# Uncertainty Analysis of 3D Line Reconstruction in a New Minimal Spatial Line Representation
## 3 Results
### Experiment on rEl-World Image Sequence
#### 3.3.2 Experimental Results  
Figure 11 shows an example of the uncertainty fusion and visualization on the real-world dataset. In Figure 11a, for example, line segments are labeled as A, B, C, and D. Taking A as an example, Figure 11b,c show the uncertainty visualization of orientation and position, respectively. Each red ellipse denotes the 95% confidence region of an estimation. The blue ellipse illustrates the confidence region after fusion. Table 2 reports lengths of the final 95% confidence region for these four line segments. By comparing the direction uncertainty of the line segments A and B on the \(\theta\) and \(\phi\) components, we can conclude that longer line segments can be better estimated. As for the line segment C, though C is much longer than A and B, C presents a larger direction uncertainty since the camera centers all lie in about a horizontal plane, which is in line with our previous analysis. Only the component of camera motion that is vertical to the reference projection plane reduces the direction uncertainty. The further line D has a larger positional uncertainty compared with A, which is also consistent with our previous analysis. Figure 11: Example of uncertainty estimation, fusion and visualization for line reconstruction. By fusing these noisy estimations, a posterior estimation with smaller uncertainty can be obtained. The uncertainty of direction and position are visualized by 95% confidence region. (**a**) Four example line segments are labelled on the image, the corresponding confidence interval length of each component are reported in Table 2. (**b**) Uncertainty estimation and fusion of direction components for A. \(\Delta\theta\) and \(\Delta\phi\) correspond to 95% confidence interval lengths of \(\theta\) and \(\phi\), respectively. (**c**) Uncertainty estimation and fusion of positional components for A. \(\Delta m_{l}\) and \(\Delta\alpha\) are the confidence interval lengths of positional components \(m_{l}\) and \(\alpha\). Figure 12 reports the number of lines that presented large uncertainties under different reprojection error thresholds. By varying the threshold of reprojection error, we counted the number of lines that passed the reprojection error rejection but presented large uncertainties. As illustrated in Figure 12, nine tests were conducted and the reprojection error threshold was reduced from 7 to 0.1 pixels. In each test, the blue bar denotes the number of lines that passed the reprojection error rejection, and the yellow bar denotes the number of lines that passed the reprojection error rejection but failed in the uncertainty rejection. By setting a smaller threshold on reprojection error, the reconstructed lines with large uncertainties can be reduced to some extent. However, lines with small reprojection errors can still be reconstructed inaccurately. Although the reprojection error was set to 0.1 pixels, there were still seven lines presenting large uncertainties. The results show that small reprojection errors do not guarantee the accuracy of the reconstructed lines when nearly degenerate cases happen. An additional test is required to reject these lines. Figure 13 shows the reconstructed 3D line map and culled maps by different methods. The first row shows the 3D line map without map culling. The second row shows the results after reprojection error based map culling. The last row shows the results culled by reprojection error and uncertainty. The first column denotes the corresponding line segments in the first image. The last two columns show the corresponding 3D line maps from different views. As noted in the description of the dataset, there are still some mismatches because camera centers all lie in about a horizontal plane. This results in some badly reconstructed lines in Figure 13b,c. Mismatches cause inconsistency in multi-view stereo and present large reprojection errors.