Nonlocal robust tensor recovery with nonconvex regularization*The research of Bai was supported in part by the National Natural Science Foundation of China under Grant No. 11971159. The research of Ng was supported in part by the HKRGC GRF 12306616, 12200317, 12300218, 12300519 and 17201020, and HKU Grant No. 104005583. The research of Zhang was supported in part by the National Natural Science Foundation of China under Grant Nos. 11801206, 11871025, Hubei Provincial Natural Science Foundation of China under Grant No. 2018CFB105, and Fundamental Research Funds for the Central Universities under Grant No. CCNU19ZN017. (28th January 2021) - Qiu, Duo; Bai, Minru; Ng, Michael K; Zhang, Xiongjun
Abstract: 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 ℓ 1 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.