Nonlocal and Nonconvex Robust Tensor Completion with Tensor Dictionary Learning for Multi-Dimensional Image Recovery

Robust tensor completion aims to recover a clean tensor from noisy and incomplete observations, where the observed tensor is corrupted by Gaussian noise and sparse noise simultaneously. Existing methods only exploit one or two priors out of global tensor low-rankness, local properties, and nonlocal self-similarity, leading to suboptimal recovery performance. In this paper, we propose a nonconvex model combined with nonlocal self-similarity and tensor dictionary learning for robust tensor completion. Specifically, by partitioning the tensor into several overlapping cubes, the similar cubes are grouped together. Then, we unfold the cubes into matrices and stack these matrices into a third-order tensor. Subsequently, the minimax concave penalty (MCP) is employed on the singular values of all frontal slices of the sub-tensors in the transformed domain to explore the low-rankness of the underlying sub-tensor. The tensor dictionary learning based on Tucker decomposition is used to explore the local patterns of the underlying sub-tensor. Moreover, the MCP is employed onto each entry of the sparse noise tensor to explore the sparsity. A proximal alternating linearized minimization algorithm is adopted to solve the resulting model. Extensive numerical experiments demonstrate that the proposed method outperforms the competing state-of-the-art methods in both visual quality and quantitative metrics.