## 4 Image Analyses
### Sunbeam Analyses
#### 4.2.2 Acquisition of Coordinates on the Pond Surface from Images and Data Preprocessing  
In the same way as the Section 4.1.7, we determined the conversion parameters in order to obtain the (x, y) coordinates on the pond surface from the pixel coordinates. The images were used after standardizations of the shooting conditions in the digital camera. As shown in Figure 17, the slope of the straight line k\({}_{\rm I}\) of the locus was obtained from a number of coordinates on the trajectory (e.g., (X\({}_{1}\), Y\({}_{1}\)) and (X\({}_{2}\), Y\({}_{2}\))) on the pond surface, giving tan\(\zeta\) = k\({}_{\rm I}\). The band width of the coordinates on lines perpendicular to the straight line of the optical locus was set to \(\pm\)1 m, whose range included the number of points of 2W\({}_{\rm I}\) = 80. Thus, the 80 points on the perpendicular line were sampled as data from the pixels in the images. To determine the pixel coordinate (X\({}_{\rm p}\), Y\({}_{\rm p}\)) corresponding to the point coordinate (X, Y) on the pond surface, we set a point on the axial center as (X\({}_{0}\), Y\({}_{0}\)) and we used the following expressions where j was scanned from \(-W_{\rm I}\) to W\({}_{\rm I}\)\(-\)1:  
\[ \begin{array}{l}\rm{X}=X_{0}-\frac{j}{W_{\rm I}}\sin\zeta\times 1\ m,\\ \rm{Y}=Y_{0}+\frac{j}{W_{\rm I}}\cos\zeta\times 1\ m,\end{array}  \tag{45}\]  
where the points (X\({}_{3}\), Y\({}_{3}\)) in Figure 17 was defined by the following equation:  
\[ \begin{array}{l}\rm{X}_{3}=X_{0}-\sin\zeta\times 1\ m,\\ \rm{Y}_{3}=Y_{0}+\cos\zeta\times 1\ m.\end{array}  \tag{46}\]  
From the above (X, Y), the pixel values (X\({}_{\rm p}\), Y\({}_{\rm p}\)) on the image are obtained by the method of [PERSON] et al. [17]. Next, a band-shaped RGB data matrix was constructed as follows. In the longitudinal direction, matrix elements corresponded to (X\({}_{0}\), Y\({}_{0}\)) along the optical locus of sunbeam on the pond surface were aligned. In the transverse direction, (X, Y), derived from (X\({}_{0}\), Y\({}_{0}\)), along the perpendicular line to the axial center were aligned. The band-shaped RGB data thus obtained might also include influences from fallen leaves on the pond surface. To avoid this effect, each pixel with the linear RGB values of R > 10 was excluded from the analyses. Figure 17: Sunbeam coordinates on the pond surface images used to calculate the integrated data for image analyses. Likewise, since the band-shaped RGB data matrix was denser than the original pixel data matrix, there were some matrix elements which the pixel data did not correspond to. In order to avoid the vacant elements in the band-shaped matrix, we carried out spline interpolations and then used the matrix of band-shaped RGB data with fully occupied elements. From thus obtained band-shaped data (band width: \(2W_{1}=80\)), in order to obtain the integrated value along the axis perpendicular to the axial center, \(\pm 10\) point-ranges, i.e., range with \(\pm 0.25\) m from the center, were used. The increment of the distance \(\Delta I=10/W_{1}^{2}\times 1\) m was multiplied and the sum was taken to obtain the integral along the transverse directions in the band-shaped RGB matrix. Figure 18 shows an example of the data (14:35 (JST) of 5 June 2021). In Figure 18, the integrated RGB values are depicted for the x-coordinate on the pond surface in the relative coordinate system, i.e., the coordinate system seen from the observation point. Likewise, in this figure, the range of sunbeam trajectory is \(-4.12\) m \(<\) x \(<\)\(-1.83\) m, and the data for image analyses were obtained by subtraction from the integrated RGB data by baselines obtained as the regression lines passing through the averaged values of RGB integrated data of 5 points just below x \(=-4.12\) m and those of 10 points just above x \(=-1.83\) m. The RGB integral values obtained from other sunbeam images were also subjected to the same processing, the base was subtracted and then analyzed.