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Chapter 7 Stochastic Level Sets rarefaction fan shock E Var E Var PC SC MC MCL Figure 7.3: Comparison of the expected value and variance of the resulting phase field for the rarefaction fan and the shock, two classical tests for level set propagation. The figure shows the comparison of the four discretizations of the stochastic phase field equation. preconditioned phase field (MC), and a Monte Carlo simulation of the original level set implementation (MCL). The comparison is performed on two typical tests for level set evolution, the evolution of a cosine curve in the inward and outward direction and the evolution of an edge of a square in the inward and outward direction. Furthermore, we demonstrate the extension of the proposed method to three spatial dimensions on the Stanford bunny data set [149]. In contrast to other publications dealing with mean curvature motion [138], we use the Stanford bunny and apply the preconditioned phase field equation with stochastic speed on it. In all numerical experiments, we set W = 2.5h, where h is the grid spacing and in the absence of a curvature dependent speed, we set b = 1.25h. For the evolution of the cosine (see Fig. 7.2), the challenge is the development of a shock [138], when the curve moves inward. Due to the stochastic velocity, we used a uniformly distributed speed a with E(a) = 1.0 and Var(a) = 0.04, i.e. a ∼ U [1 − 0.2 √ 3,1 + 0.2 √ 3], the position of the shock is uncertain, and the discretization has to be adequate in a vicinity of the possible shock positions. For the numerical experiments, we use a spatial resolution of 129 × 129, a polynomial chaos in one 86
7.4 Numerical Experiments Figure 7.4: Expected value color-coded by the variance for the Stanford bunny after shrinkage under an uncertain speed in the normal direction. Red indicates regions with a high variance and green regions with low variance. In addition, we show one slice of the variance. random variable with order two and apply 30 steps with step size 0.1h. Furthermore, we computed 20 time steps of the reinitialization equation (7.24) with step size 0.2h after every time step. The polynomial chaos coefficients of the speed are set to a 1 = 1, a 2 = 0.2, and a 3 = 0, such that the expansion fulfills E(a) = 1 and Var(a) = 0.04. It is visible from Fig. 7.2 that the methods based on the stochastic preconditioned phase field formulation lead to the same results. Only the discretization of the level set method leads to deviating results. This is due to the reinitialization of the level set via Fast Marching [138]. The Fast Marching method assumes that the level set values at a grid point near the interface, the trial nodes (see [138]) is the signed distance to the interface. This is not true in the presence of a shock due to the crossing of the zero level set. For deterministic level sets this error can be neglected, even for stochastic level sets the expected value is accurate. However, for the stochastic part of the solution (the variance in Fig. 7.2), which is orders of magnitude smaller than the expected value, this error becomes relevant. Thus, it is more precise to use the reinitialization via (7.24) in the presence of a shock. When the interface moves outward, we have a rarefaction fan (see [138]), because one point on the zero level set is the closest point for multiple points away from the interface. The same problem as for the shock arises and again, the reinitialization via (7.24) is the better method. The second test is the evolution of one edge of a square in the inward and outward direction depicted in Fig. 7.3. Again, we have the development of a shock when the curve moves inward and of a rarefaction fan when the curve moves outward. The last test is the contraction of the Stanford bunny under an uncertain velocity. Again, we used the speed E(a) = 1.0 and Var(a) = 0.04 from the previous tests. Fig. 7.4 shows the results. For the Stanford bunny, we have the evolution of a 3D object. We use a method for the visualization of 3D stochastic images from Section 5.4 by visualizing the expected value color-coded by the variance. As expected, we see a high variance of the contour in regions with high curvature. This is due to the development of shocks, when the contour moves inside these regions. 87
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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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