Segmentation of Stochastic Images using ... - Jacobs University

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.
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