# An Unsupervised Image Denoising Method Using a Nonconvex Low-Rank Model with TV Regularization
## 1 Introduction
Recent literature [23; 24] has shown that nonconvex functions provide a better estimation accuracy and variable selection consistency and can provide more accurate rank approximations than the nuclear norm, avoiding its singular value penalty problem. Some specific works have been proposed by a number of scholars. For example, [PERSON] et al. [25] proposed a nonconvex formula consisting of the upper-bound trace norm and the l1-norm, which further restored the low rank of the data matrix; [PERSON] et al. [26] extended a nonconvex paradigm to a sparse group feature selection and proposed an efficient algorithm for large-scale problems. Nonetheless, these nonconvex models require further optimization in terms of efficiency, robustness, and low-rank recovery accuracy. In image denoising problems, an image can be viewed as a matrix where the low-rank part represents the structure and texture information, and the sparse part represents the noise information in the image. Therefore, it is feasible to apply the RPCA model to the domain of image denoising. However, RPCA methods based on nonconvex function approximations are less applicable to the domain of image denoising. To address the efficiency and robustness issues of previous nonconvex models, more efficient image denoising schemes have been devised. We design an unsupervised image denoising method based on the nonconvex \(\gamma\)-norm and RPCA. Firstly, the \(\gamma\)-norm overcomes the shortcomings of the nuclear norm overestimation and pays more attention to the singular values in the matrix. Compared to other convex functions, the low-rank matrix approximation performs better. Secondly, the \(\gamma\)-norm has a strong robustness because it is less sensitive to outliers. In practical applications, there are often some outliers in the matrix, and the \(\gamma\)-norm can deal with these outliers more robustly. Finally, the \(\gamma\)-norm can be processed by using various optimization algorithms, such as the proximal gradient descent [27], the iterative threshold shrinkage algorithm [28], and the augmented Lagrange multiplier algorithm [29]. These algorithms improve the convergence properties and denoising effectiveness of the overall denoising model. In other aspects, to address the structural smoothness of the images and further improve the denoising performance, we introduce the TV regularization and \(l_{1}\)-norm, respectively. We use the augmented Lagrangian multiplier method [29] to construct Lagrangian functions to solve the final mathematical model, transform the constrained problem into an unconstrained problem, and find the global optimal solution by alternating the updates of variables. The contributions of this work are summarized as follows:  
1. We design an unsupervised image denoising method based on nonconvex \(\gamma\)-norm and RPCA. To the best of our knowledge, this should be the first application of the nonconvex \(\gamma\)-norm to image denoising. The use of the \(\gamma\)-norm avoids the problem of overpunishing large singular values by the nuclear norm and provides a high robustness and rank approximation. Combined with the associated solution algorithm, the overall model is highly efficient and converges fast, which facilitates the overall denoising model in quickly discarding matrix elements with large approximation errors and providing a high estimation accuracy. 2. The denoising effect is facilitated by combining the \(l_{1}\)-norm and a TV norm regularization. The \(l_{1}\)-norm can effectively enhance the sparsity of the noise, while the TV norm can exploit the sparsity of the image in the gradient domain to further enhance the smoothness of the image while preserving the edge information of the denoised image. 3. Our method does not require any labeled images for training, and its unsupervised denoising principle makes it easy to generalize to different scenarios for application.