# A 3D Estimation Method Using an Omnidirectional Camera and a Spherical Mirror
## 6 Simulation
### Discussion of Room Model Simulation  
Figure 11b indicates that the proposed method achieves minor errors in many locations within the mirror image, suggesting that estimating the 3D information with reasonable accuracy is possible. However, there are two main reasons for the lower accuracy observed in some cases. One reason is the difference in appearance between different viewpoints. The direct image captured by the omnidirectional camera and the mirror image reflected by the spherical mirror makes it possible to have different appearances from different positions. These differences correspond to the motion disparity of a camera in stereo, etc. For example, if we focus on the trash can in the lower right corner of the mirror in the figure, its appearance will differ from the trash can reflected in the mirror. As a result, the error increases at the corresponding position in Figure 11b. Figure 11: Distance and error estimation from the simulated scene: (**a**) The estimation map by the proposed method. The map is visualized by changing the hue linearly. (**b**) The error map. The map is visualized by varying the brightness linearly. The map is obtained by the difference between the ground truth at each pixel and the result estimated by the proposed method. The errors are larger at the edge and the center of the mirror image region. Figure 9: Input Images in Comparetion Simulation in Room Model: (**a**) The shooting omnidirectional image. The mirror image is in the center. (**b**) The GT image of the shooting image. Another factor contributing to the lower accuracy is the difference in reflection positions within the mirror image region. Figure 12 illustrates a graph showing the change in estimation accuracy based on the angle between the optical axis and the ray to the camera at the camera center on the spherical mirror. The graph reveals that the Mean Absolute Error (MAE) increases as the angle decreases or increases. These regions correspond to the center and the edges of the specular region. Consequently, these areas in the figure display more prominent errors. To analyze this accuracy discrepancy, a parameter called \(h\) is introduced, which represents the size of the line segment perpendicular from the camera to the reflected ray from the object to the reflection point in 3D space (refer to Figure 13). It is defined by Equation (14). \[h=d_{s}\|\mathbf{v_{i}}-\mathbf{v_{r}}(\mathbf{v_{i}}^{T}\mathbf{v_{r}})\| \tag{14}\]  
Figure 14 presents the variation in \(h\) with the angle between the optical axis and the ray to the camera at the camera center on the spherical mirror. The horizontal axis represents \(\phi\) (angle in degrees), and the vertical axis represents \(h\) (distance in meters). When comparing this graph with Figure 12, it appears to be inverted. These two figures demonstrate that the estimation accuracy is related to \(h\). Smaller values of \(h\) indicate more significant changes in the 3D position when shifting one pixel on the omnidirectional image. In other words, the estimated 3D position undergoes significant variations when the matching is shifted by a single pixel, leading to reduced estimation accuracy. As observed in Figure 13, the central and edge regions of the mirror image in the omnidirectional image (the boundary between the omnidirectional image and the non-omnidirectional image part) exhibit small values of \(h\). Based on this information, it is possible to assume that the proposed method's accuracy is guaranteed when \(h\) is larger than 0.4 m, which corresponds to regions other than the central and edge regions. The range where the above accuracy is guaranteed is the area of the side of the system due to the characteristics of the optical design. In addition, the accuracy of 3D information estimation decreases as the depth value increases. Considering these characteristics, this method is effective for the 3D measurement of tubular objects.