Finding Corresponding Points Based on Bayesian Triangulation

In this paper, we consider the problems of finding corresponding points from multiple perspective projection images (the correspondence problem), and estimating the 3-D point from which these points have arisen (the triangulation problem). We pose the triangulation problem as that of finding the Bayesian maximum a posteriori estimate of the 3-D point, given its projections in N images, assuming a Gaussian error model for the image point co-ordinates and the camera parameters. We solve this by an iterative steepest descent method. We then consider the correspondence problem as a statistical hypothesis verification problem. Given a set of 2-D points, UndeT the hypothesis that the points are in correspondence, the MAP estimate of the 3-D point is computed. Based on the MAP estimate, we derive a statistical test for verifying this hypothesis. To find sets of corresponding points when multiple points in each of N images are given, we propose a method that does the Bayesian triangulation and hypothesis verification on each N-tuple of points, selecting those that pass the hypothesis test. We characterize the performance of the Bayesian triangulation in terms of the average distance of the triangulated 3D point from the true 3-D point, and of the point correspondence method in terms of its misdetection and false alarm rates.