Robust Tensor Completion via Dictionary Learning and Generalized Nonconvex Regularization for Visual Data Recovery

Robust tensor completion, which aims to recover a tensor from partial observations corrupted by Gaussian noise and sparse noise simultaneously, has a wide range of applications in visual data recovery. The existing approaches make use of convex or nonconvex relaxation based on transformed tensor nuclear norm, which may be challenged since only the global low-rankness of the underlying tensor data is utilized. In order to explore the global and local patterns simultaneously, in this paper, we propose a nonconvex model for robust tensor completion by combining the dictionary learning and nonconvex regularization. For the sake of exploring the global low-rankness, a family of nonconvex functions are employed onto the singular values of all frontal slices of the underlying tensor in the transformed domain. The dictionary learning is utilized to elucidate the local patterns of the underlying tensor data. Moreover, a family of nonconvex functions are used onto each entry of the sparse noise, which can obtain sparser solutions compared with tensor ℓ
₁
norm. A proximal alternating linearized minimization algorithm is designed to solve the proposed model, whose convergence is established under very mild conditions. Extensive numerical experiments on multispectral images, video, and magnetic resonance imaging datasets show that the proposed model outperforms other state-of-the-art approaches.