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Chapter 2 Image Segmentation and Limitations Figure 2.5: Left: The initial (noisy) US image treated as input for the Ambrosio-Tortorelli approach. Middle: The smooth Ambrosio-Tortorelli approximation of the initial image. Right: The corresponding phase field, i.e. the approximation of the edge set of the smoothed image. derivatives [17]. In the following θ : D → IR is a test function. For the fidelity energy, we get: d dε E ∫ f id[u + εθ] ∣ = 2(u + u 0 )θ dx . (2.17) ε=0 For the other energies, we get similar results. The Euler-Lagrange equations of (2.15) are D −∇ · (µ(φ 2 + k ε )∇u ) + u = u 0 ( 1 −ε∆φ + 4ε + µ ) 2ν |∇u|2 φ = 1 4ε . (2.18) This is a system of two coupled elliptic PDEs. We seek u,φ ∈ H 1 (D) as the weak solutions of these Euler-Lagrange equations. An implementation solves both equations alternately, letting either u or φ vary alternatingly until they reach a fixed point as the joint solution of both equations. Fig. 2.5 shows an exemplary result of the Ambrosio-Tortorelli segmentation approach. 2.3.2 Γ-Convergence As already stated, the Ambrosio-Tortorelli functional approximates the Mumford-Shah functional. We show that the Ambrosio-Tortorelli functional converges in a variational sense towards the Mumford-Shah functional. This variational convergence is called Γ-convergence [14]: Definition The sequence of functionals F n : X → IR Γ-converges to the functional F if 1. For every x ∈ X and for every sequence x n converging to x ∈ X, F(x) ≤ liminf n→∞ F n(x n ) . (2.19) 2. For every x ∈ X there exists a sequence x n converging to x ∈ X such that F(x) ≥ limsupF n (x n ) . (2.20) n→∞ The proof of the Γ-convergence of a function sequence consists of two steps: First, we have to prove (2.19) for all sequences and then we have to construct a sequence fulfilling (2.20). This last step is the challenging task when proving Γ-convergence [14]. Using the definition of Γ-convergence and the space GSBV introduced in Section 2.1, the following theorem from [14, 17] holds. 14
2.3 Mumford-Shah and Ambrosio-Tortorelli Segmentation Theorem 2.1. Define Ẽ AT : L 1 (D) × L 1 (D) → IR + by { EAT (u,φ) if (u,φ) ∈ H Ẽ AT (u,φ) := 1 (D) × H 1 (D),0 ≤ φ ≤ 1 +∞ otherwise (2.21) and G : L 1 (D) × L 1 (D) → IR + by { EMS (u) if u ∈ GSBV(D) and φ = 1 almost everywhere G(u,φ) = +∞ otherwise. (2.22) If k ε = o(ε), then Ẽ AT Γ-converges to G(u,φ) for ε → 0. The convergence of the Ambrosio-Tortorelli energy towards the Mumford-Shah energy enables us to use the coupled pair of PDEs obtained as Euler-Lagrange equations of the Ambrosio-Tortorelli energy and to solve this with very small ε. The result is a phase field that is close to the characteristic function of the edge set of the Mumford-Shah functional. 2.3.3 Edge Continuity and Edge Consistency The classical Mumford-Shah model and the Ambrosio-Tortorelli approximation lack a step linking edges. This step is necessary to enforce the detection of closed contours in the images. Otherwise, the appearance of partially detected, breaking up contours is possible, see Fig. 2.5. For example, Erdem et al. [49] introduced such a step for the Ambrosio-Tortorelli model. The idea is to use a modified diffusion coefficient in the image equation. This modified coefficient does not depend on the phase field exclusively, but contains information about the continuity and directional consistency of the detected edges. To be more precise, Erdem et al. [49] proposed to use the equation −∇ · (µ((cφ) 2 + k ε )∇u ) + u = u 0 , (2.23) instead of the first equation of (2.18). The additional factor c is the product of the two factors from the directional consistency c dc and the edge continuity c h , i.e. c = c dc · c h . (2.24) If c < 1, the diffusivity decreases, allowing to form new edges in the image, whereas c > 1 leads to an increased diffusivity, allowing to smooth away unwanted edges. Directional Consistency The directional consistency tries to judge the quality of the detected edges based on information from surrounding pixels. The idea is that an edge is reliable if the gradients of the image for pixels in directions perpendicular to the edge are in parallel. For inaccurately detected edges, e.g. due to noise, these gradients are typically not aligned. To do so, Erdem et al. [49] introduced (c dc ) i = ζ dc i + 1 − ζ i dc (2.25) φ i for all pixels i ∈ I , where ζi dc measures the alignment of the gradients. This factor increases the diffusion, if the image gradients around the detected edge are not aligned. As the feedback measure for the alignment of the gradients they proposed to use ( ( )) 1 ζi dc = exp ε dc |η s | ∑ ∇v j∈η s i · ∇v j − 1 , (2.26) 15
Page 1: Segmentation of Stochastic Images u
Page 5: Abstract The task of segmentation,
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Page 18 and 19: Chapter 2 Image Segmentation and Li
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6.1 Random Walker Segmentation on S
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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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