Precise Matching of PL Curves in $R^N$ in the Square Root Velocity Framework

TL;DR

This paper rigorously analyzes the SRVF framework for curve comparison in R^N, establishing theoretical foundations, density of piecewise linear curves, and providing a precise algorithm for optimal curve matching and geodesic computation.

Contribution

It provides a detailed theoretical foundation for the SRVF method, including the structure of quotient spaces and an exact algorithm for matching piecewise linear curves.

Findings

The set of piecewise linear curves is dense in the space of absolutely continuous curves under the SRVF metric.

An exact algorithm for optimal matching of piecewise linear curves is developed.

The algorithm allows precise computation of geodesics in the quotient space.

Abstract

The square root velocity function (SRVF), introduced by Srivastava et al, has proved to be an effective way to compare absolutely continuous curves in $R^N$ modulo reparametrization. Several computational papers have been published based on this method. In this paper, we carefully establish the theoretical foundations of the SRVF method. In particular, we analyze the quotient construction of the set of absolutely continuous curves modulo the group (or in some cases, semigroup) of reparametrizations, proving an important theorem about the structure of the closed orbits required in this quotient construction. We observe that the set of piecewise linear curves is dense in the space of absolutely continuous curves with respect to the SRVF metric. Finally, given two piecewise linear curves, we establish a precise algorithm for producing the optimal matching between these curves. This also results in a precise determination of the geodesic between the points in the quotient space corresponding to these curves. In the past, this geodesic has only been approximated using the method of Dynamic Programming. We show examples resulting from this algorithm.