Segmentation of Stochastic Images using ... - Jacobs University

Chapter 7 Stochastic Level Sets
Figure 7.5: Mean of the CT data set (left) and the liver data set (right) for the segmentation test.
7.5 Segmentation of Stochastic Images Using Stochastic Level Sets
Level set evolution with an uncertain speed can be useful in applications, e.g. in physical applications
where level sets track the interface between materials. Often, the material parameters are not known
exactly and the interface speed can be modeled dependent on the random material parameters.
The focus of this thesis is on the application of segmentation methods. This is why we use this new
concept for stochastic level sets at the moment for segmentation only. The author used level sets for
the modeling of physical effects, the evaporation of water during radiofrequency ablation [10,11,13].
It is possible to use stochastic level sets in this context, because the material parameters can be
modeled as random variables [87, 128] which leads to an uncertain interface speed.
For segmentation, we investigate three segmentation methods based on level sets for stochastic
extensions: gradient-based segmentation, geodesic active contours, and Chan-Vese segmentation.
Other segmentation methods based on level sets can also be suitable for stochastic extensions.
7.5.1 Gradient-Based Segmentation of Stochastic Images
Gradient-based segmentation of an image u : D → IR is introduced in Section 2.4.2 and given by
φ t + v(1 − bκ)|∇φ| = 0 , (7.26)
where the function v is called stopping function, because this function controls the stopping of the
level set evolution at the desired boundaries. Often, the function v is given by
v = 1/(1 + |∇u|) , (7.27)
where u is the image to segment. Typically, the level set is initialized as the signed distance function
of a small circle inside the object to segment. There is no theoretical justification of this method
besides the observation that the level set speed is close to zero at sharp edges due to the reciprocal
dependence between image gradient and speed. We replace the classical image u(x) by a stochastic
image u(x,ω). The equation for stochastic gradient-based segmentation is
and the speed is a stochastic quantity, too:
φ t (t,x,ω) + v(t,x,ω)(1 − bκ(t,x,ω))|∇φ(t,x,ω)| = 0 (7.28)
v(t,x,ω) =
1
1 + |∇u(t,x,ω)|
. (7.29)
88