Nonconvex TVq-Models in Image Restoration: Analysis and a Trust-Region Regularization Based Superlinearly Convergent Solver

Michael Hintermueller*

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 proven, and a Newton-type solution algorithm is introduced and its global as well as local superlinear convergence towards a stationary point of a locally regularized version of the problem is established. The potential non-positive 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.

Mathematics Subject Classification: 65K05 65F22, 90C26, 94A08

Keywords: Image restoration, compressed sensing, nonconvex regularization, total variation, nonconvex programming, generalized Newton method, trust-region method, superlinear convergence

Minisymposion: Noise Estimation, Model Selection and Bilevel Optimization