On the complexity of Mumford-Shah type regularization, viewed as a relaxed sparsity constraint

Details On the complexity of Mumford-Shah type regularization, viewed as a relaxed sparsity constraint On the complexity of Mumford-Shah type regularization, viewed as a relaxed sparsity constraint

2010-01-18

arxiv.org/abs/1001.2952v1

IEEE Transactions on Image Processing 19 (2010), no. 10, 2787-2789

Mathematics

Optimization and Control

6 pages

Scientific paper

10.1109/TIP.2010.2048969

We show that inverse problems with a truncated quadratic regularization are NP-hard in general to solve, or even approximate up to an additive error. This stands in contrast to the case corresponding to a finite-dimensional approximation to the Mumford-Shah functional, where the operator involved is the identity and for which polynomial-time solutions are known. Consequently, we confirm the infeasibility of any natural extension of the Mumford-Shah functional to general inverse problems. A connection between truncated quadratic minimization and sparsity-constrained minimization is also discussed.

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