Total Bounded Variation Regularization as a Bilaterally Constrained Optimization Problem

It is demonstrated that the pre-dual for problems with total bounded variation regularization terms can be expressed as bilaterally constrained optimization problem. Existence of a Lagrange multiplier and an optimality system are established. This allows 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 semi-smooth Newton method is analyzed and superlinear convergence is proved. As a by-product it is obtained 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.