# Fundamental Matrix Computing Based on 3D Metrical Distance
## 2 Previous Research  
Lots of works have been done on the fundamental matrix computation in the past 30 years [2; 7; 8; 9; 10; 11; 12]. It's a well and deeply developed area in computer vision; however, there still are some papers on the fundamental matrix published every year. Some traditional methods yield good results, which means very low projective errors to the epipolar lines. This average distance to the epipolar lines is a universal or almost unique criterion to judge the quality of the fundamental matrix. The five-point method [4; 5] is the most popular method that finds the possible solutions for relative camera pose between two calibrated views based on five given corresponding points. The algorithm consists of computing the coefficients of a tenth degree  
Figure 1: Two-view stereo to demonstrate the epipolar theory. polynomial in closed form and subsequently finding its roots. [PERSON] et al. [13] proposed polynomial eigenvalue solutions to the five-point and six-point relative pose problems, which can be solved using the standard efficient numerical algorithms. It is somewhat more stable than solutions proposed by [PERSON] [4; 5] and [PERSON] [14]. Another five-point method [15] was proposed to estimate the fundamental matrix between two non-calibrated cameras from five correspondences of rotation invariant features. Three of the points have to be co-planar and two of them in general position. The solver, combined with Graph-Cut RANSAC, was superior to the seven and eight-point algorithms both in terms of accuracy and the needed sample number on the evaluated 561 publicly available real image pairs. [PERSON] [11] proposed a novel technique for estimating the fundamental matrix based on Least Absolute Deviation (LAD), which is also known as the \(L_{1}\) norm method. [PERSON] et al. [16] proposed a parameterization for the nonlinear optimization which includes the rank 2 constraint. Double quaternion and a scalar are adopted as the parameter set. [PERSON] et al. A two-point fundamental matrix method [17] is even proposed. It makes use of the epipolar theory to estimate the fundamental matrix based on three corresponding epipolar lines instead of seven or eight corresponding points. [PERSON] et al. [12] introduce a novel rank constraint, \(F=A+AT\) and \(rank(A)=3\), on collections of fundamental matrices in multi-view settings. Then iterative re-weighted least squares and alternate direction method of multiplier (ADMM) are used to minimize a \(L1\)-cost function. Reference [18] developed the algorithm MAPSAC to obtain the robust MAP estimate of an arbitrary manifold to compute \(F\). But by its open source code, the SfM result is not stable which means that the reconstructed scene varies from one to another totally different one. As [19] points out, the use of symmetric epipolar distance should be avoided. The error criterion, i.e., the Kanatani distance (REK) [20], is the most effective error criterion found in their experiment for use as a distance function during the outlier removal phase of the \(F\) estimation. Some methods [21; 22] searched for corresponding epipolar lines using points on the convex hull of the silhouette of a single moving object. These methods fail when the scene includes multiple moving objects. Reference [23] extends previous work to scenes having multiple moving objects by using the \"Motion Barcodes\", a temporal signature of lines. Reference [24] developed a new method for calculating the fundamental matrix combined with the feature lines. In [24], the camera orientation elements and relative orientation elements are used as the parameters of the fundamental matrix, and the equivalent relationships are deduced based on the horizontal and vertical feature lines. Recently, after overwhelming studies of deep learning by neural networks, fundamental matrix estimation without correspondences is proposed by novel neural network architectures in an end-to-end manner [25].