An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach

David Mumford,Peter W. Michor

doi:10.1016/j.acha.2006.07.004

David Mumford, Peter W. Michor

Open Access

https://doi.org/10.1016/j.acha.2006.07.004

Copy DOI

Abstract

Here shape space is either the manifold of simple closed smooth unparameterized curves in R 2 or is the orbifold of immersions from S 1 to R 2 modulo the group of diffeomorphisms of S 1 . We investigate several Riemannian metrics on shape space: L 2 -metrics weighted by expressions in length and curvature. These include a scale invariant metric and a Wasserstein type metric which is sandwiched between two length-weighted metrics. Sobolev metrics of order n on curves are described. Here the horizontal projection of a tangent field is given by a pseudo-differential operator. Finally the metric induced from the Sobolev metric on the group of diffeomorphisms on R 2 is treated. Although the quotient metrics are all given by pseudo-differential operators, their inverses are given by convolution with smooth kernels. We are able to prove local existence and uniqueness of solution to the geodesic equation for both kinds of Sobolev metrics. We are interested in all conserved quantities, so the paper starts with the Hamiltonian setting and computes conserved momenta and geodesics in general on the space of immersions. For each metric we compute the geodesic equation on shape space. In the end we sketch in some examples the differences between these metrics.

Highlights

Shape space is either the manifold of simple closed smooth unparameterized curves in R2 or is the orbifold of immersions from S1 to R2 modulo the group of diffeomorphisms of S1
The common theory which links these two points of view is the study of the various ways in which the space of simple closed curves can be endowed with a Riemannian metric
Anal. 23 (2007) 74–113 perspective, a Riemannian metric leads to geodesics, curvature and diffusion and, hopefully, to an understanding of the global geometry of the space

Summary

Introduction—multiple Riemannian metrics on the space of curves

Both from a mathematical and a computer vision point of view, it is of great interest to understand the space of simple closed curves in the plane. Its inverse L−1 is an integral operator whose kernel has a simple expression in terms of arc length distance between 2 points on the curve and their unit normal vectors For this metric, we work out the geodesic equation and prove that the geodesic flow is well posed in the sense that we have local existence and uniqueness of solutions in Imm(S1, R2) and in Bi. we discuss a little bit a scale invariant version of the metric Gimm,n. This metric gives a quotient metric on Emb(S1, R2) and Be which we will denote by Gdciff,n(h, k) In this case, the inverse L−1 of the operator defining the metric is an integral operator with a kernel given by a classical Bessel function applied to the distance in R2 between 2 points on the curve. The second example takes a fixed ‘cigar-shaped’ curve C and compares the unit balls in the tangent space TCBe given by the different metrics

The Hamiltonian approach

The setting

The Hamiltonian vector field mapping

The geodesic equation

The momentum mapping for a G-isometric group action

Metrics and momenta on the group of diffeomorphisms

Conserved momenta for GΦ

Horizontality for GΦ

The horizontal geodesic equation

Special case: the metric GA

Special case: the conformal metrics

Special case: the smooth scale invariant metric GSI