Robust Image Recovery via Affine Transformation and $L_{2,1}$ Norm

In this paper, we propose a novel robust algorithm for image recovery via affine transformations and the $L_{2,1}$ norm. To be robust against miscellaneous adverse effects such as occlusions, outliers, and heavy sparse noise, the new algorithm integrates affine transformations with low-rank plus sparse decomposition, where the low-rank component lies in a union of disjoint subspaces, so the distorted or misaligned images can be rectified to render more faithful image representation. In addition, the $L_{2,1}$ norm is employed to remove the correlated samples across the images, enabling the new approach to be more resilient to outliers and large variations in the images. The determination of the variables involved and the affine transformations is cast as a convex optimization problem. To alleviate the computational complexity, the Alternating Direction Method of Multipliers (ADMM) method is utilized to derive a new set of recursive equations to update the optimization variables and the affine transformations iteratively in a round-robin manner. The convergence of the developed updating equations is addressed and experimentally validated as well. The conducted simulations demonstrate that the new algorithm is superior to the state-of-the-art works in terms of accuracy on some public databases.