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Chapter 7 Stochastic Level Sets Figure 7.1: Stochastic level sets do not have a fixed position where φ(x) = 0. Instead, there is a band with positive probability that the level set is equal to zero, i.e. the position of the zero level set is random and it is possible to estimate the PDF of the interface location in the normal direction of the expected value of the interface (lower right corner). which ensures that ψ is a signed distance function to the interface because of φ = −tanh √ 2W , we get the final version of the stochastic level set equation ( ( 1 − |∇ψ| 2 )√ 2 ψ t + a|∇ψ| = b ∆ψ + tanh ψ ( ) ) ∇ψ √ − |∇ψ|∇ · , (7.18) W 2W |∇ψ| where we omitted the dependence of the function ψ on time t, spatial position x, and random event ω. Following [143], where the deterministic equivalent of this equation was derived, the right hand side of this function serves as an integrated reinitialization scheme for the level set ψ. Thus, further reinitialization is not required for deterministic level sets. 7.1.1 Interpretation of Stochastic Level Sets Having (7.18) at hand, we have to interpret the result of the level set motion with random speed. Due to the random variable/field that controls the speed of the level set motion the position of the zero (and all other) level sets is a random quantity, too. A possibility to estimate the influence of the random speed component on the level set motion is to calculate the probability that the zero level set is at a specific position. Furthermore, we can calculate the whole band with positive probability that the zero level set is located there, i.e. where P(φ(x) = 0) > 0 (7.19) holds. In the normal direction of the expected value E(φ) = 0 of the zero level set location, we can estimate the PDF of the interface position (see Fig. 7.1). Remark 16. Using Gaussian random variables in combination with stochastic level sets, we end up with a nonzero probability for the interface location in the whole domain. This is due to the infinite support of Gaussian random variables. Thus, we limit the following investigations to a polynomial chaos in uniform random variables. We denote a random variable X uniformly distributed in the interval [a,b] by X ∼ U [a,b]. Uniform random variables have a compact support, leading to a band with finite thickness for the potential interface location. n 82
7.2 Discretization of the Stochastic Level Set Equation Stochastic Signed Distance Functions It is desirable to use signed distance functions as level sets in the stochastic context, too. A stochastic signed distance function has to be a classical signed distance function for every realization ω ∈ Ω. Theorem 7.1. A stochastic signed distance function fulfills E(|∇φ|) = 1 and Var(|∇φ|) = 0. Proof. The first property is ensured by the second property by ∫ ∫ E(|∇φ(x)|) = |∇φ(x,ω)|dω = 1 dω = 1 , (7.20) Ω Ω ∫ ∫ Var(|∇φ(x)|) = (|∇φ(x,ω)| − 1) 2 dω = 0 dω = 0 . (7.21) Ω Ω 7.2 Discretization of the Stochastic Level Set Equation For the numerical tests, we discretize (7.18) using the explicit Euler scheme for the discretization of the time derivative. The time discrete version of (7.18) is ( ( √ 2 ψ t+τ = ψ t +τ −a|∇ψ t | + b ∆ψ t + W ( 1 − |∇ψ t | 2) ( ) )) tanh √ ψt ∇ψ − |∇ψ t t |∇ · 2W |∇ψ t , (7.22) | where ψ t is the phase field at time t. The spatial discretization is done via a uniform grid like in Section 5.1. The stochastic part is discretized using the polynomial chaos. Thus, we have to build numerical schemes for the gradient norm, the curvature, and the hyperbolic tangent that deal with polynomial chaos expansions. Gradient Norm and Laplacian The gradient norm is computed using finite difference schemes. The directional derivatives are computed using central differences in the interior of the domain and forward resp. backward differences at the domain boundary. The necessary computations of the square and the square root are performed using the methods from Section 3.3. The Laplacian is computed as the sum of the second directional derivatives, which we compute using central differences in the interior and forward resp. backward differences at the boundary. Hyperbolic Tangent We compute the hyperbolic tangent using tanhx = 1 − 2(exp(2x) + 1) −1 . Thus, we use the computation of the exponential function in the polynomial chaos from Section 3.3 and other methods from there, i.e. we compute the Galerkin projection of the hyperbolic tangent on the polynomial chaos. Curvature The computation of the curvature is the most critical process for the computation of the update, because the update is done in the whole domain, not in a narrow band around the zero level set. This makes it necessary to compute a stable curvature even in regions with a high curvature. These regions arise in simple settings, e.g. when the level set is initialized as a circle. The curvature in the midpoint of the circle goes to infinity. We use a method for the stable curvature computation proposed by Sun 83
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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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