# An Unsupervised Image Denoising Method Using a Nonconvex Low-Rank Model with TV Regularization
## 1 Introduction  
Due to the transmission channel's limitations, it is easy to introduce noise during the process of image shooting, which has a significant negative impact on the image's visual effect. Noise introduces random or stochastic variations into the images, which may blur, distort, or even hide useful information [1]. Moreover, the effect of noise removal is also directly related to the subsequent processing of the image, such as image segmentation, object recognition, and edge extraction. Therefore, the topic of image denoising is of great significance in scientific research, the engineering field, and medical clinical diagnosis [2]. As the name suggests, image denoising aims to preserve as much of the original information in the image as possible while minimizing noise. Over the past decades, many efficient denoising methods have emerged. Spatial domain denoising methods [3]directly act on the source image and include classical filtering methods such as mean filtering [4], median filtering [5], and Wiener filtering [6]. While these methods are easy to implement, they are less adaptable and stable, and the denoised images are not sharp enough. Transform domain denoising methods [7] can effectively avoid image distortion by transforming the image into the frequency domain for processing, including Fourier transform [8], discrete cosine transform [9], and wavelet transform [10], but such methods usually have a high complexity and uncertainty. Moreover, deep-learning-based denoising methods enhance the denoising effect to some extent due to their strong feature representation capabilities, but they face higher training-set requirements and more time-consuming computational cost [11]. Robust principal component analysis (RPCA) is a typical representative matrix decomposition method that decomposes a matrix into two matrices, namely a low-rank matrix and a sparse matrix [12]. This idea can effectively avoid complex computational procedures and enhance the robustness of the processing process. It has a wide range of applications in image processing, machine learning, earth science, and medicine [13]. For example, [PERSON] [14] proposed an adaptive weighted rank reduction (AWRR) method and applied it to synthetic and real seismic data to test its efficiency; [PERSON] et al. [15] used a weighted Schatten p-norm minimization to denoise impulse noise in medical clinical images; [PERSON] et al. proposed a new Laplacian regularized RPCA framework, where the \"robust\" aspect came from the introduction of a sparse term [16] and a novel process monitoring approach using the structured joint sparse canonical correlation analysis (SJSCCA) [17]; [PERSON] et al. [18] proposed a spatiotemporal structured sparse RPCA method for moving-object detection, which imposed spatial and temporal regularization on the sparse component in the form of graph Laplacians; [PERSON] et al. [19] proposed a method that used manifold constraints to maintain the local geometric structure and introduced nonconvex joint sparsity to capture the global progressive sparse structure. Among these methods and other RPCA image processing methods that have not been mentioned, we find that most of them are optimized on the basis of the original RPCA algorithm, namely, the optimization of the sparse matrix. Most rank function methods for approximating low-rank matrices still use the standard nuclear norm for approximation. Of course, there are some improvements to the nuclear norm, such as the weighted nuclear norm [20], the singular value threshold [21], and the truncated nuclear norm [22]. However, the nuclear norm can overpenalize large singular values of the matrix during the computation, resulting in the nuclear norm minimization problem not obtaining an optimal solution, which greatly affects the performance of the associated method.