A nonconvex variational model is introduced which contains the q-"norm," q (0, 1), of the gradientof the underlying image in the regularization part together with a least squarestype datafidelity 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 ofthe support of the reconstruction and the classical convex total variation model. In the discretesetting, existence of a minimizer is proved, and a Newton-type solution algorithm is introduced andits global as well as local superlinear convergence toward a stationary point of a locally regularizedversion of the problem is established. The potential nonpositive definiteness of the Hessian of theobjective during the iteration is handled by a trust-regionbased regularization scheme. The performanceof the new algorithm is studied by means of a series of numerical tests. For the associatedinfinite dimensional model an existence result based on the weakly lower semicontinuous envelopeis established, and its relation to the original problem is discussed.