## 2 Super-Resolution Methods
### Infrared Polarization Super-Resolution Reconstruction Model  
To obtain the high-resolution image \(I_{SR}\), the low-resolution image \(I_{LR}\) is used. It is assumed that \(I_{SR}\) can be linearly represented as \(\alpha\) using the over-complete basis vector space \(D_{SR}\). According to the theory of root sparse coding primitives, we can infer that the following formula holds when \(\alpha\) is sufficiently sparse:  
\[\widetilde{I}_{SR}=D_{SR}\alpha\quad\exists\alpha\in\mathbf{R}^{\mathbf{k}}\; with\;\|\alpha\|_{0}<<k \tag{5}\]  
The over-complete basis vector \(D_{SR}\), which can also be referred to as an over-complete dictionary set, is utilized. The objective is to minimize the Euclidean distance between the reconstructed image \(\widetilde{I}_{SR}\) and the actual high-resolution image, while ensuring that \(\alpha\) satisfies the sparsity condition. This can be formulated as an optimization problem:  
\[\min_{\alpha}\|\alpha\|_{0}\quad s.t.\quad\|D_{SR}\alpha-I_{SR}\|_{2}^{2}<\varepsilon \tag{6}\]  
Formula (6) satisfies the condition that the square of the 2-norm is smaller than the error \(\varepsilon\), rather than just the 2-norm being smaller than the error \(\varepsilon\). This choice is justified by previous research which has shown that in the field of super-resolution reconstruction, penalizing high-frequency errors by using the square of the 2-norm yields better results [26; 27; 28]. When performing super-resolution tasks with the input of image \(I_{LR}\), it is evident that Formula (6) does not have enough known optimization solutions. In this context, the existence of dictionary \(D_{LR}\) is assumed. This dictionary shares the sparse representation \(\alpha\) with \(D_{SR}\), and solving Formula (6) can be equivalently stated as the following equation:  
\[\min_{\alpha}\|\alpha\|_{0}\quad s.t.\quad\|GD_{LR}\alpha-GI_{LR}\|_{2}^{2}<\varepsilon \tag{7}\]  
In the equation above, \(G\) denotes an edge extraction operator, responsible for generating edge features for Formula (7). We observe that the pseudo-polarization information is primarily concentrated in the high-frequency portion. Precisely reconstructing the missing high-frequency content in the targeted high-resolution image holds the utmost significance. Extracting the high-frequency components from low-resolution images proves advantageous for the reconstruction of high-resolution images [30]. Formula (7) corresponds to an L0 problem, which can be approximated through an L1 problem using the Lagrange multiplier method:  
\[\min_{\alpha}\|GD_{LR}\alpha-GI_{LR}\|_{2}^{2}+\lambda\|\alpha\|_{1} \tag{8}\]  
In Formula (8), \(\lambda\) is an adjustable parameter that is associated with the noise variance of the input image \(I_{LR}\). The derivation process will be conducted later on. With the given image \(I_{LR}\) and over-complete dictionary sets \(D_{SR}\) and \(D_{SR}\), a set of optimal sparse representations \(\alpha^{*}\) is obtained by solving Formula (8). Subsequently, the reconstructed image \(\widetilde{I}_{SR}\) can be obtained by applying it to Formula (5). The resulting image satisfies the constraint stated in Formula (6). We introduce adjustable parameter \(\lambda\) into Formula (8), which is derived and analyzed using the Maximum A Posteriori (MAP) theory. According to the MAP theory, the squared term of the L2 norm in Formula (8) can be interpreted as the maximum likelihood estimate of \(\alpha\). Additionally, part of the L1 norm can be considered as the Laplace prior of \(\alpha\). Without taking into account the edge extraction operator \(G\), this prior can be expressed as follows:  
\[\alpha^{*}=\text{ar}\ g\underset{\alpha}{\text{max}}\ p(I_{LR}\ \mid\ D_{LR},\alpha)p(\alpha) \tag{9}\]  
The \(\alpha\) variable in Formula (9) follows a Laplace distribution with the parameter (0, \(b\)).