# Uncertainty Analysis of 3D Line Reconstruction in a New Minimal Spatial Line Representation
## 1 Introduction
### Related Work  
Degeneracy and map culling in 3D line reconstruction. Spatial lines can be reconstructed from observations with known poses and further can be used to determine the poses of new cameras. Different from triangulation for points, line segment correspondences between two views do not have any geometry constraints. The line can be solved uniquely by intersecting two projection planes. We can't measure the accuracy by reprojection error as in two-view triangulation for 3D points. Using more observations and minimizing the sum of reprojection errors in all observed images is a convention of estimating a spatial line. To make things worse, degeneracy for line triangulation is far more severe than for point triangulation [18]. In degenerate cases, lines will be reconstructed with large uncertainties. To deal specifically with the degeneracy, the detected line segments that are nearly-aligned (\(\leq\)10\({}^{\circ}\)) with the epipolar line were labeled as degenerated cases in [13] and [19]. Ref. [17] avoided degeneracy by measuring the distance between the reconstructed 3D line and the camera baseline. Ref. [20; 21] detected degenerate cases by measuring the angular span of all projection planes' normal vectors. However, these methods specify degeneracy by setting thresholds on some parameters with personal experience. Small reprojection errors and large baselines do not guarantee a well-reconstructed line. Most of the existing line reconstruction work does not consider the degeneracy specifically. To get a reliable line map, lines seen from less than three viewpoints or in less than 25% of the frames from which they were expected to be seen were discarded in [6]. In [PERSON]'s series of work on 3D line reconstruction [14; 16], clustering approaches [22; 23] were applied to generate the final line-based 3D model. Ref. [15] proposed to eliminate outliers by line grouping, a representative line was estimated for each group and these lines that do not form a group with at least one other line were discarded. In the group-based methods, some lines that are no outliers but different 3D lines are merged into one line. Though effective to some extent, these methods generally need sufficient observations to ensure a good line thus not suitable for limited views. Many poorly localized lines will present in the final line map if we simply set a threshold on the reprojection error. Subsequent motion estimation and mesh generation will suffer from these badly reconstructed lines. Uncertainty of 3D reconstruction. Uncertainty is a common criterion used to measure the reliability of estimations. The estimation and propagation of uncertainty have been well studied mathematically [18; 24]. The uncertainties of reconstruction parameters can be obtained by propagating the uncertainties of input parameters, i.e., detected features in images. Depth uncertainty of point reconstruction has been well-studied [25; 26; 27; 28; 29]. Recently, [30] proposed an efficient algorithm for uncertainty propagation which works with large scale 3D reconstruction. In comparison, the research on the uncertainty of line reconstruction is much less. Ref. [31] analyzed the stereo line reconstruction and demonstrated that a line can be more accurately reconstructed as an intersection of two planes than reconstructing pair of endpoints. To be more general, [32] presented a performance analysis of reconstructing a 3D line from two arbitrary perspective views. A spatial line was represented by three direction cosines and a point on the line. Inverse perspective equations were derived to describe 3D line error. Simulation experiments were conducted to illustrate the effect of multiple factors on errors in line reconstruction. Ref. [33] analyzed error generation and propagation in a multilayer feature graph including line segments, and used them as observation error models in the extended Kalman filter. Uncertainty of 2D line segment extraction in images and 3D line segment extraction by RGB-D sensors are investigated.