In many inverse problems, one has to recover functions that are piecewise smooth separated by lower dimensional interfaces. For these problems, it is important to choose a regularization technique that will respect and preserve the discontinuities of the function values, as well as control the geometric regularity of the interfaces. Notable examples of this class of regularization include minimizing the total variation of the function and minimizing the surface area of the interfaces. In this talk, I'll review recent advances in this area, with illustrating examples from image restoration and segmentation, elliptic inverse problems, and medical image tomography problems.
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