A Half-Quadratic Approach to Mixed Weighted Smooth and Anisotropic BV Regularization for Inverse Ill-Posed Problems with Applications to Signal and Image Restoration

Abstract

Detection-estimation type penalizers have been widely used to regularize inverse ill-posed problems in which it is known that the solution may present discontinuities ( J. Idier, Bayesian Approach to Inverse Problems, John Wiley & Sons, (2008)). For the case of quadratic penalty functionals it is known that the detection-estimation problem can be reformulated as a non-convex penalization problem.
Although this approach is somewhat formally simpler, finding the corresponding global minimizer is usually a computationally challenging task, specially in high dimensional problems, such as those in image processing. At this step, a duality criterion between the non-quadratic and half-quadratic optimization becomes extremely useful to greatly reduce the computational cost (J. Idier, Convex half-quadratic criteria and interacting auxiliary variables for image restoration, IEEE Transactions on image Processing, 10(7):1001-1009, (2001)).
In this article we will consider general Tikhonov-Phillips regularization methods where the penalizers are given by mixed spatially varying weighted convex combinations of L2 and BV functionals.
Both isotropic and anisotropic BV diffusion cases will be considered. We will use the above mentioned non-convex reformulation plus a non-quadratic half-quadratic approach to attack the problem of approximating the global minimizers of those functionals. The associated optimization problems will then be recast by means of a duality argument as half-quadratic optimization problems. Numerical results in signal and image restoration problems will be shown.