## 2 Methods
### Adaptive Regularization Algorithm Based on Maximum a Posteriori Estimation  
#### 2.2.1 Related Algorithm  
The algorithm in this paper is proposed on the basis of the maximum a posteriori estimation, which is described in detail below. The implication of the super-resolution reconstruction algorithm based on maximum a posteriori estimation is that if the sequence of low-resolution images is known; the posteriori probability of high-resolution images is maximized as follows:  
\[\hat{x}_{MAP}=\operatorname*{argmax}_{x}\Bigl{[}\Pr\bigl{\{}x\bigl{|}y\bigr{\}} \Bigr{]} \tag{4}\]  
According to the Bayesian formula, the following formula is obtained:  
\[\hat{x}_{MAP}=\operatorname*{argmax}_{x}\Biggl{[}\frac{\Pr\bigl{\{}x\bigl{|}y \bigr{\}}\Pr\bigl{\{}x\bigr{\}}}{\Pr\bigl{\{}y\bigr{\}}}\Biggr{]} \tag{5}\]  
Take the logarithm to the right side of Formula (5):  
\[\hat{x}_{MAP}=\operatorname*{argmax}[\log\Pr\bigl{\{}y\bigl{|}x\bigr{\}}+ \log\Pr\bigl{\{}x\bigr{\}}] \tag{6}\]  
where \(\log\Pr\bigl{\{}y\bigl{|}x\bigr{\}}\) is the logarithm of the maximum likelihood function; \(\log\Pr\bigl{\{}x\bigr{\}}\) is the logarithm of the prior probability of \(x\). If the image noise is assumed to be Gauss noise with mean values of 0 and variance of \(\sigma_{k}^{2}\), the overall probability function for estimating \(y_{k}\) by \(x\) is:  
\[\Pr\bigl{(}y_{k}\bigl{|}x\bigr{)}=\prod_{\forall x,y}\frac{1}{\sigma_{k}\sqrt {2\pi}}\exp\Biggl{(}-\frac{\bigl{(}\hat{y}_{k}-y_{k}\bigr{)}^{2}}{2\sigma_{k}^ {2}}\Biggr{)} \tag{7}\]  
In Formula (7), \(\hat{y}_{k}\) simulates low-resolution images and is given by \(\hat{y}_{k}=A_{k}\hat{x}\), where \(A_{k}\) is a fuzzy matrix. Assuming that the images are independent of each other, the probability density function of low-resolution image sequences can be expressed as Formula (8). \[\Pr\bigl{(}y_{k}\bigl{|}x\bigr{)}=\prod_{\forall k}\Biggl{[}\prod_{\forall x,y }\frac{1}{\sigma_{k}\sqrt{2\pi}}\exp\Biggl{(}-\frac{\bigl{(}\hat{y}_{k}-y_{k} \bigr{)}^{2}}{2\sigma_{k}^{2}}\Biggr{)}\Biggr{]} \tag{8}\]  
A common prior probability estimation method is to smooth the image, that is, to punish the adjacent pixel units with large differences in the image, and its probability density function can be assumed to be in the following form:  
\[\Pr\bigl{(}x\bigr{)}=\exp\biggl{[}-\frac{1}{\lambda}\bigl{\|}Qx\bigr{\|}^{2} \biggr{]} \tag{9}\]  
where \(\lambda\) is a control parameter to control the peak of probability density distribution. Substitute Formula (9) into Formula (6):  
\[\hat{x}_{MAP}=\operatorname*{argmax}\Biggl{[}\log\frac{N}{\sigma_{K}\sqrt{2 \pi}}-\sum\frac{\bigl{(}\hat{y}_{k}-y_{k}\bigr{)}^{2}}{2\sigma_{k}^{2}}-\frac{ 1}{\lambda}\bigl{\|}Qx\bigr{\|}^{2}\Biggr{]} \tag{10}\]  
In Formula (10), the first term in the middle bracket on the right side of the equal sign is a constant term, which can be eliminated directly; then the negative sign of the other two terms ischanged to a positive sign, and assuming that the noise of each image has the same variance, the above maximization problem can be transformed into the following minimization problem:  
\[\hat{x}_{MAP}=\text{argmin}\left[\sum{(\hat{y}_{k}-y_{k})^{2}}+\frac{2\sigma^{2 }}{\lambda}\|Qx\|^{2}\right] \tag{11}\]  
Set \(\alpha_{1}=\frac{2\sigma^{2}}{\lambda}\); the above formula can be written as:  
\[\hat{x}_{MAP}=\text{argmin}\left[\|y-Ax\|^{2}+\alpha_{1}\|Qx\|^{2}\right] \tag{12}\]  
where \(\alpha_{1}\) is the regularization parameter. The necessary condition for minimizing the above formula is that the partial differential value of \(\|y-Ax\|^{2}+\alpha_{1}\|Qx\|^{2}\) with respect to \(x\) is zero, as Formula (13). \[(A^{T}A+\alpha_{1}Q^{T}Q)x=A^{T}y \tag{13}\]  
According to Formula (12), regularization parameter \(\alpha_{1}\) controls the relative contribution of \(\|y-Ax\|^{2}\) and \(\|Qx\|^{2}\) in the process of solving, and constrains the degree of distortion and smoothness of reconstructed images. If \(\alpha_{1}\) is too small, the problem of noise is not solved well, and the reconstructed image will still be distorted.