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Optimal selection of the regularization function in a generalized total variation model. Part II: Algorithm, its analysis and numerical tests
2016, Hintermüller, Michael, Rautenberg, Carlos N., Wu, Tao, Langer, Andreas
Based on the generalized total variation model and its analysis pursued in part I (WIAS Preprint no. 2235), in this paper a continuous, i.e., infinite dimensional, projected gradient algorithm and its convergence analysis are presented. The method computes a stationary point of a regularized bilevel optimization problem for simultaneously recovering the image as well as determining a spatially distributed regularization weight. Further, its numerical realization is discussed and results obtained for image denoising and deblurring as well as Fourier and wavelet inpainting are reported on.
Optimal selection of the regularization function in a generalized total variation model. Part I: Modelling and theory
2016, Hintermüller, Michael, Rautenberg, Carlos N.
A generalized total variation model with a spatially varying regularization weight is considered. Existence of a solution is shown, and the associated Fenchel-predual problem is derived. For automatically selecting the regularization function, a bilevel optimization framework is proposed. In this context, the lower-level problem, which is parameterized by the regularization weight, is the Fenchel predual of the generalized total variation model and the upper-level objective penalizes violations of a variance corridor. The latter object relies on a localization of the image residual as well as on lower and upper bounds inspired by the statistics of the extremes.