Un-reduction

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Abstract

This paper provides a full geometric development of a new technique called un-reduction, for dealing with dynamics and optimal control problems posed on spaces that are unwieldy for numerical implementation. The technique, which was originally concieved for an application to image dynamics, uses Lagrangian reduction by symmetry in reverse. A deeper understanding of un-reduction leads to new developments in image matching which serve to illustrate the mathematical power of the technique.

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Riemannian geometries on spaces of plane curves

Peter W. Michor, David Mumford (2006)

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

Peter W. Michor, David Mumford (2006)

Here shape space is either the manifold of simple closed smooth unparameterized curves in \(\mathbb R^2\) or is the orbifold of immersions from \(S^1\) to \(\mathbb R^2\) modulo the group of diffeomorphisms of \(S^1\). We investige 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 \(\mathbb R^2\)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.

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Evolutions equations in computational anatomy

Michael Miller, Felipe Arrate, Laurent Younes (2009)

One of the main purposes in computational anatomy is the measurement and statistical study of anatomical variations in organs, notably in the brain or the heart. Over the last decade, our group has progressively developed several approaches for this problem, all related to the Riemannian geometry of groups of diffeomorphisms and the shape spaces on which these groups act. Several important shape evolution equations that are now used routinely in applications have emerged over time. Our goal in this paper is to provide an overview of these equations, placing them in their theoretical context, and giving examples of applications in which they can be used. We introduce the required theoretical background before discussing several classes of equations of increasingly complexity. These equations include energy minimizing evolutions deriving from Riemannian gradient descent, geodesics, parallel transport and Jacobi fields.