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, under 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 3-D point from the true 3-D point, and of the point correspondence method in terms of its misdetection and false alarm rates.