Ambrosio-Tortorelli Segmentation of Stochastic Images: Model Extensions, Theoretical Investigations and Numerical Methods

摘要

We discuss an extension of the Ambrosio-Tortorelli approximation of the Mumford-Shah functional 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. 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 a stochastic smoothed version of the original image and a stochastic phase field for the edge set. For the discretization of these equations we utilize the generalized polynomial chaos expansion and the generalized spectral decomposition (GSD) method. In contrast to the simple classical sampling technique, this approach allows for an efficient determination of the stochastic properties of the output image and edge set by computations on an optimally small set of random variables. Also, we use an adaptive grid approach for the spatial dimensions to further improve the performance, and we extend an edge linking method for the classical Ambrosio-Tortorelli model for use with our stochastic model. The performance of the method is demonstrated on artificial data and a data set from a digital camera as well as real medical ultrasound data. A comparison of the intrusive GSD discretization with a stochastic collocation and a Monte Carlo sampling is shown.