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Total bounded variation regularization as a bilaterally constrained optimization problem. (English) Zbl 1055.94504

Summary: It is demonstrated that the predual for problems with total bounded variation regularization terms can be expressed as a bilaterally constrained optimization problem. Existence of a Lagrange multiplier and an optimality system are established. This allows us to utilize efficient optimization methods developed for problems with box constraints in the context of bounded variation formulations. Here, in particular, the primal-dual active set method, considered as a semismooth Newton method, is analyzed, and superlinear convergence is proved. As a by-product we obtain that the Lagrange multiplier associated with the box constraints acts as an edge detector. Numerical results for image denoising and zooming/resizing show the efficiency of the new approach.

MSC:

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
49M29 Numerical methods involving duality
65K05 Numerical mathematical programming methods
65K10 Numerical optimization and variational techniques

Software:

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