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We propose a new algorithmic approach to the non-smooth and non-convex Pottsproblem (also called piecewise-constant Mumford-Shah problem) for inverseimaging problems. We derive a suitable splitting into specific subproblems thatcan all be solved efficiently. Our method does not require a priori knowledgeon the gray levels nor on the number of segments of the reconstruction.Further, it avoids anisotropic artifacts such as geometric staircasing. Wedemonstrate the suitability of our method for joint image reconstruction andsegmentation. We focus on Radon data, where we in particular consider limiteddata situations. For instance, our method is able to recover all segments ofthe Shepp-Logan phantom from $7$ angular views only. We illustrate thepractical applicability on a real PET dataset. As further applications, weconsider spherical Radon data as well as blurred data.

Joint Image Reconstruction and Segmentation Using the Potts Model