Segmentation of Stochastic Images using Level Set Propagation with Uncertain Speed

We present an approach for the evolution of level sets under an uncertain velocity leading to stochastic level sets. The uncertain velocity can either be a random variable or a random field, i.e. a spatially varying random quantity, and it may result from measurement errors, noise, unknown material parameters or other sources of uncertainty. The use of stochastic level sets for the segmentation of images with uncertain gray values leads to stochastic domains, because the zero level set is not a single closed curve anymore. Instead, we have a band of possibly infinite thickness which contains all possible locations of the zero level set under the uncertainty. Thus, the approach allows for a probabilistic description of the segmented volume and the shape of the object. Due to numerical reasons, we use a parabolic approximation of the stochastic level set equation, which is a stochastic partial differential equation, and discretized the equation using the polynomial chaos and a stochastic finite difference scheme. For the verification of the intrusive discretization in the polynomial chaos we performed Monte Carlo and Stochastic Collocation simulations. We demonstrate the power of the stochastic level set approach by showing examples ranging from artificial tests to demonstrate individual aspects to a segmentation of objects in medical images.