Regularizing properties of the Mumford-Shah functional for imaging applications

The Mumford-Shah functional was originally introduced for image denoising and segmentation problems, and is of interest because it provides a regularization of image edges in addition to images. Recently, this functional has emerged as a regularization technique for several imaging applications, such as x-ray tomography, electric impedance tomography, image deblurring and SPECT. In this context of operator equations it is necessary to understand its regularization properties and to determine its range of applicability. Following the approach of Rondi and Santosa, we exploit an L-infinity-constraint on the images, however in contrast to this approach we achieve convergence results not only for the images but also for the edge sets in the sense of sigma-convergence introduced by Maso et al. The analysis exploits an assumption on the decay properties of the fidelity term. Under the above two conditions, we establish the stability of the Mumford-Shah regularization for perturbations in the data. Moreover we present a parameter choice rule which ensures, that the reconstructed images and edges converge to the true image and its edges as the noise level goes to zero. We demonstrate the applications of the Mumford-Shah regularization to some linear and nonlinear imaging problems, namely image deblurring, x-ray tomography and two-dimensional diffuse optical tomography.