Robust Tensor Recovery Using Second-Order Difference-Induced Adaptive Nuclear Norm
Tensor data, such as hyperspectral images and videos, are often degraded by mixed noise, including Gaussian noise, sparse corruption, and outliers. In this paper, we propose a robust tensor recovery model based on second-order difference-induced adaptive tensor nuclear norm regularization. The underlying clean tensor is represented by a representative coefficient tensor and a learned orthogonal basis along the third mode, so that global low-rank correlations can be characterized in a compact and data-adaptive coefficient domain rather than in a fixed transform space. To incorporate local smoothness into the same representation, tensor nuclear norm penalties are imposed on the spatial second-order difference tensors of the representative coefficients. Compared with conventional first-order total variation, the proposed regularizer models local curvature variations and the correlations among second-order difference patterns, which helps reduce staircase artifacts while preserving structural details. A Hybrid Ordinary–Welsch fidelity term and an $\ell _{1}$ -norm sparse error term are further incorporated to improve robustness against mixed noise. The resulting optimization problem is solved by an ADMM-based algorithm. Experiments on hyperspectral image and video denoising demonstrate that the proposed method consistently improves PSNR and ERGAS under all tested noise settings while achieving competitive SSIM values.