Robust Tensor Recovery Using Second-Order Difference-Induced Adaptive Nuclear Norm

Abstract

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.