On the complexity of mumford-shah-type regularization, viewed as a relaxed sparsity constraint

Authors:
Boris Alexeev;Rachel Ward
Affiliations:
Department of Mathematics, Princeton University, Princeton, NJ;Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, New York, NY
Venue:
IEEE Transactions on Image Processing - Special section on distributed camera networks: sensing, processing, communication, and implementation
Year:
2010

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Abstract

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.