Shape Analysis of Framed Space Curves

Abstract

In the elastic shape analysis approach to shape matching and object classification, plane curves are represented as points in an infinite-dimensional Riemannian manifold, wherein shape dissimilarity is measured by geodesic distance. A remarkable result of Younes, Michor, Shah and Mumford says that the space of closed planar shapes, endowed with a natural metric, is isometric to an infinite-dimensional Grassmann manifold via the so-called square root transform. This result facilitates efficient shape comparison by virtue of explicit descriptions of Grassmannian geodesics. In this paper, we extend this shape analysis framework to treat shapes of framed space curves. By considering framed curves, we are able to generalize the square root transform by using quaternionic arithmetic and properties of the Hopf fibration. Under our coordinate transformation, the space of closed framed curves corresponds to an infinite-dimensional complex Grassmannian. This allows us to describe geodesics in framed curve space explicitly. We are also able to produce explicit geodesics between closed, unframed space curves by studying the action of the loop group of the circle on the Grassmann manifold. Averages of collections of plane and space curves are computed via a novel algorithm utilizing flag means. arXiv:1807.03477v1 [math.DG] 10 Jul 2018 Theorem 1. 1 ([39]). The space ImmpS 1 , R 2 q{ttransl., rot., scal.u, endowed with the elastic metric g a,a is locally isometric to the Grassmann manifold Gr 2 pC 8 pS 1 , Rqq of two-dimensional planes in the vector space C 8 pS 1 , Rq, endowed with its canonical L 2 metric.