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Chapter 4 Discretization of SPDEs Figure 4.5: For an unsaturated error indicator, the appearance of hanging nodes constrained by hanging nodes (due to level transitions of more than one between neighboring elements) is possible (left). The saturation of the error indicator ensures that there are level one transitions between neighboring elements only (right). This saturation condition ensures that there is a level one transition between neighboring elements only. Furthermore, we have to avoid the refinement of coarsened elements. Otherwise it is possible to end up in a situation where an element is refined in step n, coarsened in step n + 1 and so on. A slightly modified error indicator ˜S, which we define as the minimum of the actual error indicator and the error indicator of the previous iteration, achieves this. Alternatively, the refinement of coarsened elements can be avoided by using different thresholds for coarsening and refinement [34]. 4.5.1 Combining GSD and Adaptive Grids The combination of adaptive grids with the GSD method is straightforward. We assemble the stochastic matrices in the same way as the deterministic matrices. After the solution of the system is available, we interpolate the values on the hanging and inactive nodes. The only difficulty arises for the generation of the equation system for the new stochastic basis element (equation (4.30) respectively line 7 of the algorithm). There we have to compute the scalar product 〈U i ,AU i 〉 using the adaptive matrix A. The product AU i has a different weight for the constraining nodes than the vector U i , because the matrix has additional weights from the hanging nodes at the constraining nodes. We propose to add these weighting factors to the vector U i . Conclusion We presented methods for the discretization of SPDEs. Based on sampling strategies we presented Monte Carlo simulation and stochastic collocation with full or sparse grids constructed via Smolyak’s algorithm. This thesis uses the sampling based approaches to verify the implementations of the intrusive methods. Intrusive methods do not use a sampling strategy to solve the SPDEs. Instead, they are based on a development of numerical schemes acting on random variables. Intrusive methods are the key to the efficient numerical solution of SPDEs arising in image processing, because other methods are orders of magnitude too slow or provide inaccurate results after an adequate period. We presented the SFEM and the GSD method that tries to speed up the solution process on the SFEM by constructing an optimal, problem dependent, subspace. With this chapter, we have the fundamentals at hand to develop the concept of stochastic images and to design image processing operators acting on these stochastic images. 46
Chapter 5 Stochastic Images As described in Section 2.5, noise corrupts classical images. The repeated acquisition of the same scene does not give identical images, because the noise typically is a stochastic quantity. Furthermore, applying segmentation methods to two randomly chosen samples from the same scene yield different results due to the noise. To model the noise of the acquisition process, we identify pixels by random variables, i.e. identify images by random fields. Assuming that these stochastic images fulfill mild regularity assumptions (H 1 -regularity in the spatial dimensions and L 2 -regularity in the stochastic dimensions), they are elements of the tensor product space H 1 (D) ⊗ L 2 (Ω) introduced in Section 3.1. We discretize this tensor product space using the polynomial chaos expansion introduced in Section 3.3 and finite elements or finite differences for the spatial dimensions. This chapter combines the methods presented so far to introduce the concept of stochastic images. Preusser et al. [130] introduced stochastic images, but used a pointwise product space, which is a subspace of the tensor product space H 1 (D) ⊗ L 2 (Ω). We compare both approaches at the end of this chapter. 5.1 Polynomial Chaos for Stochastic Images It is popular in PDE based image processing to model an image f : D → IR on a domain D ⊂ IR d , d = 2,3 using a finite element space and a representation f (x) = ∑ i∈I f i P i (x) , (5.1) where f i ∈ IR is the value of the ith pixel from the pixel set I and P i the shape function (e.g. tent function) of the ith pixel (see e.g. [17]). In a stochastic image, a single pixel no longer has a fixed value. Instead, it depends on a vector of random variables ξ (ω) = (ξ 1 (ω),...,ξ n (ω)) and on a random event ω ∈ Ω. Note that it is possible to combine the concept of stochastic images with other spatial discretizations, e.g. finite difference schemes. Then we have a pointwise representation f (x i ,ξ ) and apply an interpolation rule for positions located between pixel positions. Following [130], we obtain the representation of an image whose pixel values are random variables from (5.1) by replacing the fixed f i by random variables f i (ξ ): f (x,ξ ) = ∑ i∈I f i (ξ )P i (x) . (5.2) Fig. 5.1 shows a schematic sketch of this idea. Note that we omit denoting the dependence of ξ on ω to simplify the notation. The polynomial chaos expansion (3.31) approximates any second order random variable f i (ξ ) by a weighted sum of orthogonal multidimensional polynomials. This yields f (x,ξ ) = ∑ i∈I ∑ N α=1 f i αΨ α (ξ )P i (x) (5.3) as the representation of stochastic images, i.e. images whose pixels are random variables, discretized using finite elements for the spatial dimensions. Using finite differences, the value at a pixel is f (x i ,ξ ) = ∑ N α=1 f i αΨ α (ξ ) (5.4) 47
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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
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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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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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