Un-reduction

TL;DR

Un-reduction is a novel geometric technique that reverses Lagrangian reduction by symmetry, facilitating the numerical solution of complex dynamics and optimal control problems, with applications in image matching.

Contribution

The paper introduces un-reduction, a new geometric method that reverses Lagrangian reduction, enhancing the handling of complex dynamical systems and optimal control problems.

Findings

Un-reduction improves numerical implementation of dynamics on complex spaces.

Application to image matching demonstrates the technique's effectiveness.

Provides a deeper understanding of symmetry-based reduction methods.

Abstract

This paper provides a full geometric development of a new technique called un-reduction, for dealing with dynamics and optimal control problems posed on spaces that are unwieldy for numerical implementation. The technique, which was originally concieved for an application to image dynamics, uses Lagrangian reduction by symmetry in reverse. A deeper understanding of un-reduction leads to new developments in image matching which serve to illustrate the mathematical power of the technique.