# Single-Lens Imaging Spectral Restoration Method Based on Gradient Prior Information Optimization
## 3 Methods
In the field of mathematics, inversion processes and deconvolution are both considered inverse problems, which are generally ill posed. In other words, a slight perturbation can lead to a severe deviation in the final solution. To address this issue, more prior information is needed for constraints to be imposed, and appropriate solution methods must be chosen to obtain stable approximate solutions. This approach is known as regularization. In 1992, [PERSON] et al. proposed the total variation (TV) regularization model, initially applied to image denoising problems and later widely used in various image restoration tasks. The expression is as follows:  
\[\|f\|=\sum_{i}\sqrt{\left(D_{x}f\right)_{i}^{2}+\left(D_{y}f\right)_{i}^{2}} \tag{9}\]  
In the above equation, \(D_{x}\) and \(D_{y}\) are first-order gradient operators in the \(x\) and \(y\) directions, respectively, and \(i\) represents the pixel position. MTV (multi-task total variation) is a multi-channel model that calculates gradients for each pixel individually, expressed as [28]:  
\[\|f\|=\sum_{i}\left(\sum_{k}\sqrt{\left(D_{x}f\right)_{i,k}^{2}+\left(D_{y}f \right)_{i,k}^{2}}\right) \tag{10}\]  
where \(k\) represents the spectral segment. Using MTV, we can restore the aliased spectral imaging of a single-lens imaging system, which can be expressed as:  
\[\min_{f}\frac{1}{2}\|Hf-G\|_{2}^{2}+a\|f\| \tag{11}\]  
where \(a\) represents the regularization coefficient, which balances the weights between the first and second terms. To solve the above equation, the alternating direction method of multiplier (ADMM) [29; 30; 31] can be applied, which usually solves the optimization problem, as follows:  
\[G(u,v)=H(u,v)F(u,v)+N(u,v) \tag{4}\]0 \[G(u,v)=H(u,v)F(u,v)+N(u,v) \tag{4}\]1  
where \(x\in R^{n},z\in R^{m}\) are two variables that need to be optimized and \(A\in R^{p\times n}\), \(B\in R^{p\times m}\), and \(C\in R^{p\times n}\). If functions \(f(x)\) and \(g(z)\) are convex in the solution of the above expression, variables \(x\) and \(z\) can be separated, which means that the optimization problem can be decomposed into two separate optimization problems for the two variables. The optimization process involves alternating the optimization of these variables until the optimal solution is obtained. The augmented Lagrangian function for the objective function above is formulated by introducing a quadratic penalty term and is given as follows:  
\[G(u,v)=H(u,v)F(u,v)+N(u,v) \tag{4}\]2where \(\mu\) represents the penalty parameter, which takes a positive value, and \(y\) is also a positive parameter, serving as a dual variable. The process of alternating optimization can be described as optimizing \(x\) and \(z\), and then iteratively updating \(y\); it can be expressed as follows:  
\[G(u,v)=H(u,v)F(u,v)+N(u,v) \tag{4}\]3 \[G(u,v)=H(u,v)F(u,v)+N(u,v) \tag{4}\]4 \[G(u,v)=H(u,v)F(u,v)+N(u,v) \tag{4}\]5  
Scaling \(y\), we define \(b=y/\mu\), which results in:  
\[G(u,v)=H(u,v)F(u,v)+N(u,v) \tag{4}\]6  
Therefore, the iterative process can be updated as follows:  
\[G(u,v)=H(u,v)F(u,v)+N(u,v) \tag{4}\]7 \[G(u,v)=H(u,v)F(u,v)+N(u,v) \tag{4}\]8 \[G(u,v)=H(u,v)F(u,v)+N(u,v) \tag{4}\]9  
The above reconstruction model only provides prior information for the spatial dimension and does not consider prior information for the spectral dimension. Therefore, the key to the restoration algorithm is how to introduce effective spectral prior information. The total variation (TV) model mainly deals with gradient information, so spectral gradient information is introduced as prior information. In terms of spatial gradient information, regions with large gradient values represent the edges of an image. However, the total variation constraint is essentially the L1 norm of the image gradient, which may lead to some larger gradient values (such as edge information) not being well preserved, resulting in a certain degree of edge blurring in the reconstructed image. Therefore, we optimize gradient information by implementing the following improvements:  
\[P=