# Uncertainty Analysis of 3D Line Reconstruction in a New Minimal Spatial Line Representation
## 1 Introduction
### Related Work
A 3D line reconstruction using uncertain projective geometry was proposed in [34]. The results need to be improved because only geometry information is used. Although a series of qualitative measures have been proposed to achieve better reconstruction, quantitative measurements of line reconstruction accuracy are still missing. Representations of 3D lines. The widely adopted representation methods of spatial lines can be divided into two types according to the number of parameters: the linear over-parametrization and the nonlinear minimal four-parameter [2]. The linear over-parametrization representations are suitable for expressing transformation and projection, while the nonlinear minimal four-parameter representations are more favorable for nonlinear optimization. For the first type, a 3D line can be represented by two endpoints [5; 6; 7; 8], the closest point and direction [35], pair of points or pair of planes [18], or Plucker lines [19; 36; 37; 38; 12]. For the nonlinear minimal four-parameter representations, a 3D line was represented by the intersection of two certain planes in [39], then lines parallel to both planes can not be represented. The orthonormal representation of spatial lines was proposed in [1]. There is no internal gauge freedom in the orthonormal representation thus well-suited for optimization. Ref. [10] utilized the orthonormal representation to deal with parameter redundancy in the backend of visual SLAM. In [2], the Plucker lines were transformed into four-vector by Cayley transform [40] to automatically guarantee that the parameters of lines satisfied the Plucker constraints. In both methods, spatial lines are represented in other spaces or by relation with other spaces, the orientation and location are expressed indirectly. Ref. [41] proposed to use two angles and a 2D coordinates to represent a 3D line. Two angles were used to indicate the direction of the line in 3D space. The location was indirectly indicated by a 2D coordinates on the plane that passes through the origin and is perpendicular to the direction. For the special case in [21], a 3D line on the reference frame is parameterized by an angle to specify the direction and a variable \(\lambda\) to specify the location on the projection plane. Redundant spatial line representation methods are not suitable for uncertainty analysis. The redundancy in the representation leads to singular covariance matrices and thus there is no proper probability density function (PDF). Besides, a 3D line has four degrees of freedom, hence it is not easy to visualize the uncertainty by the confidence region. As we know, the confidence region of a one-dimensional variable is a one-dimensional interval, the confidence region of a two-dimensional variable is a two-dimensional ellipse, and the confidence region of a three-dimensional variable is a three-dimensional ellipsoid. Variables with dimensions greater than four are more difficult to visually represent their confidence regions. In order to intuitively visualize the uncertainty and easily evaluate the reliability of the reconstruction line, we expect a minimal spatial line representation that meets two conditions: uses non-redundant form to represent the direction and position of the line in three-space; each component corresponds to a motion type of the line in three-space, then the uncertainty can be interpreted as confidence interval lengths of all components.