# Evolutionary Optimization for Robust Epipolar-Geometry Estimation and Outlier Detection
## 2 Related Works  
Random sample consensus (RANSAC) techniques are popular approaches in fields of robust estimation [15]. RANSAC is a method to estimate the parameters of a mathematical model from a set of observations that contain outliers, assuming that the quality of the model and the observations are inter-dependent. More precisely, RANSAC aims at determining the optimal model from an outlier-free sample set of correspondences by maximizing the support size of the model. The inliers that support the model are detected as correspondences whose residuals from the estimated model are less than a given threshold. To find an outlier-free sample set, successive random sampling is performed. To ensure, with probability \(p\), that at least one outlier-free set of \(m\) correspondences is drawn from a dataset containing \(\epsilon\) percent of inliers, at least \(k\) sample sets should be drawn such that \(k\geq\log(1-p)/\log(1-\epsilon^{m})\). This also means that within RANSAC, approximately, \(\log(1-p)^{-1}\) good models are generated before the confidence \(p\) is achieved. By this definition, six major questions are involved in RANSAC-like techniques. (i) Is maximizing the support cardinality a robust objective function when no information about the rate of the outliers is available? (ii) How does one handle a large number of required samples in cases where \(\epsilon\) is very small? (iii) Why does the algorithm not take advantage of the probability of generating good hypotheses before reaching the termination criterion; i.e., why not include the feedback of previous samples in the sampling procedure? (iv) How can the robustness of the estimated model be ensured against the influence of noise since it is relying on a minimal (just enough) subset of inliers? (v) How does one control the effect of degenerate sample sets, which naturally maximize the support cardinality? (vi) Does the threshold used to detect the inliers reflect the uncertainty of the estimated model as well? Some of these questions are answered by different variants of RANSAC, which are discussed here. Unlike the standard RANSAC, there are improved variants, which use robust objective functions to determine the support cardinality. In m-estimator sample consensus (MSAC), inliers are scored based on their fitness, and outliers are scored with a non-zero constant penalty [16]. Maximum likelihood estimation sample consensus (MLESAC) maximizes the log-likelihood of the solution via the RANSAC process by assuming that outliers are distributed uniformly and residuals distribute a Gaussian function over inliers [17]. Maximum a posterior estimation sample consensus (MAPSAC) is also a refined version of MLESAC with Bayesian parameter estimation [18]. These objective functions make models with similar inlier scores more distinguishable. However, they make certain assumptions about the distribution of the residuals either for inliers or outliers. Besides, they still score the models and detect the final inliers by applying a hard threshold to the residuals. Generally, this threshold is determined from the standard deviation of the residuals themselves. Assuming that the noise in data points follows a Gaussian distribution \(\mathrm{N}(0,\sigma)\) and that the residuals are expressed as point-model distances, then the residuals follow a Chi-square distribution. Therefore, the threshold can be expressed as \(\chi^{-1}(p)\sigma^{2}\), where \(\chi(p)\) is the cumulative Chi-square distribution with one degree-of-freedom at probability \(p\) as the fraction of the inliers to be captured (e.g., 0.95). However, this assumption is valid when ignoring the uncertainty of the model itself. Besides, estimating the standard deviation \(\sigma\) at any RANSAC iteration is another problem. One of the most common methods is to estimate this variable from the median of the residuals on the potential inliers that support the best tentative model. Therefore, it might not be robust against outliers [19].