique is to solve the Euler-Lagrange equations resulting from this energy. For the computation of the Euler- Lagrange equations, we have to compute the first variation of the above energies using the Gâteaux 13
Page 1: Segmentation of Stochastic Images u
Page 5: Abstract The task of segmentation,
Page 8 and 9: 7.5 Segmentation of Stochastic Imag
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
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Page 58 and 59: Chapter 5 Stochastic Images
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6.1 Random Walker Segmentation on S
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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 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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