# A Variational Neural Network Based on Algorithm Unfolding for Image Blind Deblurring
## 3 Methodology
### Model Design and Structure  
Since the \(L_{0}\) regularizer typically sets many weights to zero, resulting in greater sparsity, we have decided to replace the existing regularizer with it. The \(L_{0}\) regularizer is a widely used technique in image restoration that suppresses noise while preserving detail, particularly edges and structures. By constraining the gradient of the image, it prevents excessive smoothing or blurring, encouraging the model to retain finer details. It penalizes non-zero gradients, which helps preserve edge information, enhance clarity, and maintain structural integrity in the restored image, thus minimizing artifacts and preventing detail loss. This ultimately improves robustness in restoration tasks. Based on Equation (3), it can be expressed as follows:  
\[ \begin{split} E\Big{(}\{g_{i}\}_{i=1}^{C},k\Big{)}=\sum\limits_{i= 1}^{C}\Big{(}\frac{1}{2}\|k*g_{i}-v_{i}\|_{2}^{2}+\lambda_{i}\|g_{i}\|_{0} \Big{)}+\frac{\gamma}{2}\|k\|_{2}^{2}\\ \|k\|_{1}=1,\ k\geq 0\end{split}  \tag{5}\]  
where \(\{g_{i}\}_{i=1}^{C}=\
abla u\), \(\{v_{i}\}_{i=1}^{C}\) is the feature map of the blurred image obtained by convolution and ReLU activation function. By introducing auxiliary variables \(\{z_{i}\}_{i=1}^{C}=\{g_{i}\}_{i=1}^{C}\), using HQS, we can obtain the following:  
\[ \begin{split} E\Big{(}\{g_{i},z_{i}\}_{i=1}^{C},k\Big{)}=\sum \limits_{i=1}^{C}\Big{(}\frac{1}{2}\|k*g_{i}-v_{i}\|_{2}^{2}+\lambda_{i}\|z_{i }\|_{0}+\frac{\xi_{i}}{2}\|g_{i}-z_{i}\|_{2}^{2}\Big{)}+\frac{\gamma}{2}\|k\|_ {2}^{2}\\ \|k\|_{1}=1,\ k\geq 0\end{split}  \tag{6}\]  
where \(\xi_{i}\) is a penalty parameter. Algorithm 1 summarizes the iterative algorithm for estimating the blur kernel. Here, \(\delta\) represents the impulse function, while \(\mathcal{F}(\cdot)\) and \(\mathcal{F}^{-1}(\cdot)\) denote the fast Fourier transform and its inverse, respectively. \(H\!\mathit{e}u_{\sigma}(\cdot)\) is a thresholding method proposed by [PERSON] et al. [21] that aims to approximate the hard-thresholding formula in compressed sensing. In contrast to the approach in [PERSON]'s model [22], we utilize a hard-thresholding function and introduce the \(L_{0}\) regularizer as a constraint. Figure 1: The flowchart of the algorithm unfolding. To improve image detail restoration, we utilize the following energy functional:  
\[E(u)=\frac{1}{2}\|k*u-y\|_{2}^{2}+\eta\sum_{i=1}^{N}\Phi_{i}(G_{i}*u) \tag{7}\]  
where \(\Phi_{i}\) is a trainable activation function, \(G_{i}\) is a convolution kernel designed based on the Gabor filters [23], \(N\) represents the number of channels, and \(\eta\) is a penalty parameter. Gabor filters effectively represent frequency domain structures while preserving time domain information, and they possess multi-scale and multi-directional selectivity. This allows for the effective capture of image features across different scales and orientations, making them suitable for initializing the convolution kernel. They can be expressed as follows:  
\[ \begin{gathered} G(p,q)=\frac{\left\|k_{p,q}\right\|^{2}}{\sigma^{2 }}e^{-(\frac{\|k_{p,q}\|^{2}\|^{2}\|^{2}}{2\sigma^{2}})}[e^{\tilde{k}_{p,q}^{2} }-e^{-\sigma^{2}/2}]\\ z=(x,y),\;\;k_{p,q}=k_{q}e^{\tilde{k}_{p}},\;k_{q}=\frac{(\frac{ \gamma}{2})}{\sqrt{2}^{(q-1)}},\;k_{p}=u\frac{\gamma}{\eta},\;q=0,1\dots,Q,\;p= 0,1,\dots,P\end{gathered}  \tag{8}\]  
where \(\sigma=2\pi\), \(p\) and \(q\) represent the direction and scale parameters, respectively, which control the rotation angle and frequency of the convolutional kernel. The proximal gradient descent method is employed to solve Equation (7), which is expressed as follows:  
\[u^{t}=u^{t-1}-\left(\mu\overset{\sim}{k}*\left(k*u^{t-1}-y\right)+\eta\sum_{i =1}^{N}\overset{\sim}{G_{i}}*\phi_{i}^{t}\left(G_{i}^{t}*u^{t-1}\right)\right) \tag{9}\]  
where \(\overset{\sim}{\cdot}\) is its corresponding conjugate, \(t\) represents the iteration index, and \(\sum_{i=1}^{N}\phi_{i}^{t}(\cdot)=\Phi_{i}^{t}(\cdot)\) denotes the Gaussian radial basis function (\(RBF\)).