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.

翻译：张量鲁棒主成分分析（TRPCA）是实现低秩张量恢复的经典方法，通过均匀收缩每个张量奇异值来最小化张量秩的凸替代函数。然而，对于真实世界的视觉数据，大奇异值比小奇异值承载着更显著的信息。本文基于张量可调对数范数提出了一种非凸TRPCA（N-TRPCA）模型。与TRPCA不同，N-TRPCA能够自适应地更多收缩小奇异值、更少收缩大奇异值。此外，TRPCA假设整个数据张量具有低秩性，然而自然视觉数据在实际中难以满足这一假设，限制了TRPCA从噪声图像和视频中恢复边缘与纹理细节的能力。为此，我们将非局部自相似性融入N-TRPCA，进一步提出一种非凸与非局部TRPCA（NN-TRPCA）模型。具体而言，将相似的非局部图像块分组为张量，并利用N-TRPCA对每个组张量进行恢复。由于同一组内的图像块高度相关，所有组张量均具有强低秩特性，从而提升了恢复性能。实验结果表明，所提出的NN-TRPCA在视觉数据恢复方面优于现有TRPCA方法。演示代码可在https://github.com/qguo2010/NN-TRPCA获取。