Segmentation of Stochastic Images using ... - Jacobs University

Chapter 7
Stochastic Level Sets
Level sets are widely used in applications ranging from computer vision [148] over material science
to computer-aided design [138] for the tracking and representation of moving interfaces arising
e.g. in the simulation of radiofrequency ablation [13]. Dervieux and Thomasset [40] and Sethian
and Osher [121, 138] introduced level sets in the form used today. The main idea is to embed the
moving interface as the zero level set of a higher-dimensional function φ. The moving boundary is
then equivalent to a propagation of the level sets of the function φ over time. The actual position of
the boundary at time t is reconstructed from the function φ by tracking the zero level set at time t.
Level sets are used for the segmentation of images as well. They are more flexible in comparison
to a parametrization of the boundary used e.g. for snakes [76]. In addition, advanced segmentation
methods like geodesic active contours [30, 82], an energy minimization method, are based on
level sets.
When we try to combine a level set based segmentation approach with stochastic images, we end
up with a stochastic velocity for the level set propagation, i.e. we have to solve a hyperbolic SPDE.
The development of numerical methods for hyperbolic SPDEs is an active research field. To the
best of the authors knowledge, there is no method available in the literature that can be applied to
the stochastic level set equation. The use of classical methods, like upwinding schemes [138], is
not possible, because they are based on the sign of the propagation speed, which is in the stochastic
context a random variable, too. Thus, we use a parabolic approximation of the level set equation.
This enables us to use the methods developed in the previous chapters.
Due to the importance of the level set equation in other applications besides the segmentation of
images, this chapter is split into two parts. First, we present the derivation of the parabolic approximation
of the stochastic level set equation along with the numerical discretization. Furthermore,
we present numerical tests showing the applicability of the discretization. The second part of this
chapter deals with the application of the stochastic level set equation for image segmentation. We
introduce stochastic extensions of three widely used segmentation methods based on the level set
equation: gradient-based segmentation, geodesic active contours, and Chan-Vese segmentation.
7.1 Derivation of a Stochastic Level Set Equation
The discretization of the classical level set equation is based on techniques for the discretization of
hyperbolic conservation laws. The discretization of hyperbolic SPDEs is still a challenging task. To
the best of the authors knowledge, there are two possibilities, which are less accurate [92] or timeconsuming
[147]. Thus, we focus on a parabolic approximation of the level set equation to avoid
the numerical problems related to the hyperbolic level set version. The parabolic stochastic level set
equation is based on the work of Sun and Beckermann [143] for the classical level set equation. The
stochastic level set equation is derived from the equation
φ(y(t,ω),t,ω) = 0 almost sure in Ω , (7.1)
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