# Fundamental Matrix Computing Based on 3D Metrical Distance
## 1 Introduction  
Nowadays, the demand for 3D graphical models in computer graphics, virtual reality and communication, etc., and their application are increasing rapidly. Much progress in multi-view stereo (MVS) algorithms has been made in recent years, both in terms of precision and performance. Fundamental matrix \(F\) computation is a basic problem in multi-view geometry for the reconstruction of 3D scenes. Once \(F\) is obtained, the projective reconstruction of a scene or object can be inferred from the fundamental matrix [1]. Usually, the eight-point method (8-pt) [2; 3], seven-point method (7-pt), five-point method (5-pt) [4; 5], and gold distance method (Gold) [6] are all based on the RANSAC (RANdom Sample And Consensus) framework. If picture quality and feature matching are precise enough, accurate results can be achieved; however, the estimation from the seven-point method is sensitive to noise. The five-point method makes use of all kinds of constraints but is not very robust. These methods often include a non-linear optimization as the last step, which is time-consuming. The sampling method is based on the distance from feature points to epipolar lines on the image to select the best \(F\) from the candidate \(F\). The \(F\) with least average distance wins and is output as the final result. So the quality of \(F\) is determined by this distance number. According to the epipolar theory in the multi-view geometry of computer vision, as demonstrated in Figure 1, all the epipolar lines, \(l_{1},l_{2}\), intersect at a single point, epipole, which is \(e\) or \(e^{\prime}\) at each image plane. An epipolar line such as \(l_{1}\) or \(l_{2}\) is the intersection of an epipolar plane with the image plane. The epipolar plane, e.g., \(O_{1}O_{2}X_{i}\), is a plane containing the baseline, e.g., \(O_{1}O_{2}\). The epipole is the point of intersection of the line joining the camera centers (the baseline) with the image plane. Equivalently, the epipole is the image in one view of the camera centre of the other view. E.g. \(e\) is the projection of \(O_{2}\) on the left view in Figure 1. It is also the vanishing point of the baseline (translation) direction. This geometry relation between two images for \(X_{i^{\prime}}\) [\(u,u^{\prime}\)] is linked through \(F\). Our method also utilizes the RANSAC framework to compute fundamental matrix \(F\) like most of the other ordinary algorithms. Although every result after RANSAC could have very small average distance from matching feature pixel to the epipolar lines, sometimes the reconstruction result is still unstable if the reconstructed scene is viewed by the other different standard. For example, epipoles of different final best samplings distribute anywhere but not concentrate in a small region. Even the epipoles could be located either within the image or out of the image. Sometimes, the epipole of each computation from \(F\) varies a long distance between them; however, both \(F\)s have very small average distance to epipolar lines. When we utilized this \(F\) to reconstruct the two-view stereo, sometimes the reconstructed 3D scene was abnormal and occasionally kind of projective. This means the average distance to epipolar lines is not the only, nor very robust, criteria to decide the fundamental matrix. Actually, a fundamental matrix with very small average distance to epipolar lines sometimes led to very poor 3D reconstruction results, such as two-view reconstruction based on \(F\). This problem is critical in affine and projective reconstruction in which there is no meaningful metric information about the object space. A new error selection criterion is proposed in this paper to solve the problem that \(F\)s with very small average distance to epipolar lines sometimes result in very bad 3D metrical reconstruction or even result in totally wrong results. The advantage of our criterion is that this method can visually present the 3D reconstructed scene to indicate the quality of \(F\).