Nonlocal robust tensor recovery with nonconvex regularization

The robust tensor recovery problem consists in reconstructing a tensor from a sample of entries corrupted by noise, which has attracted great interest in a wide range of practical situations such as image processing and computer vision. In this paper, we study robust tensor recovery for third-order tensors with different degradations, which aims to recover a tensor from partial observations corrupted by Gaussian noise and sparse noise simultaneously. In contrast to traditional approaches based on the tensor nuclear norm penalty for the low-rank component and the tensor L₁ norm penalty for the sparse component, we propose a nonlocal robust low-rank tensor recovery model with nonconvex regularization (NRTRM) to explore the global low-rankness and nonlocal self-similarity of the underlying tensor. The NRTRM method is first to extract similar patched-tubes to form a third-order sub-tensor. Then a class of nonconvex low-rank penalties and nonconvex sparse penalties are employed to explore the low-rank component and the sparse corruptions for such sub-tensor, respectively. Moreover, a proximal alternating linearized minimization algorithm is developed to solve the resulting model in each group and its convergence is established under very mild conditions. Extensive numerical experiments on both multispectral images and video datasets demonstrate the superior performance of NRTRM in comparison with several state-of-the-art methods.