Shape of Elastic Strings in Euclidean Space

Abstract

We construct a 1-parameter family of geodesic shape metrics on a space of closed parametric curves in Euclidean space of any dimension. The curves are modeled on homogeneous elastic strings whose elasticity properties are described in terms of their tension and rigidity coefficients. As we change the elasticity properties, we obtain the various elastic models. The metrics are invariant under reparametrizations of the curves and induce metrics on shape space. Analysis of the geometry of the space of elastic strings and path spaces of elastic curves enables us to develop a computational model and algorithms for the estimation of geodesics and geodesic distances based on energy minimization. We also investigate a curve registration procedure that is employed in the estimation of shape distances and can be used as a general method for matching the geometric features of a family of curves. Several examples of geodesics are given and experiments are carried out to demonstrate the discriminative quality of the elastic metrics.

A Computational Model of Multidimensional Shape

“…Thus, we need a single cycle γ , the simplest choice being a cycle that goes once around the circle. In this case, the shape model of this paper coincides with the model of elastic strings studied in Mio et al (2009).…”

“…This model can be computed quite efficiently (Schmidt et al 2006), however, the lack of elasticity sometimes forces the model to rely on somewhat unnatural curve correspondences. Mio et al (2007, 2009) constructed a shape space of elastic strings equipped with a 1-parameter family of geodesic metrics; planar curves were studied in Mio et al (2007) and an extension to curves in Euclidean space of any dimension in Mio et al (2009), where an alternative, more robust computational approach is employed. These metrics are indexed by a parameter that controls how much resistance a curve offers to stretching or compression relative to deformation by bending.…”

“…These metrics are indexed by a parameter that controls how much resistance a curve offers to stretching or compression relative to deformation by bending. The elastic model of Younes (1998), Younes et al (2007) is isometric to a special case of the model of Mio et al (2007, 2009), however, the formulation of Younes et al (2007) for this special case has several computational advantages. The shape space of Joshi et al (2007) is also equivalent to a special case of the general model of Mio et al (2007, 2009), but in this case the computational costs for closed curves are comparable.…”

“…The elastic model of Younes (1998), Younes et al (2007) is isometric to a special case of the model of Mio et al (2007, 2009), however, the formulation of Younes et al (2007) for this special case has several computational advantages. The shape space of Joshi et al (2007) is also equivalent to a special case of the general model of Mio et al (2007, 2009), but in this case the computational costs for closed curves are comparable. In Mio et al (2009), the arc-length model of Klassen et al (2004) is also interpreted as a limit case of elastic curves, as the resistance to stretching and compression becomes progressively larger.…”

“…The shape space of Joshi et al (2007) is also equivalent to a special case of the general model of Mio et al (2007, 2009), but in this case the computational costs for closed curves are comparable. In Mio et al (2009), the arc-length model of Klassen et al (2004) is also interpreted as a limit case of elastic curves, as the resistance to stretching and compression becomes progressively larger. Other metrics in the space of curves have been investigated by Michor and Mumford (2007), including their organization and hierarchy.…”