Two-View Geometry Estimation by Random Sample and Consensus

The problem of model parameters estimation from data with a presence of outlier measurements often arises in computer vision and methods of robust estimation have to be used. The RANSAC algorithm introduced by Fishler and Bolles in 1981 is the a widely used robust estimator in the field of computer vision. The algorithm is capable of providing good estimates from data contaminated by large (even significantly more than 50%) fraction of outliers. RANSAC is an optimization method that uses a data-driven random sampling of the parameter space to find the extremum of the cost function. Samples of data define points of the parameter space in which the cost function is evaluated and model parameters with the best score are output. This thesis provides a detailed analysis of RANSAC, which is recast as time-constrained optimization – a solution that is optimal with certain confidence is sought in the shortest possible time. Next, the concept of randomized cost function evaluation in RANSAC is introduced and its superiority over the deterministic evaluation is shown. A provably optimal strategy for the randomized cost function evaluation is derived. A known discrepancy, caused by noise on inliers, between theoretical prediction of the time required to find the solution and practically observed running times is traced to a tacit assumptions of RANSAC. The proposed LO-RANSAC algorithm reaches almost perfect agreement with theoretical predictions without any negative impact on the time complexity. A unified method of estimation of model and its degenerate configuration (epipolar geometry and homography of a dominant plane) at the same time without a priori knowledge of the presence of the degenerate configuration (dominant plane) is derived. Next, it is shown that using oriented geometric constraints that arise from a realistic model of physical camera devices, saves non-negligible fraction of computational time. No negative side effect are related to the application of the oriented constraints. An algorithm exploiting (possibly noisy) match quality to modify the sampling strategy is introduced. The quality of a match is an often freely available quantity in the matching problem. The approach increases the efficiency of the algorithm while keeping the same robustness as RANSAC in the worst-case situation (when the match quality is unrelated to whether a correspondence is a mismatch or not). Most of the algorithms in the thesis are motivated by (and presented on) estimation of a multiview geometry. The algorithms are, however, general robust estimation techniques and can be easily used in other application areas too.