## 2 Materials and Methods
### Modified Image Reconstruction Algorithm  
In addition to the algorithm itself, the input signal can also improve the accuracy of imaging [33; 34]. In this field, the rays graph can be regarded as the input of spatial interpolation algorithm. The color and length are two basic graphic features of a certain propagation ray. For red rays which pass through the defective area, compared with longer rays, the ellipse corresponding to a shorter ray is more likely to overlap the defect area. [PERSON] et al. [25] increased the weight of shorter rays, and achieved better imaging results than that of equivalent weight strategy, especially improving the image quality near the sensor, because the rays between adjacent sensors are usually the shortest. Therefore, making full use of the graphic elements in the graph of propagation rays is helpful to improve the accuracy of the spatial interpolation algorithm. By considering this issue, a modified image reconstruction method is presented in this paper. The basis of the proposed method is to segment long rays into short rays in order to make every propagation ray more precise and more conducive to subsequent spatial interpolation. As shown in Figure 2, the algorithm consists of two parts: ray segmentation by elliptical neighborhood (RSEN for short) and corresponding spatial interpolation by segmented ellipse (SISE for short). In the RSEN algorithm, each original ray is segmented several times by iteration. At each iteration, a certain ray is divided into two rays with equal length, and the velocity determined for that ray is calculated by the elliptical neighborhood corresponding to the ray. In the SISE algorithm, based on a group of segmented rays, a segmented ellipse is constructed to estimate the value of surrounding grid cells. Estimating the new value of the segmented ray is the key step in the RSEN method, and an ellipse is used to define the neighborhood range. As shown in Figure 3a, those propagation rays passing through the gray ellipse, such as \(L1\) and \(L2\), can be used to calculate the value of segmented ray. The long axis of the ellipse is the segmented ray, and the shape of the elliptical neighborhood can be expressed as,  
\[c_{1}=b_{1}/a_{1} \tag{1}\]  
where \(a_{1}\) is the long axis of the ellipse, and \(b_{1}\) is the short axis of the ellipse. Therefore, \(c_{1}\) is the control coefficient of the elliptical shape. For convenience of calculation, each original ray is discretized into a set of points. If any point of a certain ray is in the range of the ellipse, it indicates that the ray passes through the ellipse. Whether a certain point is in the elliptical zone can be identified as follows,  
\[f(x,y)=\left\{ \begin{array}{l}1D^{2}x_{b_{1}}/{a_{1}}^{2}+D^{2}y_{a_{1}}/{b_{ 1}}^{2}\leq 1\\ 0D^{2}x_{b_{1}}/{a_{1}}^{2}+D^{2}y_{a_{1}}/{b_{1}}^{2}>1\end{array} \right. \tag{2}\]  
where \(D_{xb1}\) is the distance between a certain discretized point and the short axis of the ellipse, and \(D_{ya1}\) is the distance between a certain discretized point and the long axis of the ellipse. If \(f(x,y)=1\)  
Figure 1: **(a)** An example of the rays graph; **(b)** Gridding of the imaging area; **(c)** Illustration of the EBSI (ellipse-based spatial interpolation) method. it means that the point is within the ellipse. Then, the original ray that the point belongs to is recorded. When all the rays have been identified, the estimated value of the segmented ray can be calculated as  
\[v=\sum_{k=1}^{M}v_{R_{k}}/M \tag{3}\]  
where \(v\) is the estimated value of the segmented ray, \(v_{Rk}\) is the velocity value of a certain ray in the elliptical neighborhood, and \(M\) is the total number of the recorded rays. An original ray will be segmented many times to get a more precise rays graph for spatial interpolation, and the termination condition for iterative segmentation can be given as,  
\[l_{Ri}\leq(l_{\min}/2) \tag{4}\]  
where \(l_{Ri}\) is the length of a certain ray after segmentation, and \(l_{min}\) is the shortest length of all original propagation rays.