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# Geometry, Imaging and Computing

## Volume 2 (2015)

### Number 3

### Precise matching of PL curves in $\mathbb{R}^N$ in the square root velocity framework

Pages: 133 – 186

DOI: http://dx.doi.org/10.4310/GIC.2015.v2.n3.a1

#### Authors

#### Abstract

The representation of curves by their square root velocity functions (SRVF) provides a useful and computationally effective way to make a metric space out of the set of all absolutely continuous curves modulo reparametrization. The first part of this paper establishes some important theoretical properties of this method, proving the completeness of this metric space, characterizing the exact nature of the closed orbits under reparametrization, and proving the existence of optimal matchings between pairs of curves, provided that at least one of the two curves is piecewise linear.

The second part of the paper develops a computational algorithm that produces a precise optimal matching between any two piecewise linear curves in $\mathbb{R}^N$, with respect to the SRVF framework. This method is demonstrated on several examples.