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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 a...

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