# Evolutionary Optimization for Robust Epipolar-Geometry Estimation and Outlier Detection
## 5 Experimental Results and Discussion
### Performance of Inlier Classification with Adaptive Thresholding  
In order to verify the performance of the proposed thresholding method for inlier classification (Section 4.4), a stereo pair from the Multiview dataset at the baseline of 20 m was used. To eliminate the effect of other components of the algorithm, such as sampling and the objective function, no outlier was introduced to the images; i.e., the dataset was outlier-free. The results from our method were compared with those of the median-based and covariance-based algorithms. To this end, each algorithm was applied 500 times. In any trial, a minimal sample set of eight points was randomly drawn, and the fundamental matrix was estimated with the normalized eight-point algorithm. For our algorithm, the technique of Section 4.4 was applied to determine the inlier thresholds. For the median-based algorithm, the robust standard deviation of residuals was defined as \(\sigma=1.4826\big{(}1+5/(n-8)\big{)}\sqrt{median_{i}(r_{i})}\), \(i=\{1,\ldots,n\}\)  
Figure 5: Performance of sampling methods on the Church dataset as the outlier ratio increases: (**a**) percentage of outlier-free sample sets, (**b**) estimation accuracy. Since the dataset was outlier-free, \(n\) was equal to the total number of matches, and \(r_{i}\) was the residual of any correspondence with respect to the estimated fundamental matrix. Then, the inlier threshold was calculated as 1.96\(\sigma\). For the covariance-based algorithm, which is the core component of Cov-RANSAC, first, the uncertainty of the estimated model was used to narrow down the total set of matches to the set of potential inliers. Then, the median-based algorithm was applied to the potential inliers for determining the inlier threshold. In order to evaluate the performance of these algorithms, the fraction of runs (from a total of 500 runs) in which a correspondence was classified as inlier was calculated, namely the inlier probability of that match. The inlier probabilities (sorted in ascending order) are shown in Figure 6. Knowing that the dataset was outlier-free, the inlier probability for all of the points was ideally one. Our thresholding algorithm resulted in the most stable and robust solutions. That is, for 92% of the points, the inlier probability was higher than 0.9. For the median-based algorithm, this percentage was only 15%. The covariance-based algorithm had poor performance in comparison with both of the other methods. Our algorithm resulted in inlier ratios higher than 90% for more than 88% of the runs. However, the median-based and covariance-based algorithms yielded inlier ratios greater than 90% at only 32% and 15% of the runs, respectively. To investigate the reason behind the performance of each algorithm, the threshold values determined at each run are illustrated versus the run index as black points in Figure 7. Then, the true residuals, which are residuals of the matches from the true fundamental matrix, were calculated. Note that the true fundamental matrix is the one based on which the matches are synthesized. The maximum  
Figure 6: Inlier probability of correspondences obtained using (**a**) our adaptive thresholding method: (**b**) median-based algorithm; (**c**) covariance-based algorithm. and the median of the true residuals are shown in Figure 7 as the red and green lines, respectively. Then, the parameter \(\sigma=1.4826\big{(}1+5/(n-8)\big{)}\sqrt{median_{i}(r_{i})}\) using true residuals was used to calculate the true median-based threshold. The blue line illustrates this value in Figure 7.  
The true median-based threshold was slightly higher than the true maximum residual, and it would be an ideal choice of threshold only if the fundamental matrix were perfect. However, in these tests, the fundamental matrix was calculated from a minimal sample set of eight correspondences, which had different values of noise and did not necessarily have an ideal spatial configuration either.