Factorization model with total variation regularizer for image reconstruction and subgradient algorithm

This paper concerns the reconstruction of images in which the pixels of images are missing and the observations are corrupted by noise. By leveraging the approximate low-rank and gradient smoothing prior information of images, we propose a factorization model with the total variation (TV) and a weakly convex surrogate of column ${\ell }_{2,0}$-norm regularizers. This model avoids the computation cost of SVDs required by those models of full matrix variables, and moreover, the TV regularizer accounts for the edge structure of the target image, and the weakly convex surrogate of column ${\ell }_{2,0}$-norm of factor matrices considers the rough upper estimation for the true rank. For the proposed nonconvex and nonsmooth model, we develop an efficient subgradient algorithm, and prove that any cluster point of its iterate sequence is a stationary point and the cost value sequence converges to a critical value. Numerical experiments are conducted on color images and hyperspectral images with pixel missing and observation that is corrupted by Gaussian or impulse noise. Numerical comparisons with seven state-of-art methods for color image reconstruction and one deep leaning method for hyperspectral image restoration validate the efficiency of the proposed method.