Non-local self-similarity (NLSS) is widely used as prior information in an image restoration method. In particular, a low-rankness-based prior has a significant effect on performance. On the other hand, a number of color extensions of NLSS-based grayscale image restoration methods have been developed. These extensions focus on the pixel-wise correlation among color channels. However, a natural color image also has a complex dependency, known as an inter-channel dependency, among local regions from different color channels. As a result, color artifacts appear in a denoised image obtained by using the existing methods. In this paper, we propose a novel non-local and inter-channel dependency-aware prior called the weighted tensor nuclear norm (WTNN). The proposed prior is derived by incorporating inter-channel dependency to low-rank-based NLSS prior. The WTNN is a low-rankness-of-the-third-order patch tensor, and we apply it to the tensors constructed with non-local similar patches. It enables us to naturally represent the higher-order dependencies among similar color patches. We propose an image denoising algorithm using the WTNN and image restoration algorithm by using a non-trivial generalization of this algorithm. The experimental results clearly show that the proposed WTNN-based color image denoising and restoration algorithms outperform state-of-the-art methods.
Low-rank tensor recovery has attracted much attention among various tensor recovery approaches. A tensor rank has several definitions, unlike the matrix rank—e.g., the CP rank and the Tucker rank. Many low-rank tensor recovery methods are focused on the Tucker rank. Since the Tucker rank is nonconvex and discontinuous, many relaxations of the Tucker rank have been proposed, e.g., the sum of nuclear norm, weighted tensor nuclear norm, and weighted tensor schatten-p norm. In particular, the weighted tensor schatten-p norm has two parameters, the weight and p, and the sum of nuclear norm and weighted tensor nuclear norm are special cases of these parameters. However, there has been no detailed discussion of whether the effects of the weighting and p are synergistic. In this paper, we propose a novel low-rank tensor completion model using the weighted tensor schatten-p norm to reveal the relationships between the weight and p. To clarify whether complex methods such as the weighted tensor schatten-p norm are necessary, we compare them with a simple method using rank-constrained minimization. It was found that the simple methods did not outperform the complex methods unless the rank of the original tensor could be accurately known. If we can obtain the ideal weight, p=1 is sufficient, although it is necessary to set p<1 when using the weights obtained from observations. These results are consistent with existing reports.
Low-rank tensor recovery has attracted much attention among various tensor recovery approaches. A tensor rank has several definitions, unlike the matrix rank-e.g. the CP rank and the Tucker rank. Many low-rank tensor recovery methods are focused on the Tucker rank. Since the Tucker rank is nonconvex and discontinuous, many relaxations of the Tucker rank have been proposed, e.g., the tensor nuclear norm, weighted tensor nuclear norm, and weighted tensor Schattenp norm. In particular, the weighted tensor Schatten-p norm has two parameters, the weight and p, and the tensor nuclear norm and weighted tensor nuclear norm are special cases of these parameters. However, there has been no detailed discussion of whether the effects of the weighting and p are synergistic. In this paper, we propose a novel low-rank tensor completion model using the weighted tensor Schatten-p norm to reveal the relationships between the weight and p. To clarify whether complex methods such as the weighted tensor Schatten-p norm are necessary, we compare them with a simple method using rank-constrained minimization. It was found that the simple methods did not outperform the complex methods unless the rank of the original tensor could be accurately known. If we can obtain the ideal weight, p = 1 is sufficient, although it is necessary to set p < 1 when using the weights obtained from observations. These results are consistent with existing reports.
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