Towards non-iterative closest point: Exact recovery of pose for rigid 2D/3D registration using semidefinite programming

We describe a framework for globally solving the 2D/3D registration problem with unknown point correspondences. We give two mixed-integer nonlinear program (MINP) formulations of the problem to simultaneously solve for pose and correspondence when there are multiple 2D images. We further propose convex relaxations of both of the original MINP to semidefinite program (SDP) that can be solved efficiently by interior point methods. Under certain situations, we prove that the convex programs exactly recover the solution to the original nonconvex 2D/3D registration problem. Though the problem of point registration occurs frequently in many applications, we focus on registration of 3D models of coronary vessels with their 2D projections obtained from multiple intra-operative fluoroscopic images. To this effect, one of the MINP estimates the point correspondences between multiple 2D images in accordance with epipolar constraints while maintaining cycle consistency. Another natural extension that augments the objective of the program with feature descriptors of a particular point is also presented. We demonstrate the success of the convex programs in such application in both simulated and real data. The main idea of this work, of exploiting the special structure of the MINP and of the 3D rotation space to devise convex relaxations, can be applied to many other registration problems as well.