Image restoration with a half-quadratic approach to mixed weighted smooth and anisotropic bounded variation regularization

Abstract

The problem of restoring a signal or image is often tantamount to approximating the solution of a linear inverse ill-posed problem. In order to find such an approximation one might regularize the problem by penalizing variations on the estimated solution. Among the wide variety of methods available to perform this penalization, the most commonly used is the Tikhonov-Phillips regularization, which is appropriate when the sought signal or image is expected to be smooth, but it results unsuitable whenever preservation of discontinuities and edges is an important matter. Nonetheless, there are other methods with edge preserving properties, such as bounded variation (BV) regularization. However, these methods tend to produce piecewise constant solutions showing the so called “staircasing effect” and their numerical implementations entail great computational effort and cost. In order to overcome these obstacles, we consider a mixed weighted Tikhonov and anisotropic BV regularization method to obtain improved restorations and we use a half-quadratic approach to construct highly efficient numerical algorithms. Several numerical results in signal and image restoration problems are presented.