## 3 Results  
### Measurement Analysis and Focal Length Calculation  
The specifications of the optical system used in this study are given in Table S1. Six lenses were used in this system. The first lens has a reflective surface on its back. The first lens plays the role of a converter, converting the wide field angle of an item to a narrow field angle. The optical system beyond the second lens is of the retro-focus type, and forms an image of a ray passing through the first lens on the image sensor.  
Figure 4 shows the pattern images taken with the system. The patterns were installed 90\({}^{\circ}\) apart, as shown in Figure 3, and there are four patterns in the image, as shown in Figure 4. The omni-directional system is generally designed to have the maximum FOV at the shorter side of the image sensor. Thus, the image obtained at the image sensor has the same number of vertical and horizontal pixels, as shown in Figure 4. The calculation of the number of pixels in the pattern helps determine \(Y_{T}\) using Equation (1). The incident angle, \(\theta\), for the pattern can also be easily determined owing to the uniformity of the pattern. For an accurate calculation of \(Y_{T}\), however, we need to identify the screen center. The center of the image sensor and that of the optical system may not coincide due to assembly error within the system and the image sensor. Figure 5, an enlarged image of the central obscuration in Figure 4, shows a photograph of the screen center obtained from an extension line of the pattern shown in Figure 4. The center of the screen obtained using this method would be at 744 pixels crosswise and 807 pixels lengthwise from the upper left of the screen.  
Figure 6 shows an enlarged image of Point 1 in Figure 4. We used Adobe Photoshop (Adobe Photoshop 2016, Adobe Systems, San Jose, CA, USA) to determine the pattern's cutoff point. The pattern has alternating black and green rectangles of the same size; hence, we can calculate the pattern height at the edge point where the \"G\" (green) value of the pixel information changes rapidly, as shown in Figure 6. The pattern height here would be \(y_{T_{i}}\) in Equation (1).  
Figure 3: Photograph of the measurement system.  
Figure 4: Recorded image.  
Figure 5: Enlarged image in Photoshop.  
Table S2 lists the calculated pattern heights (\(Y_{T}\)) for the pixel sizes given in Table S1 obtained after calculating \(Y_{T}\) for the pixels at the edges of the pattern. Table S3 lists the pattern heights used to obtain \(\theta\), the FOV of the ray incident on the optical system. \(X\), \(Y_{T}\), and \(Y_{B}\) in Table S3 refer to the distance from the center of the system to the point where the pattern is measured, the height from the bottom of the pattern to the particular pattern shape, and the height from the base to the optical system, respectively. \(Y\)\(\gamma\) is the difference between \(Y_{T}\) and \(Y_{B}\), from which \(\theta\) can be calculated. \(DIY\) is the distortion calculated from each \(\theta\) value with the optical system software (Code V, Synopsys, Santa Clara, CA, USA). The \(Y_{T}\) values in Table S3 are the mean values of those presented in Table S2.  
All the variables required to calculate the EFL are thus compiled and the EFL can be calculated by substituting the values from Table S3 into Equation (3). Table S4 lists the EFL values calculated from the FOV (\(\theta\)) and the distortion (\(DIY\)) in Table S3 and \(Y_{T}\) in Table S2 for the images of four identical patterns. In addition, Figure 7 shows a histogram of the EFLs calculated from the measurements tabulated in Table S4. The average EFL is 0.3739 mm. The calculated EFL is slightly greater than the theoretical value, as can be seen in Figure 7.  
Figure 6: Enlarged image at Point 1 in Figure 4.  
Figure 7: Histogram of the measured EFLs.