Tensor Robust PCA with Nonconvex and Nonlocal Regularization

TL;DR

This paper introduces a nonconvex and nonlocal tensor robust PCA model that adaptively shrinks singular values and leverages nonlocal self-similarity, significantly improving visual data recovery from noisy images and videos.

Contribution

It proposes a novel nonconvex TRPCA based on the tensor adjustable logarithmic norm and integrates nonlocal self-similarity to enhance recovery performance.

Findings

Outperforms existing TRPCA methods in visual data recovery

Effectively preserves edges and textures in noisy images and videos

Demonstrates superior recovery quality through experiments

Abstract

Tensor robust principal component analysis (TRPCA) is a classical way for low-rank tensor recovery, which minimizes the convex surrogate of tensor rank by shrinking each tensor singular value equally. However, for real-world visual data, large singular values represent more significant information than small singular values. In this paper, we propose a nonconvex TRPCA (N-TRPCA) model based on the tensor adjustable logarithmic norm. Unlike TRPCA, our N-TRPCA can adaptively shrink small singular values more and shrink large singular values less. In addition, TRPCA assumes that the whole data tensor is of low rank. This assumption is hardly satisfied in practice for natural visual data, restricting the capability of TRPCA to recover the edges and texture details from noisy images and videos. To this end, we integrate nonlocal self-similarity into N-TRPCA, and further develop a nonconvex and nonlocal TRPCA (NN-TRPCA) model. Specifically, similar nonlocal patches are grouped as a tensor and then each group tensor is recovered by our N-TRPCA. Since the patches in one group are highly correlated, all group tensors have strong low-rank property, leading to an improvement of recovery performance. Experimental results demonstrate that the proposed NN-TRPCA outperforms existing TRPCA methods in visual data recovery. The demo code is available at https://github.com/qguo2010/NN-TRPCA.