Intrinsic Riemannian metrics on spaces of curves: theory and computation

Abstract

This chapter reviews some past and recent developments in shape comparison and analysis of curves based on the computation of intrinsic Riemannian metrics on the space of curves modulo shape-preserving transformations. We summarize the general construction and theoretical properties of quotient elastic metrics for Euclidean as well as non-Euclidean curves before considering the special case of the square root velocity metric for which the expression of the resulting distance simplifies through a particular transformation. We then examine different numerical approaches that have been proposed to estimate such distances in practice and in particular to quotient out curve reparametrization in the resulting minimization problems.

A square root velocity framework for curves of bounded variation

“…Then, the distance between two curves c 1 and c 2 with square root velocity transforms q 1 and q 2 is defined as d(c 1 , c 2 ) = q 1 − q 2 L 2 . It is also possible to regard the space of all smooth regular curves as a manifold with the Riemannian structure inherited from L 2 (I; R d ) via the mapping R. This differential geometric point of view of shape analysis has been studied and discussed for instance in [2,3,15,19].…”