  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.
