# Uncertainty Analysis of 3D Line Reconstruction in a New Minimal Spatial Line Representation
## 2 The Proposed Method
### Conversion between the Proposed Line Representation and the Plucker Line
#### Uncertainty Analysis  
The orientational error tends to be much larger than the positional error in the presence of large levels of noise in image planes [32]. To determine the direction of a line, we can split it into three steps: Firstly, the normal vectors of projection planes \(\mathbf{n}_{c}^{1}\) and \(\mathbf{n}_{c}^{2}\) are computed. This step is affected by noise from camera calibration (in this work, we assume images are calibrated and ignore this item) and line segment extraction. Given two endpoints and constant Gaussian noise that causes displacement of line segment extraction, the computed normal vectors tend to be more accurate for longer line segments. This is the reason why most works ignore short line segments. Secondly, the normal vectors are transformed into the world coordinate system. This step is affected by noise on the rotation estimation of camera orientation. In pose estimation, images that are not well tracked should not be used to triangulate for 3D lines. Thirdly, the direction is computed by the cross product of two normal vectors in world coordinates. This step is affected by the relative orientation of normal vectors. The orientation differences of these two normal vectors come from relative movements vertical to the reference projection plane. Though relative movements along the reference projection plane do generate disparity, it contributes nothing to the direction determination. Degeneracy happens when relative movement is nearly along the projection plane. We can conclude that the quality of the estimated direction by triangulation is related to many factors: ego-motion of views, the accuracy of pose estimation and line segment extraction.  
When discussing the positional error, we assume that the direction is known. An illustration of the distribution of the reconstructed line with fixed direction is given in Figure 6. A projection plane is determined by the camera's optical center and the two endpoints of the line on the image plane. A line can be solved by intersecting two projection planes. Due to the error in camera position estimation and line detection, both projection planes are biased and finally lead to line reconstruction uncertainty. The cylinder indicated by the blue lines illustrates the distribution of the reconstructed line with a fixed direction. With fixed direction, line triangulation degenerates to a case similar to point triangulation. Both estimations only have one degree of freedom. In point triangulation, a good rule of thumb is that the angle between the rays determines the accuracy of reconstruction. While in our case of line reconstruction, the angle between the two projection planes determines the accuracy (size of the distribution). This also explains why further lines tend to preserve larger positional errors. The angle between the two projection planes decreases as the distance between the line and the camera centers increases. By intersecting the cylinder with a plane that passes through the origin and is perpendicular to the direction, the area of the section implies the uncertainty of the line position. \(\Delta\alpha\) and \(\Delta m_{l}\) in Figure 6 implies the position accuracy in the proposed representation.