Multi-Dimensional Visual Data Restoration: Uncovering the Global Discrepancy in Transformed High-Order Tensor Singular Values

The recently proposed high-order tensor algebraic framework generalizes the tensor singular value decomposition (t-SVD) induced by the invertible linear transform from order-3 to order-d ( $d\gt 3$ ). However, the derived order-d t-SVD rank essentially ignores the implicit global discrepancy in the quantity distribution of non-zero transformed high-order singular values across the higher modes of tensors. This oversight leads to suboptimal restoration in processing real-world multi-dimensional visual datasets. To address this challenge, in this study, we look in-depth at the intrinsic properties of practical visual data tensors, and put our efforts into faithfully measuring their high-order low-rank nature. Technically, we first present a novel order-d tensor rank definition. This rank function effectively captures the aforementioned discrepancy property observed in real visual data tensors and is thus called the discrepant t-SVD rank. Subsequently, we introduce a nonconvex regularizer to facilitate the construction of the corresponding discrepant t-SVD rank minimization regime. The results show that the investigated low-rank approximation has the closed-form solution and avoids dilemmas caused by the previous convex optimization approach. Based on this new regime, we meticulously develop two models for typical restoration tasks: high-order tensor completion and high-order tensor robust principal component analysis. Numerical examples on order-4 hyperspectral videos, order-4 color videos, and order-5 light field images substantiate that our methods outperform state-of-the-art tensor-represented competitors. Finally, taking a fundamental order-3 hyperspectral tensor restoration task as an example, we further demonstrate the effectiveness of our new rank minimization regime for more practical applications. The source codes of the proposed methods are available at https://github.com/CX-He/DTSVD.git.