Bootstrapping Globally Optimal Variational Calculus Solutions

Abstract

Whereas in a coordinate-dependent setting the Euler-Lagrange equations establish necessary conditions for solving variational problems, uniqueness and global optimality are generally hard to prove, and often times they may not even exist. This is also true for variational problems on Lie groups, with the Euler-Poincaré equation establishing necessary conditions. This article therefore reviews several cases where unique globally optimal solutions can be guaranteed, and establishes a "bootstrapping" process to build new globally optimal variational solutions on larger spaces from existing ones on smaller spaces. This general theory is then applied to several topics such as optimal framing of curves in three-dimensional Euclidean space, optimal motion interpolation, and optimal reparametrization of video sequences to compare salient actions.