## 2 Super-Resolution Methods
### Establishment of Dictionary Set  
Section 2.1 discusses the dictionary, called \(D_{LR}\), which is mentioned as having a shared sparse representation with \(D_{SR}\). To achieve our desired solution, we need to construct a joint dictionary. The optimization problem for this joint dictionary is represented in the downward form:  
\[\min_{D_{LR},D_{SR},u_{U}}\frac{1}{C_{1}}\left\|GD_{LR}a_{U}-GT_{LR}\right\|_{2} ^{2}+\frac{1}{C_{2}}\left\|D_{SR}a_{U}-T_{SR}\right\|_{2}^{2}+\lambda(\frac{1} {C_{1}}+\frac{1}{C_{2}})\left|a_{U}\right|_{1} \tag{16}\]  
The variables \(C_{1}\) and \(C_{2}\) in the formula above are used to balance the costs of the Dictionary-based Super-Resolution \(D_{SR}\) and Dictionary-based Low-Resolution \(D_{LR}\) methods. Since the reconstruction method requires a sliding window, the dictionary set is also built using a sliding window approach with a window size of 5 and step size of 1. However, this practice leads to inconsistent matrix sizes, so we up-sample the low-resolution image before building the dictionary set. The values of \(C_{1}\) and \(C_{2}\) are set to the number of atoms corresponding to \(D_{LR}\) and \(D_{SR}\), respectively. In our case, \(C_{2}\) is set to 25 and \(C_{1}\) is set to 100, as our edge extraction operator \(G\) is defined as the first and second derivatives in the x and y directions. Additionally, we further simplify Formula (16):  
\[\begin{array}{l}\min_{D_{U},u_{U}}\left\|D_{U}a_{U}-T_{U}\right\|_{2}^{2}+ \lambda\left|a_{U}\right|_{1}\\ D_{U}=\left[\begin{array}{cc}\frac{1}{10}GD_{LR}\\ \frac{1}{5}D_{SR}\end{array}\right]&T_{U}=\left[\begin{array}{cc}\frac{1}{10 }GT_{LR}\\ \frac{1}{5}T_{SR}\end{array}\right]\end{array} \tag{17}\]  
According to [33], to optimize the problem of solving Formula (17), it is necessary to constrain \(D_{U}\). This ensures the existence of both \(d_{U,i}^{\text{rank }(D_{U})\times 1}\) and \(a_{U}\) atomic sets within \(D_{U}\), which leads to consistent results for \(d_{U,i}^{\text{rank }(D_{U})\times 1}\cdot a_{U}\) and approaching 0 for \(\left|a_{U}\right|_{1}\). As a result, it is crucial to normalize the atomic sets of \(D_{U}\) and impose constraints before training. Representation of the entire dictionary set can be expressed using the following formula, with K being the total number of atomic sets in the dictionary set, set to 512 during training:  
\[D_{U}^{*}=\underset{D_{U},u_{U}}{\text{argmin}}\|D_{U}a_{U}-T_{U}\|_{2}^{2}+ \lambda\left|a_{U}\right|_{1}\quad\text{s.f.}\left\|d_{u,i}^{rank(a_{U})\times 1 }\right\|_{2}^{2}\leq 1,i=1,2,3,\ldots K \tag{18}\]  
Figure 1: Change in the overlap coefficient in the sliding window super-resolution process. The red window represents one step of the rebuilding process, while the yellow window represents another step. The overlapping portion carries a weight of 2, whereas the non-overlapping section holds a weight of 1.  
The formula is not convex on both \(D_{U}\) and \(\alpha_{U}\). However, it is convex on one of them. To address this, we follow the approach proposed in [32]. We first initialize \(D_{U}\) and solve \(\alpha_{U}\) using the feature symbol search algorithm. Then, we fix \(\alpha_{U}\) and solve the duality problem of \(\alpha_{U}\) using the conjugate gradient method. Solving the duality problem offers the advantage of generating fewer variables during the optimization process, thus making it faster.