Image reconstruction by Mumford-Shah regularization with a priori edge information

Abstract

The Mumford-Shah functional has provided an important approach for image denoising and segmentation. Recently, it has been applied to image reconstruction in fields such as X-ray tomography and electric impedance tomography. In this thesis we study the applicability of the Mumford-Shah model to a setting, where a priori edge information is available and reliable. Such a situation occurs for example in biomedical imaging, where multimodal imaging systems have received a lot of interest. The regularization terms in the Mumford-Shah functional force smoothness of the image within individual regions and simultaneously detect edges across which smoothing is prevented. We propose to divide the edge penalty into two parts depending on the a priori edge information. We investigate the proposed model for well-posedness and regularization properties under an assumption of pointwise boundedness of the underlying image. Furthermore, we present two variational approximations that allow numerical implementations. For one we prove that it Gamma converges to a special case of our proposed model, the other we motivate heuristically. The resulting algorithm alternates between an image reconstruction and an image evaluation step. We illustrate the feasibility with two numerical examples.