# Salt and Pepper Noise Removal with Multi-Class Dictionary Learning and L0 Norm Regularizations
## 1 Introduction  
Images may be corrupted by the salt and pepper noise when they are acquired by imperfect sensors or transmitted in unideal channels [1]. This noise introduces the sharp and sudden disturbances in the images and the noise value usually equals to either the minimal or maximal pixel value. To remove the impulse noise, traditional methods include spatial domain approaches such as median [1] or adaptive median filtering (AMF) [2], and transform domain approaches such as wavelet denoising [3; 4]. The former ones try to distinguish noise from meaningful image structures with some pre-defined filtering, owing fast computation but limited performance. The latter ones explore the image sparsity in transform domains, e.g., wavelets and contourlets, to distinguish noise but has to carefully choose appropriate basis functions to represent different image features. More comparisons can be found in a recent review on impulse noise removal [5]. Recently, the transform domain sparsity is incorporated into sparse image reconstruction models [6; 7; 8] to significantly improve the image quality. The improvement on denoising, however, is still unsatisfactory since the sparsity is limited by pre-defined dictionaries or transforms which may not capture different image structures [8; 9; 10; 11]. The state-of-the-art approaches introduce the adaptive dictionary learning [8; 9; 10; 11] to provide sparser representations, leading to boosted performance for salt and pepper noise removal. However, the commonly used redundant dictionary learning algorithm, K-SVD [12], is relatively time consuming [13], which may slow down the iterative image reconstruction [8; 9; 14; 15] or lose optimal sparsity when partial image patches are used for fast training [12]. By exploring the intrinsic self-similarity in images, patch-based nonlocal operator (PANO) [16], which is originated fromthe well-known block-matching and 3D filtering [17], has been shown to provide adaptive sparse representations and achieve high quality image reconstructions in impulse noise removal [10; 11; 16]. Yet, PANO may be suboptimal since similar patches are sparsified with pre-defined wavelets. Therefore, it is highly expecting to find new dictionary (or transform) that enables adaptive sparse representation and fast computation while performs well in impulse noise removal. Even with a proper dictionary, the sparse image reconstruction model for impulse noise removal should be carefully defined. Generally, a restored image is obtained by minimizing a model that balance the sparsity and data fidelity [10; 11; 18]. For instance, as a convex approximation of the sparsity, the commonly used L\({}_{1}\) norm may bias to penalize small sparse coefficients than large ones, and, thus, suffers from losing weak signal details in the reconstruction problems, such as image reconstruction [19; 20; 21], biological spectrum recovery [22; 23; 24], and fault detection [25]. Therefore, it is meaningful to incorporate other norms to restore more weak signals. Beyond the sparsity constraint in the sparse image reconstruction models [8; 9; 10; 11; 16], the data fidelity, which models the impulse noise with the L\({}_{1}\) norm, may not be appropriate. It was recently found that the L\({}_{0}\) norm, that model the impulse noise, can be interpreted from the Bayesian view of maximum a posteriori probability [18] and can further improve the denoising performance. To overcome these limitations of current approaches, in this paper, salt and pepper noise removal is improved from two aspects: (1) sparser representation with fast transform; and (2) better regularization of the sparsity and data fidelity.