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Chapter 6 Segmentation of Stochastic Images Using Elliptic SPDEs Figure 6.10: Nonzero pattern of the SFEM matrix for the smoothed stochastic image using n = 5 random variables and a polynomial degree p = 3. A black dot denotes a block that has a nonzero stochastic part, thus having the sparsity structure of a classical FEM matrix. for all β ∈ {1,...,N}, where M α,β ,L α,β ,S α,β and T α,β are blocks of the system matrix, defined as (M α,β ) i, j = E (Ψ α Ψ β ) ∫ (S α,β ) i, j = E (Ψ α Ψ β ) ∫ D D P i P j dx , ∇P i · ∇P j dx (6.26) and ( ( L α,β ) i, j = ∑ k T α,β ) i, j = ∑ k ( ∑E Ψ α Ψ β Ψ γ) ∫ (˜φ 2 ) k γ γ ∫ ∑E γ ( Ψ α Ψ β Ψ γ) u k γ D D ∇P i · ∇P j P k dx , P i P j P k dx . (6.27) Here, (˜φ 2 ) k γ denotes a coefficient of the polynomial chaos expansion of the Galerkin projection of φ 2 onto the image space (cf. Section 3.3.3). The right hand side vector of the phase field equation is ∫ (A α ) i = Γ ∫ Ψ α dΠ D ⎧ ∫ 1 ⎪⎨ 4ε P i dx = D ⎪⎩ 1 4ε P i dx if α = 1 , 0 else . (6.28) Note that the expectations of the products of stochastic basis functions involved above are again the components of the lookup table introduced in Section 3.3.5. The deterministic integrals can be precomputed, because they are needed several times during the assembling of the system matrix. Analog to the classical finite element method the systems of linear equations can be treated by an iterative solver like the method of conjugate gradients [67]. The general block structure of an SFEM matrix is depicted in Fig. 6.9 and the sparsity structure on the block level for five random variables and a polynomial degree of three is depicted in Fig. 6.10. In addition, the matrix generation can be accelerated using lookup tables. The memory consumption is enormous, because the matrix has N 2 -times the storage requirement of the deterministic matrix, where N is the dimension of the polynomial chaos. Thus, we use the GSD method for the solution to avoid the generation of the SFEM matrix. 70
6.2 Ambrosio-Tortorelli Segmentation on Stochastic Images Stochastic Generalization of the Edge Linking Step The edge linking step from Section 2.3 can be applied on the stochastic Ambrosio-Tortorelli model, too. We introduce an additional coefficient c for the image equation. This coefficient is a random field, i.e. c ∈ H 1 (D) × L 2 (Ω). The modified image equation in the stochastic context reads −∇ · (µc(x,ξ )(φ(x,ξ ) 2 + k ε )∇u(x,ξ ) ) + u(x,ξ ) = u 0 (x,ξ ) . (6.29) The random field c is composed of the stochastic generalizations of the edge continuity and the edge consistency step. Thus, c is c(x,ξ ) = c dc (x,ξ ) · c h (x,ξ ) , (6.30) whereas these quantities are ( (c dc (ξ )) i = ζ (ξ ) dc) + 1 − ( ζ (ξ ) dc) i i φ(ξ ) i ( ( )) ( ζ (ξ ) dc) = exp ε dc 1 i |η s | ∑ ∇u i (ξ ) · ∇u j (ξ ) − 1 j∈η s 1 (c h (ξ )) i = 1 + α ( ) φ(ξ ) i − φ(ξ ) 2 . i (6.31) To calculate c dc and c h , it is necessary to use the calculations for random variables approximated in the polynomial chaos presented in Section 3.3.3. The only quantity for which a stochastic generalization is not obvious is the orthogonal edge direction ∇u ⊥ i . This direction is needed, because we have to sum up over pixels perpendicular to the image gradient in the second equation of (6.31). This perpendicular direction is also a stochastic quantity, but it is not possible to sum up in a stochastic direction. To overcome this, we use the direction E ( (∇u) ⊥) and neglect the error due to the inaccurate direction. This is similar to the upwinding problem for stochastic equations [92]. Remark 14. Erdem et al. [49] proposed additional feedback measures for textures and the local scale. These measures are not included here, but can be generalized in a similar fashion. 6.2.3 Results In the following, we demonstrate the performance and advantages of the stochastic extension of the Ambrosio-Tortorelli segmentation approach. We use three data sets which cover a broad range of possible input data. Furthermore, we compare the results of the stochastic Ambrosio-Tortorelli model with the adaptive extension for the spatial dimensions and the stochastic version of the edge linking step. Thus, the organization of this section is the following: First, we demonstrate the method on three data sets. Then we show the results of the combination of the stochastic method with an adaptive grid approach for the spatial dimensions. Finally, we demonstrate that the stochastic method benefits from the idea of edge linking [49]. The first input image data set consists of M = 5 samples from the artificial “street sequence” [99]. The second data set consists of M = 45 image samples from ultrasound (US) imaging of a structure in the forearm, acquired within two seconds. The third data set contains ten images of a scene acquired with a digital camera 2 . Note that we do not consider the street sequence as an image sequence here. Instead, we use five frames as samples of the noisy and uncertain acquisition of the same object. From the samples, we compute the polynomial chaos representation using n = 5 (digital camera), n = 10 (US), respectively n = 4 (street scene) random variables with the method described 2 Thanks to PD Dr. Christoph S. Garbe for providing the data set. 71
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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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Chapter 2 Image Segmentation and Li
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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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