# Fundamental Matrix Computing Based on 3D Metrical Distance
## 5 Experiment
### Features Matching  
Figure 2 shows the features extracted by SIFT on dataset _Dino_ and _ZhangLan_. Many different stereo matching methods are fully and deeply researched [29]. Thus, our algorithm adopts SIFT [30] for the feature matching \([u,u^{\prime}]\) of two images. It is the first step to compute \(F\). Here, Euclidean distance is the criterion to measure the features' difference. The ratio of the nearest distance over the second nearest distance was set to be less than threshold \(T_{matching}=0.55\). Otherwise it will not be labeled as a feature.  
Actually, the other feature matching algorithms such as Harris, BRISK, SURF, ORB, etc., can still work. We choose SIFT only because it is typical and good enough to compare with other \(F\) computation methods, not because it is the best or most accurate matching method. SIFT has some mismatching features most of the time. Because of the RANSAC  
Figure 2: Feature matching for F matrix computation. (**a**) _Dino_. (**b**) _ZhangLan_. The matching of _Simu_ are shown in Figure 3, and were produced from a human-made scene. Different feature colors distinguish the different feature matching.  
framework used in this paper, the mismatching can be culled out by the sampling and iteration mechanism as shown in Figure 2. This is one of the differences to most of \(F\) computation methods that always carefully remove the wrong feature matching manually.  
For the matching of simulated scene \(Simu\) displayed in Figure 3, all the points in the image are perfectly matched because the scene was manually made. So the simulation matching of scene \(Simu\) in Figure 3 is not listed in Figure 2. The 3D points \(X\) and camera positions are displayed in Figure 3a and the projection image of \(X\) at the first and second viewpoint are displayed in Figure 3b and Figure 3c, respectively.  
For feature matching from SIFT in Figure 2, there must be some outliers of matching with long distances to epipolar lines or obvious re-projection error of \(X\). If outliers are validated by the criteria mentioned in this paper, the incorrect matching should be removed from the feature matching sequence and not be counted into error calculation. This is an important step to improve precision and robustness of our method.  
Figure 4 shows part of the typical epipoles and their corresponding epipolar lines chosen randomly and computed with our 3D metrical criterion for the final \(F\) under the RANSAC framework like the other traditional methods. As we can see, the epipoles are close to the epipolar lines. The distances are less than one pixel distance on average as shown in Table 1.  
Figure 3: The simulated scene. (**a**) 100 points on two planes are add some random disposition. The noise make the lines and two planes have obvious but slight vibration. The red \(\cdot\)\(\cdot\) and blue \(\cdot\)\(\cdot\)\(\cdot\) are the positions of cameras. In order to make the figure clear, we use \(\cdot\)\(\cdot\)\(\cdot\) to indicate the points. (**b,c**) are the projection of 3D points on two images w.r.t the cameras.  
Figure 4: The epipolar lines and features of \(Dino\) (**a,b**), \(ZhangLan\) (**c,d**), and \(Simu\) (**e,f**). Here, only 20 epipolar lines were drawn in case all the lines concentrated in one image are too dense for reading.