# Self-Adaptive Alternating Direction Method of Multipliers for Image Denoising
## 4 Proposed Model and Improved Algorithm
### A Collaborative Regularization Model  
In image denoising, noise can be seen as high-frequency variations that disrupt the smoothness and continuity of the image. By representing an image as a low-rank matrix (which captures the underlying structure and patterns in the image), the high-frequency noise components can often be separated from the low-rank structure during the denoising process. Images often exhibit high levels of redundancy, meaning that the information content of an image can be effectively represented with fewer parameters than the pixel count. Low-rank matrix recovery methods leverage this redundancy by approximating the image data with a low-rank matrix, which helps in denoising by emphasizing the common structure shared among pixels. In Equation (4), constraining the noise matrix \(X\) with the (1,1) norm can induce sparsity in the matrix, causing some pixel values to tend towards zero, thereby suppressing noise in the image. As images typically exhibit edge sparsity, the (1,1) norm of the matrix can to some extent help preserve the edge information in the image, thereby maintaining the image's structure. Adjacent pixels in an image commonly demonstrate similarity. To address this correlation, collaborative sparse regression can be employed to derive a shared sparse representation, thus preserving the structured sparsity within the image. By delving deeper into the  
parameters, making it challenging to choose suitable ones for specific problems. To address the issue of selecting appropriate penalty parameters, a refined version of the alternating direction method of multipliers is proposed. This approach leverages an adaptive rule and approximates the optimal penalty parameters based on iterative results rather than relying on fixed parameters. The sequence \(\mathbf{Z}_{k},\mathbf{X}_{k},\mathbf{Y}_{k}\) obtained from the ADMM satisfies  
\[\gamma^{2}||\mathbf{X}_{k+1}-\mathbf{X}_{k}||_{F}^{2}+||\mathbf{Y }_{k+1}-\mathbf{Y}_{k}||_{F}^{2} \leq(\gamma^{2}||\mathbf{X}_{k}-\mathbf{X}^{*}||_{F}^{2}+|| \mathbf{Y}_{k}-\mathbf{Y}^{*}||_{F}^{2})\] \[-(\gamma^{2}||\mathbf{X}_{k+1}-\mathbf{X}^{*}||_{F}^{2}+|| \mathbf{Y}_{k+1}-\mathbf{Y}^{*}||_{F}^{2})\]  
In Lemma 2, replacing \(\gamma\) with \(\gamma_{k}\), the sequence \(\mathbf{Z}_{k},\mathbf{X}_{k},\mathbf{Y}_{k}\) generated by where \(b_{0}=0\), \[b_{k+1}=\left\{ \begin{array}{ll}b_{k},&\text{if}\quad\frac{1}{1+\rho}\leq\frac{ ||\mathbf{Y}_{k}-\mathbf{Y}_{k+1}||_{F}}{\gamma^{2}||\mathbf{X}_{k}-\mathbf{X}_ {k+1}||_{F}}\leq 1+\rho\\ b_{k}+1.&\text{otherwise}\end{array} \right.\] (22) The steps of the S-ADMM algorithm are listed as follows. 1. Initialize \(\mathbf{X},\mathbf{Y},\gamma>0,\gamma_{k}=\gamma>0,k=0\). 2. Calculate the value of \(\mathbf{Z}_{k+1}\) such that it satisfies the following condition for any \(\mathbf{Z}\): \[\mathbf{Z}_{k+1}=\arg\min_{\mathbf{Z}}L(\mathbf{Z},\mathbf{  
10% of the pixel values were systematically set to 0, while an additional 10% underwent the introduction of random high-level noise. This meticulous contamination scheme was adopted to mimic common noise patterns observed in practical image acquisition and transmission scenarios. Subsequent to the contamination stage, the denoising process was initiated employing a repertoire of state-of-the-art algorithms, including Singular Value Thresholding (SVT), Alternating Direction Method of Multipliers (ADMM), and the Self-adaptive ADMM (S-ADMM) technique. This multi-faceted approach was undertaken to comprehensively evaluate the efficacy of our algorithm across a spectrum of noise levels and image sizes. Figure 3 represents the first grayscale image after contamination, while Figures 4-7 show the denoising effects under different algorithms. Figures 8-12 demonstrate the denoising effects under different algorithms for a larger pixel size image. The mean squared error between the denoised image and the ground truth image can be found in Tables 3 and 4. Figure 8: Contaminated grayscale image \((736\times 690)\).