Segmentation of Stochastic Images using ... - Jacobs University

7.5 Segmentation of Stochastic Images Using Stochastic Level Sets
MC SC PC
E
Var
Figure 7.8: Mean and variance of the stochastic geodesic active contour segmentation of the stochastic
CT data set. The variance is constant in the normal direction of the zero level set.
inside the image. Typically, the edge indicator is
g u =
1
(1 − εκ) , (7.34)
1 + |∇G σ ∗ u| p
where G σ is a Gaussian smoothing kernel with width σ and p ∈ {1,2}. Computing the stochastic
Euler-Lagrange equation as necessary condition for a minimum of the function is done in the same
fashion as in [30, 82], but we have to respect the outer integration over Ω. We end up with the
stochastic Euler-Lagrange equation
φ t (t,x,ω) = g u (t,x,ω)β|∇φ(t,x,ω)| − α∇g u (t,x,ω)∇φ(t,x,ω) + εκ|∇φ(t,x,ω)| , (7.35)
which is analog to the deterministic one. The parameters α,β, and γ can be freely chosen to optimize
the segmentation result. The meaning of the parameters is the following:
• α: The parameter α controls the attraction of the minima of the edge indicator g u when it is
positive. Otherwise, the level set is pushed away from the minima.
• β: The parameter β controls the shrinkage or expansion of the level set. A negative value of β
leads to a shrinkage of the level set and positive β to an expansion of the level set. Thus, this
parameter controls whether the initial level set is inside or outside of the desired contour.
• ε: The parameter ε acts as a weighting term for the curvature smoothing.
The stochastic geodesic active contour level set equation is discretized using the methods presented
in Section 3.3 and by using the stochastic preconditioned phase field equation.
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