# A 3D Estimation Method Using an Omnidirectional Camera and a Spherical Mirror
## 6 Simulation
### Discussion of Room Model Simulation
Because of the system's one-shot estimation capability, this method is expected to be effective for endoscopes that capture images of dynamic tubular objects inside the body and for surveying during the excavation of tunnels. Figure 12: Change in Estimation Accuracy against Change in Angle between Optical Axis and Camera Ray in the Spherical Mirror: MAE increases as the angle \(\phi\) of incidence at the reflection point \(\mathbf{X_{s}}\) decreases or increases. On the other hand, MAE decreases and accuracy is stable in the central area. Verification of the Effect of 3D Position Estimation Error of a Spherical Mirror on 3D Information Estimation  
The estimation results using the ground truth positions of the spherical mirrors are presented in Figure 15a, while Figure 15b displays the error compared to the ground truth. In this case, the proposed method achieves an MAE of 0.66625 [m]. These estimation results are then applied to the equation that describes the effect of the error in estimating the 3D position of the spherical mirror on the estimation of 3D information, as discussed in Section 5.3. Figure 16 illustrates the outcome of applying Equation to \(\|\mathbf{C_{s}}\|=1.03358\) [m]. It exhibits estimation results similar those of Figure 11a, indicating the potential influence of the estimation of the spherical mirror position. Furthermore, Figure 16 confirms that the effect of the spherical mirror position estimation is more prominent in the center and at the edge of the mirror image. As discussed earlier, a slight change in the position of the spherical mirror results in a significant variation in the estimated 3D position due to the small value of \(h\). Moreover, the reflection position at the mirror's edge differs significantly from that of the spherical mirror, suggesting that the estimated 3D position also undergoes significant changes due to substantial variations in the incident ray from the object. Figure 14: Change in Parameter \(h\) against Change in Angle \(\phi\): The parameters \(h\) increase as angle \(\phi\) increases or decreases. This feature is similar to the change in MAE against the angle of incidence. Figure 13: Definition of Parameter \(h\): The parameter \(h\) is defined as the size of the line segment perpendicular from the camera to the incident ray \(\mathbf{v_{r}}\) from the object to the reflection point \(\mathbf{X_{s}}\) in 3D space. Next, to examine the relative size of the sphere in the camera's field of view (FOV) and its effect on the distance error, Figure 17 shows the variation in the estimation results for changes in the Z-axis position of the spherical mirror according to Equation (13). As an example of this validation, the poster in the upper right of the mirror in Figure 9a is considered. Figure 17 shows how the estimation of 3D information changes with changes in the position of the spherical mirror when two images are taken at the exact location. The vertical dashed line represents the ground truth of the 3D information, and the horizontal dashed line represents the ground truth of the spherical mirror position. In Figure 17, the estimated values of the 3D information exhibit sharp changes at a certain point. Its point corresponds to the situation where the imaging position approaches the edge of the mirror image region due to changes in the position of the spherical mirror. If the position of the spherical mirror changes further, it goes beyond the mirror image region, leading to a significant decrease in the reliability of the estimation. On the other hand, when the imaging is conducted in the central region of the mirror image, the changes in the estimated values are smaller. In other words, when the relative size of the sphere in the camera's FOV is small, it becomes more sensitive to changes in the position of the spherical mirror, resulting in a larger impact on distance estimation errors.