A Low-Rank Tensor Decomposition Model With Factors Prior and Total Variation for Impulsive Noise Removal

Image restoration is a long-standing problem in signal processing and low-level computer vision. Previous studies have shown that imposing a low-rank Tucker decomposition (TKD) constraint could produce impressive performances. However, the TKD-based schemes may lead to the overfitting/underfitting problem because of incorrectly predefined ranks. To address this issue, we prove that the
$n$
-rank is upper bounded by the rank of each Tucker factor matrix. Using this relationship, we propose a formulation by imposing the nuclear norm regularization on the latent factors of TKD, which can avoid the burden of rank selection and reduce the computational cost when dealing with large-scale tensors. In this formulation, we adopt the Minimax Concave Penalty to remove the impulsive noise instead of the
$l_{1}$
-norm which may deviate from both the data-acquisition model and the prior model. Moreover, we employ an anisotropic total variation regularization to explore the piecewise smooth structure in both spatial and spectral domains. To solve this problem, we design the symmetric Gauss-Seidel (sGS) based alternating direction method of multipliers (ADMM) algorithm. Compared to the directly extended ADMM, our algorithm can achieve higher accuracy since more structural information is utilized. Finally, we conduct experiments on the three kinds of datasets, numerical results demonstrate the superiority of the proposed method, especially, the average PSNR of the proposed method can improve about 1~5dB for each noise level of color images.