In this paper we wish to endow the manifold M of smooth curves in R^n (either closed curves or open curves with ﬁxed endpoints) with a Riemannian structure that allows us to treat homotopies between two curves C0 and C1 as trajectories with computable lengths. If we regard a curve as all possible reparameterizations of a C^1 mapping of the unit interval into R^n, then the tangent space at a point on M corresponds to all possible non-tangential ﬂows of the underlying curve. We note that a Riemannian metric corresponds to a choice of inner-product on this tangent space. Once this is deﬁned, we can compute distances between curves and consider the natural problem of ﬁnding geodesics which will yield minimal length homotopies between C0 and C1. We begin by noting that a seemingly natural inner-product corresponding to a geometric version of 2 does not yield useful geodesics and will instead introduce a sequence of conformally modﬁed innerproducts which has interesting limit properties.
|Titolo del libro:||EUSIPCO 2004 (European Signal Processing Conference)|
|Data di pubblicazione:||2004|
|Nome del convegno:||12th European Signal Processing Conference - EUSIPCO 2004|
|Appare nelle tipologie:||4.1 Contributo in Atti di convegno|