# Uncertainty Analysis of 3D Line Reconstruction in a New Minimal Spatial Line Representation
## 3 Results
### Experiments on Simulation Data  
1.1 Experiment on Simulation Data to Validate and Visualize Uncertainty of Two-View Line Triangulation  
To validate and visualize the uncertainty of a reconstructed line by two-view triangulation, a randomly generated 3D line is projected to two cameras with known poses and intrinsic matrices. In our experiment, we set intrinsic parameters: \(f_{x}=f_{y}=1000\), \(c_{x}=640\), \(c_{y}=360\). For each trial, every component of rotation is added with Gaussian noise with a standard deviation of 0.02 deg; every component of translation is added with Gaussian noise with a standard deviation of 0.02 m; for line segment extraction, each axis of endpoints is added with Gaussian noises with a standard deviation of 0.5 pixels. This synthetic noisy data is used to solve a 3D line in the proposed representation. The test is repeated 1000 times for reliability. Figure 7 illustrates the results of the Monte Carlo test and uncertainty visualization. Figure 7a shows the estimated 95% confidence region and noisy estimations of \((\theta,\phi)\). Figure 7b shows the estimated 95% confidence region and noisy estimations of \((m_{l},\alpha)\). The statistical results verify the correctness of the uncertainty estimation in the proposed line representation.  
Figure 7: Monte Carlo test to validate the uncertainty estimation of line triangulation. (**a**) The red ellipse denotes the estimated 95% confidence region of \((\theta,\phi)\). Red “+” denotes the ground truth. Each green “+” represents a noisy estimation of \((\theta,\phi)\). (**b**) The red ellipse denotes the estimated 95% confidence region of \((m_{l},\alpha)\). Red “+” denotes the ground truth. Each green “+” represents a noisy estimation of \((m_{l},\alpha)\).  
1.2 Experiment on Simulation Data to Validate and Visualize Uncertainty of Multi-View Line Reconstruction  
As described in Section 2.5, the uncertainty of multi-view line reconstruction can be extended by two-view triangulation. In this section, we use three views as an example. Given a line projected to three cameras, we can obtain an estimation of the spatial line by (7). Three observations can construct three triangulations. Each triangulation contributes to an estimated uncertainty. The uncertainty of the final estimation is obtained by fusing these sub-uncertainties. Uncertainty of line reconstruction from more than three views can also be obtained in this manner. To validate the uncertainty estimation of thee-view line reconstruction, a randomly generated spatial line is projected to three randomly generated cameras. Camera intrinsic parameters are: \(f_{x}=f_{y}=1000\), \(c_{x}=640\), \(c_{y}=360\). Gaussian noises are added to camera poses and endpoints of observations. The standard deviation of Gaussian noise on three rotation angles is all set to 0.05 deg. The standard deviation of Gaussian noise on three camera translation components is all set to 0.005m. The standard deviation of Gaussian noise on the endpoints of the line segment is set to 0.3 pixels. For each trial, a spatial line is estimated with this noisy input. The test is repeated 1000 times for reliability. Figure 8 shows the result of Monte Carlo test on uncertainty estimation of three-view line reconstruction. Figure 8a shows the estimated 95% confidence region and noisy estimations of direction. Figure 8b shows the estimated 95% confidence region and noisy estimations of momentum. The results validate the correctness of fusion-based uncertainty estimation for three-view reconstruction. Thus it can also be extended to uncertainty estimation of multi-view reconstruction.