# Salt and Pepper Noise Removal with Multi-Class Dictionary Learning and L0 Norm Regularizations
## 2 Methods  
### Typical Sparsity-Based Impulse Noise Removal Method  
The target image is usually assumed to be sparse under a certain dictionary or transform representation. The sparsity is usually indicated by measuring the L\({}_{1}\) norm of the representation coefficients [6; 7; 8] and this norm can be used as feature selection criteria in the sparse representation-based image fusion [26; 27; 28]. In addition, the salt and pepper noise are often treated as the outlier in images and the L\({}_{1}\) norm is employed to constraint the data consistency [29]. Thus, a L\({}_{1}\)-L\({}_{1}\) regularization model is defined to remove impulse noise according to:  
\[\min_{\mathbf{x}}\|\boldsymbol{\
abla}\boldsymbol{x}\|_{1}+\lambda\| \boldsymbol{y}-\boldsymbol{x}\|_{1} \tag{1}\]  
where \(\lambda\) balances the sparsity of an image \(\boldsymbol{x}\) (under the transform \(\boldsymbol{\
abla}\)) and the data fidelity that is robust to outliers in the noisy image \(\boldsymbol{y}\).  
How to sparsify the image with a proper \(\boldsymbol{\
abla}\) heavily affects the denoising results. A typical choice is a fixed sparse representation, e.g. the finite difference, a basic form of total variations (TV) [18; 30] that models the piece-wise constant signals, and wavelets which characterize the piecewise smooth signals [6]. To better captures the image features, a dictionary [8; 9] or transform [10; 11] may be adaptively trained from the noisy image itself. Unlike the typical dictionary learning, which is time consuming in the iterative image reconstruction [8; 9], the recently-proposed PANO not only saves the training time (only several seconds) but also provides adaptively sparse representation to the image by learning the self-similarity [11]. Similar patches are grouped into 3D cubes and then sparsified with 3D wavelets. For removing salt and pepper noise, PANO significantly improved the denoisingperformance [11] and obtained better results with impulse detection [10]. With PANO, Equation (1) is turned into:  
\[\min_{x}\sum_{j=1}^{J}\lVert\mathbf{A}_{j}\mathbf{x}_{1}\rVert+\lambda\lVert\mathbf{W}(\mathbf{y }-\mathbf{x})\rVert_{1} \tag{2}\]  
where \(\sum_{j=1}^{J}\lVert\mathbf{A}_{j}\mathbf{x}\rVert_{1}\) promotes the sparsity on \(J\) groups of similar patches and \(\mathbf{W}\) is a diagonal matrix whose entries stands for weights on pixels. A reasonable weight penalizes the noisy pixels much heavier than the noise-free ones [10]. It is worth noting that the penalty function \(\sum_{j=1}^{J}\lVert\mathbf{A}_{j}\mathbf{x}\rVert_{1}\) promotes the intrinsic group sparsity [31], which is not only important in image denoising [32; 33], but also in other applications [34].  
Although the model in Equation (2) has shown promising performance in salt and pepper noise removal, it has two limitations: (1) sparsity is insufficient since 3D wavelets in PANO are still pre-defined basis; and (2) the L\({}_{1}\) norm term is only approximation of the sparsity which may lose weak image structures, e.g., small edges, in the reconstruction [19; 20; 21] or lack robustness to impulse noise [18].