Abstract

We present an extension of the classical Ambrosio-Tortorelli approximation of the Mumford-Shah approach for the segmentation of images with uncertain gray values resulting from measurement errors and noise. Our approach yields a reliable precision estimate for the segmen- tation result, and it allows to quantify the robustness of edges in noisy images and under gray value uncertainty. We develop an ansatz space for such images by identifying gray values with random variables. The use of these stochastic images in the minimization of energies of Ambrosio- Tortorelli type leads to stochastic partial differential equations for the stochastic smoothed image and a stochastic phase field for the edge set. For their discretization we utilize the generalized polynomial chaos ex- pansion and the generalized spectral decomposition (GSD) method. We demonstrate the performance of the method on artificial data as well as real medical ultrasound data.