# Multi-Color Channels Based Group Sparse Model for Image Restoration
## 1 Introduction  
Image restoration is one of the fundamental tasks in image processing and computer vision, and it aims to recover the original image \(x\) from the observed degraded image \(y\). Image restoration is an ill-posed inverse problem and can be chartered by the following Formula (1):  
\[y=Hx+n \tag{1}\]  
Here, \(x\) and \(y\) represent the original image and the degraded image respectively, \(H\) denotes the degenerate operator, and \(n\) denotes additive Gaussian noise. \(H\) is modeled according to different image processing tasks. When \(H\) is a unit matrix, Formula (1) represents an image denoising task [1, 2]. When \(H\) is a blurring operator, Formula (1) represents an image deblurring task [3]. When \(H\) is a mask, Formula (1) represents image restoration [4, 5]. When \(H\) is a random projection matrix, Formula (1) represents image compressive sensing applications [6, 7]. The image restoration task is studied in this paper. In order to overcome the ill-posed characteristic and recover a satisfied copy of the original image, image priors are used to regularize the minimal problem,  
\[\underset{x}{\operatorname{argmin}}\frac{1}{2}\|Hx-y\|_{2}^{2}+\lambda\Psi(x) \tag{2}\]  
where the first term \(\frac{1}{2}\|Hx-y\|_{2}^{2}\) represent the data fidelity term, \(\Psi(x)\) represents the regularization term of the image a priori, and \(\lambda\) is a regularization parameter. Image priorsplay an important role in image restoration, and many relevant algorithms are proposed, such as total-variation-based methods [8; 9; 10], sparse-representation-based methods [11; 12; 13; 14], nonlocal self-similarity-based methods [1; 2; 15] and deep-convolutional-neural-network-based methods [16; 17; 18]. Sparse representation is an effective technique in image processing. Sparse -representation-based algorithms can be divided into two categories, i.e., patch sparse representation (PSR) [11; 19] and group sparse representation (GSR) [12; 13; 15]. In the PSR model, the basic sparse representation unit is the image block. PSR assumes that each block of the image to be repaired can be perfectly modeled by a sparse linear combination of some fixed or learnable basic elements. These basic elements are called atoms and constitute a set called a dictionary. The PSR model not only does well in image denoising [20], but also can be applied to various other image processing and computer vision tasks [21]. However, the PSR model with an over-complete dictionary usually produces visual artifacts which are not expected in image restoration [22]. In addition, the PSR model assumes that the coded image blocks using sparse representation are independent and ignores the correlation among similar image blocks [13; 23], which results in image degradation. To solve the above problems, the GSR model uses an image group as its basic processing unit to integrate the local sparsity with the nonlocal self-similarity of images. GSR shows great potential in various image processing tasks, and many literatures are provided. [PERSON] et al. [13] proposed the traditional GSR model for image restoration, which is essentially equal to the low rank minimization model [24]. [PERSON] et al. [25] proposed an effective Gaussian scale mixing-based image restoration model under the GSR framework. [PERSON] et al. [26] proposed a group-based adaptive dictionary, which connects the GSR model with the low rank minimization model. [PERSON] et al. [14] combined the PSR and GSR models to reduce their shortcomings, respectively. [PERSON] et al. [27] introduced the group sparse residual constraints to the GSR model to further define and simplify the problem of image restoration. The traditional GSR model and the improved GSR models usually transform the RGB space of the image into YCbCr space and extract the Y (luminance) component for image processing. The Y channel in YCbCr space contains much valuable information about edge, shadow, object, texture and others. Moreover, human vision is sensitive to the change of the Y channel.