Nonconvex TV$^q$-Models in Image Restoration: Analysis and a Trust-Region Regularization--Based Superlinearly Convergent Solver

Abstract

A nonconvex variational model is introduced which contains the $\ell_q$-``norm,” $q\in (0,1)$, of the gradient of the underlying image in the regularization part together with a least squares--type data fidelity term which may depend on a possibly spatially dependent weighting parameter. Hence, the regularization term in this functional is a nonconvex compromise between the minimization of the support of the reconstruction and the classical convex total variation model. In the discrete setting, existence of a minimizer is proved, and a Newton-type solution algorithm is introduced and its global as well as local superlinear convergence toward a stationary point of a locally regularized version of the problem is established. The potential nonpositive definiteness of the Hessian of the objective during the iteration is handled by a trust-region--based regularization scheme. The performance of the new algorithm is studied by means of a series of numerical tests. For the associated infinite dimensional model an existence result based on the weakly lower semicontinuous envelope is established, and its relation to the original problem is discussed.