# Self-Adaptive Alternating Direction Method of Multipliers for Image Denoising
## 3 Preliminaries and Problem Statement  
### Low-Rank Matrix Recovery  
The application of low-rank matrix recovery in image denoising is a significant and effective technique within the field of image processing. In the field of image processing, images are often regarded as matrices with low-rank properties due to the inherent correlations and structural information among pixels. The objective of image denoising is to eliminate noise from various sources such as sensor noise, compression artifacts, or communication channel interference to enhance image quality and clarity.  
For a given matrix \(\mathbf{D}\) containing noise, the matrix \(\mathbf{D}\) can be decomposed into the sum of two matrices, i.e., \(\mathbf{D}=\mathbf{Z}+\mathbf{X}\), where the matrices \(\mathbf{Z}\) and \(\mathbf{X}\) are unknown, but \(\mathbf{Z}\) is low-rank. When the elements of the matrix \(\mathbf{X}\) obey an independent identically distributed Gaussian distribution, the classical PCA can be used to obtain the optimal matrix \(\mathbf{Z}\), i.e., to solve the following optimization problem.  
\[\min_{\mathbf{Z},\mathbf{X}}||\mathbf{Z}||_{\text{F}},\qquad\text{s.t.rank}( \mathbf{Z})\leq r,\mathbf{D}=\mathbf{Z}+\mathbf{X} \tag{1}\]  
When \(\mathbf{X}\) is a sparse large noise matrix, PCA is no longer applicable. In this case, the recovery of the low-rank matrix \(\mathbf{Z}\) is a two-objective optimization problem  
\[\min_{\mathbf{Z},\mathbf{X}}(\text{rank}(\mathbf{Z}),||\mathbf{X}||_{0}), \qquad\text{s.t.}\mathbf{D}=\mathbf{Z}+\mathbf{X} \tag{2}\]  
The bi-objective optimization problem (2) is converted into a single-objective optimization problem by introducing a trade-off factor \(\lambda(0<\lambda<1)\) as follows:  
\[\min_{\mathbf{Z},\mathbf{X}}\text{rank}(\mathbf{Z})+\lambda||\mathbf{X}||_{0}, \qquad\text{s.t.}\mathbf{D}=\mathbf{Z}+\mathbf{X} \tag{3}\]  
Since the objective function of Equation (3) is non-convex, it can be approximated by the following equation:  
\[\min_{\mathbf{Z},\mathbf{X}}||\mathbf{Z}||_{*}+\lambda||\mathbf{X}||_{1,1}, \qquad\text{s.t.}\mathbf{D}=\mathbf{Z}+\mathbf{X} \tag{4}\]  
**Remark 1**.: _The (1,1)-norm of a matrix is defined as the maximum absolute sum of each column. In a sense, this maximum value can be seen as an upper bound on the number of non-zero elements in the matrix. Therefore, minimizing the (1,1)-norm of a matrix also serves as a constraint on the number of non-zero elements, thus providing an approximation to optimizing the 0-norm of a matrix._  
**Remark 2**.: _The problem of solving Equation (4) is known as Robust Principal Component Analysis (RPCA), which provides an efficient way to solve the low-rank matrix recovery problem and can improve robustness to outliers in the data._  
**Remark 3**: _In reference [15], it is pointed out that as long as the singular values distribution of the low-rank matrix \(\mathbf{Z}\) is reasonable and the non-zero elements of the sparse matrix are uniformly distributed, the convex optimization problem (4) can recover the original low-rank matrix from any unknown error with a probability close to 1._