We present a matching criterion for curves and integrate it into the large deformation diffeomorphic metric mapping (LDDMM) scheme for computing an optimal transformation between two curves embedded in Euclidean space R d. Curves are first represented as vector-valued measures, which incorporate both location and the first order geometric structure of the curves. Then, a Hilbert space structure is imposed on the measures to build the norm for quantifying the closeness between two curves. We describe a discretized version of this, in which discrete sequences of points along the curve are represented by vector-valued functionals. This gives a convenient and practical way to define a matching functional for curves. We derive and implement the curve matching in the large deformation framework and demonstrate mapping results of curves in R 2 and R 3. Behaviors of the curve mapping are discussed using 2D curves. The applications to shape classification is shown and

Diffeomorphic image registration, where images are aligned using diffeomorphic warps, is a popular subject for research in medical image analysis. We introduce a novel algorithm for computing diffeomorphic warps that solves the Euler equations on the diffeomorphism group explicitly, based on a discretisation of the Hamiltonian, rather than using an optimiser. The result is an algorithm that is many times faster than those considered previously.

Abstract not found

The Euler equations that describe geodesics on the group of diffeomorphisms of the plane admit singular solutions in which the momentum is concentrated on curves, the so-called momentum sheets analogous to vortex sheets in the Euler fluid equations. We study the stability of straight and circular momentum sheets for a large family of metrics. We prove that straight sheets moving normally to themselves under an H 1 metric, corresponding to peakons for the one-dimensional (1D) Camassa–Holm equation, are linearly stable in Eulerian coordinates, suffering only a weak instability of Lagrangian particle paths, while most other cases are unstable but well-posed. Expanding circular sheets are algebraically unstable for all metrics. The evolution of the instabilities are followed numerically, illustrating several typical dynamical phenomena of momentum sheets.

Abstract. Diffeomorphic image registration, where images are aligned using diffeomorphic warps, is a popular subject for research in medical image analysis. We introduce a novel algorithm for computing diffeomorphic warps that solves the Euler equations on the diffeomorphism group explicitly, based on a discretisation of the Hamiltonian, rather than using an optimiser. The result is an algorithm that is many times faster than those considered previously. 1

statistics of the infinite dimensional shape manifolds). MM would like to thank Andrea Bertozzi of UCLA for her continuous advice and and support. This paper deals with the computation of sectional curvature for the manifolds of N landmarks (or feature points) in D dimensions, endowed with the Riemannian metric induced by the group action of diffeomorphisms. The inverse of the metric tensor for these manifolds (i.e. the cometric), when written in coordinates, is such that each of its elements depends on at most 2D of the ND coordinates. This makes the matrices of partial derivatives of the cometric very sparse in nature, thus suggesting solving the highly non-trivial problem of developing a formula that expresses sectional curvature in terms of the cometric and its first and second partial derivatives (we call this Mario’s formula). We apply such formula to the manifolds of landmarks and in particular we fully explore the case of geodesics on which only two points have non-zero momenta and compute the sectional curvatures of 2-planes spanned by the tangents to such geodesics. The latter example gives insight to the geometry of the full manifolds of landmarks. 1

Abstract. In this paper we consider planar conformal deformations, motivated by the warps that Wentworth Thompson used to deform images of one species into another. We study an equation for geodesic motion on the infinite dimensional Fréchet manifold Con(D, R 2) of conformal embeddings of the disk into the plane. We demonstrate that solutions may be represented as sheets, and use the sheet ansatz to derive a numerical discretization scheme. We also show that the equation admits totally geodesic solutions corresponding to scaling and translation, but not to affine transformations. 1