# Uncertainty Analysis of 3D Line Reconstruction in a New Minimal Spatial Line Representation
## 3 Results
### Experiment on Synthetic Image Sequence  
To validate the efficiency of the proposed uncertainty analysis in outlier rejection, we also test the proposed method on a synthetic image sequence. The experiment is performed on \"Living Room 2\" from ICL-NUIM dataset, which provides ground truth trajectory and depth maps. The former 90 frames are used to perform an incremental mapping. The first frame is set to be the reference frame and the detected line segments in the reference frame are tracked in the following 89 frames (after 90 frames, the scene is totally changed, line segments on the reference frame are no longer tracked. No new observations are acquired to update the lines' estimation and uncertainty). The line segments are extracted by the LSD algorithm proposed in [43]. Line segments are matched by combining the LBD [44] and relaxed epipolar constraint introduced in [16]. Once a line segment is tracked in the following images, a 3D line and its uncertainty are estimated by triangulation with the reference image. If the uncertainty is too large, the current estimation is abandoned; otherwise, the current estimation is used to update prior estimation. In this experiment, Gaussian noises are added to rotation(standard deviation \(\delta_{r}\) = 0.02 rad for each axis) and translation(standard deviation \(\delta_{t}\) = 5 \(\times\) 10\({}^{-4}\) m for each axis). We assume that the standard deviation of the endpoints is 0.5 pixels. The ground truths of line segments in the first frame are obtained by reprojecting the 2D line segment to 3D as the depths were known. Two items were computed for each estimation: the angular bias between the estimated line and the ground truth line; the mean of distances from endpoints to the ground truth line. The former item indicates the orientation accuracy and the later item indicates the positional accuracy. If the angular error is less than 10 deg and mean distance is less than 0.05 m, an estimation is regarded as well-reconstructed. We compared results of outlier rejection only with reprojection error and incorporating reprojection error with uncertainty. In reprojection error based outlier rejection, the reprojection error for an observation is defined as the sum of distances between line segment endpoints to the projected line. We set a threshold on mean reprojection error \(\tau_{rep}\) = 1 pixel, which is a very strict condition. However, as illustrated in Figure 10b,c, some lines are apparently inconsistent with the scene. In uncertainty based outlier rejection, we set thresholds on each component: \(\tau_{\theta}\) = 0.7 rad, \(\tau_{\phi}\) = 0.7 rad, \(\tau_{m_{l}}\) = 0.2 m, \(\tau_{a}\) = 0.7 rad (in a normal distribution, the length of the 95% confidence interval is approximately four times the standard deviation. To ensure that the standard deviation of each angular component does not exceed 10 degrees, we set these angle thresholds to 40 degrees (0.7 in radius)). As illustrated in Figure 10e,f, no such lines left by adding the uncertainty based outlier rejection. Table 1 reports the quality of the 3D line map after outlier rejection. The mean  
Figure 9: Uncertainty fusion by multiple triangulations. (**a**) Red ellipses denote uncertainty estimated by each triangulation, the green ellipse denotes the final uncertainty after fusion. The blue “**” represents the ground truth of **d** and the green “**” represents the finally estimated **d**. (**b**) The same case for **m** as described in (**a**). (**c**) The 95% confidence interval length of \(\theta\), \(\phi\), \(m_{l}\) and \(\alpha\) decrease along with more and more observations. angular error and distance error is computed for quality comparison. The good line ratio denotes the percentage of the well-reconstructed line in the culled map. After outlier rejection by reprojection error, 26 lines are left but only 15 lines are well-reconstructed. In this sequence, the camera presents frequent on-spot rotations, which results in many lines with large uncertainties.