Regularization Properties of Mumford-Shah-Type Functionals with Perimeter and Norm Constraints for Linear Ill-Posed Problems

In this paper we consider the simultaneous reconstruction and segmentation of a function $f$ from measurements $g=Kf$, where $K$ is a linear operator. Assuming that the inversion of $K$ is ill-posed, regularization methods have to be used for the inversion process in case of inexact data. We propose using a Mumford--Shah-type functional for the stabilization of the inversion. Restricting our analysis to the recovery of piecewise constant functions, we investigate the existence of minimizers, their stability, and the regularization properties of our approach. Finally, we present a numerical example from single photon emission computed tomography (SPECT).

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