Solving Quadratically Constrained Geometrical Problems using Lagrangian Duality

In this paper we consider the problem of solving different pose and registration problems under rotational constraints. Traditionally, methods such as the iterative closest point algorithm have been used to solve these problems. They may however get stuck in local minima due to the non-convexity of the problem. In recent years methods for finding the global optimum, based on Branch and Bound and convex under-estimators, have been developed. These methods are provably optimal, however since they are based on global optimization methods they are in general more time consuming than local methods. In this paper we adopt a dual approach. Rather than trying to find the globally optimal solution we investigate the quality of the solutions obtained using Lagrange duality. Our approach allows its to formulate a single convex semidefinite program that approximates the original problem well.