Verifying Global Minima for L2 Minimization Problems

Abstract

We consider the least-squares (L2) triangulation problem and structure-and-motion with known rotatation, or known plane. Although optimal algorithms have been given for these algorithms under an L-infinity cost function, finding optimal least-squares (L2) solutions to these problems is difficult, since the cost functions are not convex, and in the worst case can have multiple minima. Iterative methods can usually be used to find a good solution, but this may be a local minimum. This paper provides a method for verifying whether a local-minimum solution is globally optimal, by providing a simple and rapid test involving the Hessian of the cost function. In tests of a data set involving 277,000 independent triangulation problems, it is shown that the test verifies the global optimality of an iterative solution in over 99.9% of the cases.