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Chapter 7 Stochastic Level Sets and Beckermann [143] based on an idea by Echebarria et al. [46]. It is given by ( ) ∇ψ ∇ · |∇ψ| ⎛ ≈ 1 ⎝ h ψ i+1, j − ψ i, j √ (ψ i+1, j − ψ i, j ) 2 + (ψ i+1, j+1 + ψ i, j+1 − ψ i+1, j−1 − ψ i, j−1 ) 2 /16 ψ i, j − ψ i−1, j − √ (ψ i, j − ψ i−1, j ) 2 + (ψ i−1, j+1 + ψ i, j+1 − ψ i−1, j−1 − ψ i, j−1 ) 2 /16 + √ ψ i, j+1 − ψ i, j (ψ i, j+1 − ψ i, j ) 2 + (ψ i+1, j+1 + ψ i+1, j − ψ i−1, j+1 − ψ i−1, j ) 2 /16 ⎞ −√ ψ i, j − ψ i, j−1 ⎠ . (ψ i, j − ψ i, j−1 ) 2 + (ψ i+1, j−1 + ψ i+1, j − ψ i−1, j−1 − ψ i−1, j ) 2 /16 (7.23) Due to the independence of the update for the spatial positions, this finite difference scheme can be parallelized on multiple processor cores easily. Remark 17. Due to the hyperbolic tangent profile of the level sets across the interface, we have to respect a condition on the maximal curvature of the represented object. For a high curvature, the hyperbolic tangent profiles overlap for points on the interface. This leads to instabilities of the numerical schemes for the discretization. 7.3 Reinitialization of Stochastic Level Sets The right hand side of (7.18) is an integrated reinitialization of the level set function. Following [143], this reinitialization is sufficient to get accurate results for deterministic level sets. When using a stochastic velocity, we have to reinitialize all polynomial chaos coefficients, which are on different scales. Typically, the first coefficient, the expected value, is orders of magnitude bigger than the remaining coefficients. Furthermore, the coefficients of polynomials in uncoupled random variables are close to zero. During the numerical experiments, we observed that the reinitialization via (7.18) is not sufficient. Thus, we need an additional reinitialization to get accurate stochastic results. The classical reinitialization methods for level sets are not applicable in the stochastic context. The Fast Marching method [138] is based on an upwinding scheme. As discussed in Section 7.1, a stochastic upwinding scheme is not available. Iterative reinitialization via φ t = sign(φ)(1 − |∇φ|) is not possible because the signature of a stochastic quantity is not well-defined. The equation for energy minimization [90], i.e. α|E(|∇φ| − 1) 2 | + β|Var(|∇φ|)| → min, is unstable if φ converges to the stochastic signed distance function. To get a working reinitialization scheme for stochastic level sets, we use a modification of the stochastic level set equation (7.18). As already mentioned, the right hand side of this function is an integrated reinitialization. We use this equation, set the speed to zero, i.e. a = 0, and solve the equation for artificial time T . Doing this, the reinitialization equation is ( ( 1 − |∇ψ| 2 )√ 2 ψ t = b ∆ψ + tanh ψ ( ) ) ∇ψ √ − |∇ψ|∇ · . (7.24) W 2W |∇ψ| In all numerical experiments, we apply this reinitialization equation. We use for every time step of (7.18) ten reinitialization time steps of (7.24) with a time step size for the reinitialization of 0.5τ, where τ is the time step size of the original problem. 84
7.4 Numerical Experiments cosine inward cosine outward E Var E Var PC SC MC MCL Figure 7.2: Comparison of expected value and variance of the resulting phase field for the cosine test of (7.18) using the polynomial chaos (PC), stochastic collocation (SC), Monte Carlo simulation (MC), and Monte Carlo simulation of the original level set equation (MCL). 7.4 Numerical Experiments In this section, we present numerical experiments for the verification of the proposed algorithm and for the implementation of the algorithm. To validate the intrusive implementation in the polynomial chaos, we verify the results with Monte Carlo experiments and a stochastic collocation approach. To show that the phase field equation is comparable with the native level set equation, we added a Monte Carlo experiment based on the original level set equation φ t + a|∇φ| = 0 . (7.25) We are able to compare four implementations of the stochastic level set evolution: The intrusive implementation of the preconditioned phase field in the polynomial chaos (PC), a stochastic collocation approach based on the preconditioned phase field (SC), a Monte Carlo simulation of the 85
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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 3 SPDEs and Polynomial Chao
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