## 3 Image Processing
### Image Super-Resolution Reconstruction  
Super-resolution is an image restoration technique that utilizes low-resolution images to rebuild a high-resolution image. We acquired nine-frame UME pictures of each defect, so that a multiframe image reconstruction technique could be used to rebuilt high-resolution images. For an image degeneration process, the mathematical model can be described as follows:  
\[Y=DBX+E \tag{8}\]  
where \(X\), \(Y\), and \(E\) are high-resolution image, low-resolution image, and noise, respectively. \(B\) is a fuzzy convolution matrix and \(D\) is a down-sampling matrix, and we can replace \(W=DB\). It is an ill-posed problem that \(X\) was reconstructed by \(Y\) because of multisolution or non-solution. To solve this problem, a classical method named Tikhonov regularization, which is based on remaining raw data to solve a solution by satisfying a priori constraints, was used [24]. The regularization process could be divided into two methods, deterministic regularization and random regularization. In this paper, first the regularization method was performed to optimize the problem in order to make it solvable. This was done through a minimization cost function [24]:  
\[\underset{X}{\operatorname{argmin}}\|Y-WX\|^{2}+\lambda\|CX\|^{2}, \tag{9}\]  
where \(\|Y-WX\|^{2}\) is the error term, \(\lambda\) is the regularization parameter used to balance the relationship between error and high-frequency energy, \(\|CX\|^{2}\) is the penalty term, and \(C\) is a high-pass filter. In general, natural images have finite smooth continuity, and this condition is a constraint term for the convex Function (9) and recovering \(X\) by an iteration process as follows:  
\[\hat{X}_{n+1}=\hat{X}_{n}-\alpha\sum_{k=1}^{P}c_{k}\{W_{k}{}^{T}W_{k}+\lambda_{ k}C^{T}C\}\hat{X}_{k}-W_{k}{}^{T}Y_{k}\}, \tag{10}\]  
where \(\alpha\) is the step size, \(c_{k}\) is the weight coefficient of each frame image, \(\lambda_{k}\) is the regularization parameter, and \(P\) is the number of frames per image.  
Multiframe image super-resolution reconstruction based on the Tikhonov regularization step was as follows:  
Step one: Calculate mapping parameter between images.  
First, initialize affine shifting set.  
For \(i=2\):  
Figure 11: Primary resolution image and low-resolution images: (**a**) primary resolution; (**b–j**) down-sampling images.  
Register current image to first frame: compute the Lucas-Kanade optical using a Gaussian pyramid for the two images in each level.  
End.  
Step two: Divide image mesh, simulate the basic image as a high-resolution image with spine interpolation.  
Step three: Regularize adaptive super-resolution based on Tikhonov:  
Initialize output  
while _iter_ < 200 or _Variation_ < \(10^{-4}\)  
calculate \(W^{T}W\hat{X}_{k}\), \(C^{T}C\hat{X}_{k}\), and \(W_{k}^{T}Y_{k}\);  
For \(k\) = 1:\(P\)-1  
Perform (10)  
End  
Calculate _Variation_ as \(\textit{Variation}=\sum\left(I-I_{iter-1}\right)^{2}\);  
_iter++_;  
End.  
To reduce computation, we did not use the step, weight, and regularization parameter selection principle of [24], but the step was set as 0.2, the weight coefficient was 1, and the regulation parameter was 0.01. Double, triple, and quadruple resolution reconstruction images are shown in Figure 12. Also, the contrast of the three images was enhanced for defect visualization.  
Contrasting the three reconstruction results, excessive resolution reconstruction would produce jagged contour when other parameters and input images were the same. However, the image contrast of low-resolution SR results was higher than the high-resolution reconstruction. Contrasting the three pictures of Figure 12, when SR result was triple the computation and image property was the best, its resolution was \(53\times 53\). Thus, we set the resolution reconstruction number as three in this paper.