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Chapter 6 Segmentation of Stochastic Images Using Elliptic SPDEs Figure 6.3: MC-realizations of the stochastic object boundary for the stochastic liver image segmented with the stochastic random walker approach with β = 10. On the right we highlight a region of the image, where the noise in the input image influences the result. Remark 11. The classical random walker result can be interpreted as a probability map, i.e. the result of random walker segmentation is a probability for every pixel to belong to the object or not. When we apply the stochastic random walker method, the first interpretation of the result is that we compute “probabilities of probabilities”, because we are computing the probability distribution of the values at every pixel. Let us emphasize that the probability interpretation of the classical result is one possible interpretation only. Mathematically, we are computing the result of a diffusion problem, and the stochastic extension is, in this interpretation, a diffusion problem with stochastic diffusion coefficient. The results can be interpreted as the stochastic solution of the stochastic diffusion problem. The analog to the probability interpretation is that we computed the probabilities for belonging to the object in dependence on the input gray value uncertainty. The stochastic object boundary can be visualized by tracking the deterministic object boundary (the value 0.5 in the result image) for realizations of the random variables. The work of Prassni et al. [126] inspired this kind of visualization. The difference is that Prassni et al. [126] visualized the iso-lines of different probabilities, whereas we visualize the same iso-line for realizations of the stochastic image. Fig. 6.3 shows the result of such a visualization. It is possible to compute and visualize other quantities extracted from the segmentation, e.g. the volume of the segmented object. The obvious visualization of the stochastic volume is to draw the PDF of the volume. The PDF of the segmented volume can be computed from the segmentation by summing up the random variables at every pixel, because they specify the “probability” that the pixel belongs to the object. Thus, the random variable v(ξ ) specifying the objects volume is v(ξ ) = N ∑ α=1 v α Ψ α (ξ ) := ∑ x i (ξ ) . (6.12) i∈I Having a look at the PDF, it is easy to decide whether the image noise influences the segmented volume strongly or not. If the segmented volume is strongly influenced the PDF is broad, otherwise the function is narrow. Fig. 6.4 shows the PDF of the segmented volume from the street image. Moreover, the choice of the parameter β influences the profile of the PDF. A smaller β leads to a diffuse object boundary and to a broader PDF. 62
6.1 Random Walker Segmentation on Stochastic Images Figure 6.4: PDF of the area of the segmented person from the street image for β = 25 (black) and β = 50 (gray). From the PDF we judge the reliability of the segmentation, a narrow PDF indicates that the image noise influences the segmentation marginally. Remark 12. Another possibility to calculate the object volume is to count pixels with a value above 0.5 only. Thus, we compute image samples from the stochastic result via a Monte Carlo approach, threshold these samples, count the number of object pixels, and calculate the volume PDF. This method is time-consuming. The proposed method has the advantage to include the partial volume effect [21] at the boundary, because it considers pixels with a probability less than 0.5 partially. 6.1.3 Comparison with Monte Carlo Simulation and Stochastic Collocation To verify the intrusive solution of the resulting SPDE via polynomial chaos, stochastic finite elements, and the GSD method, we compared this solution with the solutions obtained via Monte Carlo sampling and a stochastic collocation approach. Fig. 6.5 shows the comparison of the expected value and the variance computed via GSD, stochastic collocation, and Monte Carlo sampling. The small difference between the variances of the three solutions might be due to the projection of the Laplacian matrix on the polynomial chaos. However, the great benefit of the GSD method is the significantly better performance. We now investigate this in detail. 6.1.4 Performance Evaluation Due to the availability of the implementation possibilities for the solution of SPDEs, we are able to compare the execution times of the approaches. We did the detailed comparison for the random walker segmentation in this thesis only, but the results generalize to the Ambrosio-Tortorelli approach, because it uses the same methods. Table 6.1 shows the comparison of the execution times of the GSD method, the Monte Carlo method, and the stochastic collocation method with Smolyak and full grid. It is easy to see that the GSD method outperforms the sampled based approaches. This supports the decision to prefer the GSD method and the finite difference method for random variables throughout this thesis. The stochastic collocation methods suffer from the “curse of dimension” [119], because the execution times grow exponentially with the number of random variables in the stochastic images. 63
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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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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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