Nonlocal image denoising via adaptive tensor nuclear norm minimization

Nonlocal self-similarity shows great potential in image denoising. Therefore, the denoising performance can be attained by accurately exploiting the nonlocal prior. In this paper, we model nonlocal similar patches through the multi-linear approach and then propose two tensor-based methods for image denoising. Our methods are based on the study of low-rank tensor estimation (LRTE). By exploiting low-rank prior in the tensor presentation of similar patches, we devise two new adaptive tensor nuclear norms (i.e., ATNN-1 and ATNN-2) for the LRTE problem. Among them, ATNN-1 relaxes the general tensor N-rank in a weighting scheme, while ATNN-2 is defined based on a novel tensor singular-value decomposition (t-SVD). Both ATNN-1 and ATNN-2 construct the stronger spatial relationship between patches than the matrix nuclear norm. Regularized by ATNN-1 and ATNN-2 respectively, the derived two LRTE algorithms are implemented through the adaptive singular-value thresholding with global optimal guarantee. Then, we embed the two algorithms into a residual-based iterative framework to perform nonlocal image denoising. Experiments validate the rationality of our tensor low-rank assumption, and the denoising results demonstrate that our proposed two methods are exceeding the state-of-the-art methods, both visually and quantitatively.