## 2 Materials and Methods
### Estimation of Blur Kernel and the Intermediate Latent Image
#### 2.2.1 Solve \(\mathbf{f}\) with a Given \(\mathbf{k}\)  
As discussed in Section 2.1, we use a notation in the form of a norm constraint \(\left\|\bigvee\mathbf{f}\right\|_{c}\) to represent the image prior \(P_{f}\) proposed in Equation (4) for simplicity. Then, the first subproblem of Equation (5) can be written as:  
\[\hat{\mathbf{f}}=\arg\min_{f}\parallel\mathbf{k}\otimes\mathbf{f}-\mathbf{g}\parallel_{2}^{2} +\alpha_{1}\parallel\bigvee\mathbf{f}\parallel_{c} \tag{6}\]  
where the result of \(\mathbf{f}\) is an intermediate variable that is used for estimating an accurate blur kernel. Furthermore, it cannot be regarded as the deblurring result because it loses multiple image details and retains only the outlines. The above mentioned minimization can be solved via the split Bergman iteration. With auxiliary variable \(\mathbf{u}=\left(u^{1},\ u^{2},u^{3},u^{4}\right)\) the cost function of Equation (6) turns into another optimization problem:  
\[\min_{f,\mathbf{u}}\parallel\mathbf{k}\otimes\mathbf{f}-\mathbf{g}\parallel_{2}^{2}+\beta_{1} \parallel\bigvee\mathbf{f}-\mathbf{u}\parallel_{2}^{2}+\alpha_{1}\parallel\mathbf{u} \parallel_{c} \tag{7}\]  
where \(\bigvee\mathbf{f}=(\bigvee\mathbf{f},\ \ \bigvee\mathbf{g},\bigvee\mathbf{g},\bigvee\mathbf{g}, \bigvee\mathbf{g})\) represents the collection of the gradient in the four orientations, and the intermediate coefficient, \(\beta_{1}\), is varied during the optimization process. The optimization of \(\mathbf{f}\) in Equation (7) converges to the solution of Equation (6) as \(\beta_{1}\) approaches to infinity. The alternating scheme of the subproblem is described as follows:  
\[\left\{ \begin{array}{l}\hat{\mathbf{u}}=\arg\min_{\mathbf{u}}\parallel\mathbf{u} \parallel_{c}+\frac{\beta_{1}}{\alpha_{1}}\parallel\bigvee\mathbf{f}-\mathbf{u} \parallel_{2}^{2}\\ \hat{\mathbf{f}}=\arg\min_{f}\parallel\mathbf{k}\otimes\mathbf{f}-\mathbf{g}\parallel_{2}^{2} +\beta_{1}\parallel\bigvee\mathbf{f}-\mathbf{u}\parallel_{2}^{2}\end{array} \right. \tag{8}\]  
where the result value of \(\hat{\mathbf{f}}\) represents the intermediate latent image. Considering the \(i\)th element of \(\mathbf{u}\) and \(\bigvee\mathbf{f}\), the \(\hat{\mathbf{u}}\)-subproblem in the first part of Equation (8) can be expressed as:  
\[\hat{\mathbf{u}}=\sum_{i\in\Omega}\left(\arg\min_{\mathbf{u}_{i}}\ 1-\delta\left(\sum_{i \in N}\left|u_{i}^{n}\right|\right)+\frac{\beta_{1}}{\alpha_{1}}\sum_{n\in N} \left(\bigvee_{n}\mathbf{f}_{i}-u_{i}^{n}\right)^{2}\right) \tag{9}\]  
where the bold variable \(\mathbf{u}_{i}=\left(u_{i}^{1},\ u_{i}^{2},u_{i}^{3},u_{i}^{4}\right)\) to be minimized under the \"\(\arg min\)\" function represents the \(i\)th element of \(\mathbf{u}\). It can be seen from the above equation that the solution of Equation (9) is able to be acquired via element-wise optimization. Figure 2: Framework of the proposed method in a coarse-to-fine scheme. The “Down sampling” step is accomplished through the Gaussian pyramid technique; the “Estimating” step is described in Section 2.2; the “Deblurring” step is discussed in Section 2.3. Let \(\Psi(\mathbf{u}_{i})\) represent the minimization problem, i.e., the \"\(\arg min\)\" function, for the \(i\)th element in the right-hand side of Equation (9), and it can be expanded as follows:  
\[\Psi(\mathbf{u}_{i})=\left\{ \begin{array}{ll}\frac{\beta_{1}}{\alpha_{1}}\sum \limits_{n\in\mathbb{N}}\left(\mathbb{V}_{n}\mathbf{f}_{i}\right)^{2}&,\mathbf{u}_{i}= \mathbf{0}\left(\ \forall n\ :u_{i}^{n}=0\right)\\ 1+\frac{\beta_{1}}{\alpha_{1}}\sum\limits_{n\in\mathbb{N}}\left(\mathbb{V}_{n }\mathbf{f}_{i}-\mathbf{u}_{i}^{n}\right)^{2}&,\mathbf{u}_{i}\
eq\mathbf{0}\left(\ \exists n\ :\ u_{i}^{n}\
eq 0\right)\end{array} \right. \tag{10}\]  
It can be seen from Equation (10) that when \(\mathbf{u}_{i}\
eq\mathbf{0}\), the inequality \(\Psi(\mathbf{u}_{i})\geq 1\) always holds. As a result, when \(\Psi(\mathbf{0})>1\), the minimum of \(\Psi(\mathbf{u}_{i})\) is 1; when \(\Psi(\mathbf{0})\leq 1\), the minimum of \(\Psi(\mathbf{u}_{i})\) is \(\Psi(\mathbf{0})\).