Efficient Block Coordinate Methods for Blind Cauchy Denoising

This paper deals with the problem of image blind deconvolution in presence of Cauchy noise, a type of non-Gaussian, impulsive degradation which frequently appears in engineering and biomedical applications. We consider a regularized version of the corresponding data fidelity function, by adding the total variation regularizer on the image and a Tikhonov term on the point spread function (PSF). The resulting objective function is nonconvex with respect to both the image and PSF block, which leads to the presence of several uninteresting local minima. We propose to tackle such challenging problem by means of a block coordinate linesearch based forward backward algorithm suited for nonsmooth nonconvex optimization. The proposed method allows performing multiple forward-backward steps on each block of variables, as well as adopting variable steplengths and scaling matrices to accelerate the progress towards a stationary point. The convergence of the scheme is guaranteed by imposing a linesearch procedure at each inner step of the algorithm. We provide some practical sound rules to adaptively choose both the variable metric parameters and the number of inner iterations on each block. Numerical experiments show how the proposed approach delivers better performances in terms of efficiency and accuracy if compared to a more standard block coordinate strategy.