# A Deconvolutional Deblurring Algorithm Based on Dual-Channel Images
## 3 The Block Layer Kernel Estimation
### The RSPI of Total Images
#### 3.2.1 Block Layer Iter-Kernel Estimation with RSPI  
First, the correlation coefficient of the two-channel image of this block-layer is calculated as RSPI: \(R_{\
abla-block}=block(\left[\
abla D(Y_{1}^{Total}),-\
abla D(Y_{2}^{Total}) \right]^{T}\left[\
abla D(Y_{1}^{Total}),-\
abla D(Y_{2}^{Total})\right])\) (function _block()_ means the area of the block in the whole images), and then the RSPI is applied to improve the block layer iter-kernel estimation model:  
\[ \begin{split} h_{1-block-layer}^{iter}=\underset{h_{block-layer} ^{iter}\in\mathcal{H}_{iter}}{\operatorname{argmin}}\left(\left\|h_{temp} \right.\oplus\
abla X_{1-block-layer}^{iter}-\
abla Y_{1-block-ker}^{iter- temp}\right\|_{2}^{2}+\lambda_{h}\|R_{\
abla-block}\|^{2}\|h_{temp}\|_{2}^{2} \right)\\ s.t.\sum h_{temp}^{pixel}=1\text{ and }1>h_{temp}^{pixel}\geq 0 \end{split}  \tag{11}\]  
where 1 signifies channel-1, \(h_{1-block-layer}^{iter}\) represents the block layer iter-kernel, \(\mathcal{H}_{iter}\) denotes the pool of allowed block layer kernel \(h_{1-block-layer}^{iter}\), \(h_{temp}\) represents the temp value of \(h_{1-block-layer}^{iter}\) in the iterative kernel estimation process, \(\
abla X_{1-Block-layer}^{iter-temp}\) is the block layer gradient of the ideal image, \(\
abla Y_{1-block-ker}^{iter-temp}\) is the temp iteration blurred image's block layer gradient image in the process, and \(h_{temp}^{pixel}\) is each pixel of the \(h_{temp}\). Expanding this expression yields:  
\[ \begin{split} h_{1-block-layer}^{iter}=\underset{h\in\mathcal{H}}{ \operatorname{argmin}}(& h_{temp}^{T}\oplus(\
abla\left(X_{1- block-layer}^{iter}\right)^{T}\
abla\left(X_{1-block-layer}^{iter}\right)+ \lambda_{h}R_{\
abla-block})\\ &-h_{temp}^{T}\
abla\left(x_{1-block-layer}^{iter}\right)^{T} \
abla Y_{1-block-ker}^{iter-temp})\\ s.t.\sum h_{temp}^{pixel}=1\text{ and }h_{temp}^{pixel}\geq 0 \end{split}  \tag{12}\]  
It is easy to see that there are many ways to solve this problem, as [16] shows. However, the usual method of maximum a posteriori probability estimation (MAP) does not accommodate both the normalization and non-negativity conditions of the kernel in a single objective function. Therefore, this paper found a probability distribution that satisfies the multivariate Dirichlet distribution by variational estimation, to fit the real distribution of the kernel based on statistical theory. In this paper, the variational Dirichlet (VD) method was used to solve Equation (12) [3]; the gradient projection algorithm [3] was used to solve this problem. This method typically uses a given step-size minimum as the iteration termination threshold. Based on experience of selection, it is usually decided to terminate the iteration when the maximum number of iterations is 20 or when the relative change in the objective function is less than \(10^{-5}\). The Dirichlet function for fitting the kernel is better than the other methods in its speed and accuracy of kernel estimation; for example, the gradient-like method and other functions (multivariate mixed Gaussian function etc.) to fit the kernel [3]. With the results obtained from the VD method, the block layer iter-kernel can be obtained. Usually, the _norm-2_ of the kernel \(\left\|h_{temp}\right\|^{2}\) is used as the regularization term in the estimation of the layer iter-kernel estimation [14]. Furthermore, it can be seen in Equation (11) that the computation of the \(\left\|h_{temp}\right\|^{2}\) regularization term adds the unit matrix to the diagonal of the \(\left\|h_{temp}\right\|^{2}\) result of the block layer iterated image. The essence of the RSPI operator involves introducing the two-channel image correlation to the \(\left\|h_{temp}\right\|^{2}\) result of the block layer iterated image. Therefore, the use of RSPI instead of the \(\left\|h_{temp}\right\|^{2}\) as the regularization term can effectively improve the kernel estimation accuracy. As shown in Table 1, three sets of simulated blur images were obtained using images from the GoPro image library.