Segmentation of stochastic images using stochastic extensions of the Ambrosio-Tortorelli and the random walker model

We discuss methods based on stochastic PDEs for the segmentation of images with uncertain gray values resulting from measurement errors and noise. Our approach yields a reliable precision estimate for the segmentation result, and it allows us to quantify the robustness of edges in noisy images and under gray value uncertainty. The ansatz space for such images identifies gray values with random variables. For their discretization we utilize generalized polynomial chaos expansions and the generalized spectral decomposition method. This leads to the stochastic generalization of the Ambrosio-Tortorelli approximation of the Mumford-Shah functional. Moreover, we present the extension of the random walker segmentation for our stochastic images, which is based on an identification of the graph weights with random variables. We demonstrate the performance of the methods on a data set obtained from a digital camera as well as real medical ultrasound data.