## 2 Materials and Methods
### Image Prior and Our Method's Framework  
From previous studies a practical framework of blind image deblurring is the blind iterative deconvolution algorithm [15; 16; 17; 18; 19]. Its fundamental principle is expressed as follows:  
\[\left(\hat{f},\hat{k}\right)=\arg\min_{f\hat{k}}\parallel k\otimes f-g\parallel _{2}^{2}+\alpha_{1}P_{f}+\alpha_{2}P_{k} \tag{2}\]  
where the first term, \(\parallel k\otimes f-g\parallel_{2}^{2}\), is the data fidelity term which indicates the prior of the additive noise \(\eta\) in Equation (1), \(P_{f}\) and \(P_{k}\) represent the priors of the original image and the blur kernel, respectively, while \(\alpha_{1}\) and \(\alpha_{2}\) are their corresponding weights.  
For the image prior model, we assume that the first-order derivatives of the image \(f\) in four directions are represented in Equation (3) by:  
\[\left\{\begin{array}{ll}\
abla_{1}f(x,\ y)=f(x,y)-f(x,y+1)\\ \
abla_{2}f(x,y)=f(x,y)-f(x+1,y)\\ \
abla_{3}f(x,y)=f(x,y)-f(x+1,y+1)\\ \
abla_{4}f(x,y)=f(x,y)-f(x-1,y-1)\end{array}\right. \tag{3}\]  
where \(x\) and \(y\) represent the coordinates of a digital image; and the operator \(\
abla_{n}\) with \(n\in N=\{1,\ 2,\ 3,\ 4\}\) indicates the image gradients in four orientations: \(0\), \(\pi/2\), \(\pi/4\), and \(3\pi/4\), respectively. In the following context, the images and blur kernels are represented by lexicographic order, which is to list every pixel in raster scan order as one long vector [26]. This process is convenient for analysis and mathematical calculations.  
In this paper, to measure the image sparsity, we utilized the nonzero measurement in the image gradient domain in the four orientations as the image prior, and we defined the prior as follows:  
\[P_{f}=\sum_{i\in\Omega}\left(1-\delta\left(\left|\
abla_{1}f_{i}\right|+ \left|\
abla_{2}f_{i}\right|+\left|\
abla_{3}f_{i}\right|+\left|\
abla_{4}f_{i }\right|\right)\right) \tag{4}\]where the subscript \(i\) represents the \(i\)th element of the image gradient domain in lexicographic order discussed above, and \(\Omega\) is the support domain of the images. Notation \(\left|\right|\) represents the magnitude. The function \(\delta(m)\) is the Kronecker delta function, which returns 0 if \(m\
eq 0\) and 1 if \(m=0\). In general, Equation (4) aims at counting the number of nonzero elements in image gradient domains in four orientations.  
In Figure 1, the first column shows an example of a clear image and a blurry one. The statistical charts in the second column constitute their corresponding image histograms. It shows that the pixel intensities are more concentrated in the blurred image, which implies that the gradient domain of the blurred image has a better sparsity than that of the clear one. The third column shows the statistical charts of nonzero values in the gradient domain in four orientations (\(\mathbb{V}_{1}+\mathbb{V}_{2}+\mathbb{V}_{3}+\mathbb{V}_{4}\)), which confirms that the frequencies of nonzero values of the blurred image in gradient domain are higher than those in the clear image.  
In the view of MAP, the estimators can get closer to the true values on the condition of more measurements. Since the size of the blur kernel is far smaller than that of the images, the observed image \(g\) supplies enough measurements for estimating the blur kernel alone [27]. Thus, the blind deconvolution problem of solving Equation (2) can be transformed into an alternating scheme, i.e., the blur kernel should be estimated at first, and then the latent sharp image can be acquired by a non-blind deconvolution method.