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Chapter 6 Segmentation of Stochastic Images Using Elliptic SPDEs Figure 6.7: Left: The object seed points (yellow) and background seed points (red) used as initialization of the stochastic random walker method. Right: The MC-realizations of the stochastic segmentation result differ significantly for different noise realizations. The other possibility to compute the volume PDF from the stochastic result is inspired by the classical method to compute the random walker result. We generate samples from the stochastic segmentation result via Monte Carlo sampling and estimate the volume of the object given by pixels with value above 0.5 on every sample. Fig. 6.8 compares the two approaches for the computation of the object’s volume. Having in mind that the “real” object volume is 60 pixels, both methods slightly overestimate the object’s volume, but the real object volume is close to the expected value (60.39 for the summation of the random variables and 60.83 for the object thresholding) of both PDFs. Figure 6.8: The PDF for both possibilities of the volume computation, the summation of the random variables (gray) and the thresholding (black). The true volume is 60 pixels. 66
6.2 Ambrosio-Tortorelli Segmentation on Stochastic Images 6.2 Ambrosio-Tortorelli Segmentation on Stochastic Images In the following, we focus on the combination of the notion of stochastic images with the segmentation approach in the spirit of Ambrosio and Tortorelli [14]. Section 2.3 introduced this approach. The author published the stochastic Ambrosio-Tortorelli extension in [3]. For the segmentation of stochastic images by the phase field approach of Ambrosio and Tortorelli, we replace the deterministic u and φ by their stochastic analogs. The stochastic energy components are then defined as the expectations of the classical energy components (cf. Section 2.3), i.e. ∫ Efid s (u) := E(E fid) = E s reg(u,φ) := E(E reg ) = E s phase (φ) := E(E phase) = ∫ Γ D ∫ ∫ Γ D ∫ ∫ Γ (u(x,ξ ) − u 0 (x,ξ )) 2 dxdΠ and we define the stochastic energy as the sum of these, i.e. D µ (φ (x,ξ ) 2 + k ε ) |∇u(x,ξ )| 2 dxdΠ ν ε |∇φ(x,ξ )| 2 + ν 4ε (1 − φ (x,ξ ))2 dxdΠ (6.13) EAT(u,φ) s = Efid s (u) + Es reg(u,φ) + Ephase s (φ) . (6.14) The Euler-Lagrange equations of the stochastic Ambrosio-Tortorelli energy are obtained from the first variation of (6.13). Since the stochastic energies (6.13) are the expectations of the classical energies (2.16), the computations are analog. For example, we get for a test function θ : D × Γ → IR d ∣ dt Es fid (u +t θ) ∣∣t=0 = d ∫ ∫ ( ) 2 ∣ ∣∣t=0 u(x,ξ ) +tθ(x,ξ ) − u 0 (x,ξ ) dx dΠ dt Γ D ∫ ∫ ( ) = 2 u(x,ξ ) − u 0 (x,ξ ) θ(x,ξ ) dx dΠ . Γ D (6.15) With analog computations for the remaining energy contributions, we arrive at the following system of SPDEs: We seek for u,φ : D × Γ → IR as the weak solutions of −∇ · (µ(φ(x,ξ ) 2 + k ε )∇u(x,ξ ) ) + u(x,ξ ) = u 0 (x,ξ ) ( 1 −ε∆φ(x,ξ ) + 4ε + µ ) 2ν |∇u(x,ξ )|2 φ(x,ξ ) = 1 4ε . (6.16) This system is analog to the classical system (2.18) in which stochastic images replace the classical images. The equations are SPDEs, because the coefficients φ(x,ξ ) 2 and |∇u(x,ξ )| 2 are random fields. Moreover, the right hand side of the first equation, u 0 (x,ξ ), is a random field. We use random fields from the tensor product space H 1 (D) ⊗ L 2 (Ω). This space enables us to use finite elements for the discretization of the spatial part and the polynomial chaos expansion for the stochastic part. Remark 13. Recently, Krajsek et al. [86] developed an extension of the Ambrosio-Tortorelli model based on Bayesian estimation theory [77]. This concept is related to the approach presented here, but limited to Gaussian random variables, whereas the approach presented here deals with arbitrary distributions with finite variance. The investigation of a link between the approaches is future work. 67
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Segmentation of Stochastic Images u
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Abstract The task of segmentation,
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7.5 Segmentation of Stochastic Imag
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Chapter 1 Introduction The developm
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Figure 1.3: This thesis combines fi
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the presentation of the stochastic
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Chapter 2 Image Segmentation and Li
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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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