# Blind Image Deblurring Based on Local Edges Selection
## 1 Introduction  
Image deblurring has long been a challenging problem. The aim of image deblurring is to recover a clear image from a blurred image. Image deblurring can be separated into non-blind and blind cases. In non-blind image deblurring, the blurring kernel is known in advance and the clear image is obtained from the blurred image and blurring kernel [1; 2; 3]. Different from non-blind image blurring, blind image deblurring aims to obtain a clear image from a blurred image when the blurring kernel is unknown. Generally, the uniform blurring process [4] is modeled by:  
\[\mathbf{y}=H\otimes x+n \tag{1}\]  
where \(y\) is the blurred image, \(H\) is the blurring kernel, \(x\) is the clear image, \(n\) is additive noise, and \(\otimes\) is the convolution operator. Blind image deblurring is an ill-posed problem, in which clear image and blurring kernel are unknown. Given a blurred image, there are countless sets of estimated results on clear images and blurring kernels. Researchers have been working for many years to estimate better results for clear image and blurring kernel [5; 6; 7; 8; 9]. Many state-of-the-art image deblurring algorithms solve the ill-posed problem within the maximum a posteriori (MAP) framework [5]. In MAP based image deblurring methods, the process of estimating clear image can be formulated as follows:  
\[p(x,H|y)=\frac{p(y|x,H)\cdot p(x)\cdot p(H)}{p(y)} \tag{2}\]where \(p(x)\) and \(p(H)\) are the probability density functions of clear image and blurring kernel. In image deblurring, \(p(y)\) is known in advance, so Equation (2) can be can be simplified as follows:  
\[p(x,H\big{|}y)\propto p(y\big{|}x,H)\cdot p(x)\cdot p(H) \tag{3}\]  
where \(p(y\big{|}x,H)\), \(p(x)\), and \(p(H)\) are the likelihood term, the priors on the clear image, and the priors on the blurring kernel, respectively [5]. According to Equation (3), the image deblurring model [6] can be summarized as follows:  
\[(x^{\star},H^{\star})=\underset{x,H}{argmin}[P(y,x,H)+Q(x)+R(H)] \tag{4}\]  
where \(P\), \(Q\), and \(R\) are the likelihood term, the regularization term based on image priors, and the regularization term based on the blurring kernel, respectively. The likelihood term is usually formulated as follows [6]:  
\[P(y,x,H)=\big{\|}y-x\otimes H\big{\|}_{2}^{2} \tag{5}\]  
The regularization term \(Q(x)\) is built based on a large number of experiments on image priors [7; 8; 9]. \(R(H)\) is often formulated by the Lp-norm of H [7; 8; 9; 10]. Numerous image deblurring algorithms have been proposed in the past years. Some of them use the sparse priors on natural images [7; 9; 10; 11; 12; 13], but the statistical priors on image or image gradients are not effective for all kinds of images [8]. The edges of an image are less sparse when the image becomes blurred [14], so some algorithms focus on building deblurring models by using the salient image edges. [PERSON] et al. [15] found the location and orientation of edges by a sub-pixel difference of Gaussians edge detector, then predicted sharp edges by propagating the maximum and minimum values along the edge profile. [PERSON] [16] estimated the blurring kernel by using the transparency on the image boundary. [PERSON] et al. [17] stated that smooth regions cannot contribute much for estimating the ground truth of the kernel, and proposed a method to extract suitable regions for blurring kernel estimation. [PERSON] et al. [18] extracted the main structure of the blurred object, then selected salient edges by shock filtering. [PERSON] and [PERSON] [19] obtained image salient edges by bilateral filtering. [PERSON] and [PERSON] [20] proposed a deblurring method in which the edges selection method is also based on shock filtering. In [21], color image gradients were added in the likelihood term instead of other operators. However, in these methods, the image edges are selected based on the global information of the image, the dense structures cannot be captured, and the edges are easy to be affected by noise [4].