Segmentation of Stochastic Images using ... - Jacobs University

Chapter 8 Segmentation of Classical Images Using Stochastic Parameters
8.3 Gradient-Based Segmentation with Stochastic Parameter
Gradient-based segmentation via a level set formulation contains one parameter b that controls the
influence of the curvature κ. Making this parameter a random variable, we end up with
φ t (t,x,ω) + v(1 − b(ω)κ(t,x,ω))|∇φ(t,x,ω)| = 0 . (8.12)
The stopping function v is v = 1
1+|∇u|
. Additionally to the necessary Galerkin projection in the
numerical scheme for the solution of the gradient-based segmentation, we have to project bκ back to
the polynomial chaos. For this, we use the standard methods presented in Section 3.3. The remaining
part of the discretization is analog to the discretization of the stochastic gradient-based segmentation
with stochastic image.
Results
We present the gradient-based segmentation with stochastic parameter using the CT data set and the
liver data set. As usual, we used the expected value as input and used one random variable and a
polynomial degree of four. For the experiment, we used a stochastic parameter b that is uniformly
distributed between 0.75 and 1.25, i.e. b ∼ U [0.75,1.25]. The parameter b controls the influence of
the curvature smoothing. A higher parameter b leads to smoother contours. This is shown in Fig. 8.6
where the contour realizations vary with respect to the curvature.
Figure 8.6: Result of the gradient-based segmentation with stochastic parameter b, i.e. with a stochastic
curvature smoothing. The upper row shows the results for the CT data set, expected
value of the image and contour realizations (left) and variance of the level set with contour
realizations (right). The lower row shows the same results for the liver data set. In
all figures we add Monte Carlo realizations of the stochastic object boundary. The red
contour corresponds to a b = 0.75, yellow to b = 1.0 and blue to b = 1.25.
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