Sectional Curvature in Terms of the Cometric, with Applications to the Riemannian Manifolds of Landmarks

  title={Sectional Curvature in Terms of the Cometric, with Applications to the Riemannian Manifolds of Landmarks},
  author={Mario Micheli and Peter W. Michor and David Mumford},
  journal={SIAM J. Imaging Sci.},
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… 
Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds
Given a finite-dimensional manifold , the group of diffeomorphisms diffeomorphism of  which decrease suitably rapidly to the identity, acts on the manifold of submanifolds of  of diffeomorphism-type
Consistent Curvature Approximation on Riemannian Shape Spaces
The variational time discretization of geodesic calculus presented in Rumpf and Wirth (2015) is extended and first and second order consistency are proven for the approximations of the covariant derivative and the curvature tensor.
Geometry of diffeomorphism groups and shape matching
The large deformation matching (LDM) framework is a method for registration of images and other data structures, used in computational anatomy. We show how to reformulate the large deformation
Riemannian cubics on the group of diffeomorphisms and the Fisher-Rao metric
We study a second-order variational problem on the group of diffeomorphisms of the interval [0, 1] endowed with a right-invariant Sobolev metric of order 2, which consists in the minimization of the
Homogeneous Sobolev Metric of Order One on Diffeomorphism Groups on Real Line
It is proved that the spaceequipped with the homogeneous Sobolev metric of order one is a flat space in the sense of Riemannian geometry, as it is isometric to an open subset of a mapping space equipped with the flat L2-metric.
Overview of the Geometries of Shape Spaces and Diffeomorphism Groups
This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of
Minimizing acceleration on the group of diffeomorphisms and its relaxation
We study a second-order variational problem on the group of diffeomorphisms of the interval [0, 1] endowed with a right-invariant Sobolev metric of order 2, which consists in the minimization of the
The geometry and curvature of shape spaces
The idea that the set of all smooth submanifolds of a fixed ambient finite dimensional differentiable manifold forms a manifold in its own right, albeit one of infinite dimension, goes back to
About simple variational splines from the Hamiltonian viewpoint
In this paper, we study simple splines on a Riemannian manifold $Q$ from the point of view of the Pontryagin maximum principle (PMP) in optimal control theory. The control problem consists in finding
Matrix-valued kernels for shape deformation analysis
The main purpose of this paper is providing a systematic study and classification of non-scalar kernels for Reproducing Kernel Hilbert Spaces (RKHS), to be used in the analysis of deformation in


Riemannian Geometries on Spaces of Plane Curves
We study some Riemannian metrics on the space of regular smooth curves in the plane, viewed as the orbit space of maps from the circle to the plane modulo the group of diffeomorphisms of the circle,
An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach
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
A Metric on Shape Space with Explicit Geodesics
This paper studies a specific metric on plane curves that has the property of being isometric to classical manifold (sphere, complex projective, Stiefel, Grassmann) modulo change of parametrization,
Riemannian Geometry
THE recent physical interpretation of intrinsic differential geometry of spaces has stimulated the study of this subject. Riemann proposed the generalisation, to spaces of any order, of Gauss's
Riemannian geometry and geometric analysis
* Established textbook * Continues to lead its readers to some of the hottest topics of contemporary mathematical research This established reference work continues to lead its readers to some of
N -particle dynamics of the Euler equations for planar diffeomorphisms
The Euler equations associated with diffeomorphism groups have received much recent study because of their links with fluid dynamics, computer vision, and mechanics. In this article, we consider the
The shape-space l. k m whose points a represent the shapes of not totally degenerate /c-ads in IR m is introduced as a quotient space carrying the quotient metric. When m = 1, we find that Y\ = S k ~
2D-Shape Analysis Using Conformal Mapping
This paper presents an efficient method for computing the unique shortest path, the geodesic of shape morphing between each two end-point shapes, and shows how the group of diffeomorphisms of S1 acts as a group of isometries on the space of shapes and can be used to define shape transformations, like for instance ‘adding a protruding limb’ to any shape.
Landmark matching via large deformation diffeomorphisms
Conditions for the existence of solutions in the space of diffeomorphisms are established, with a gradient algorithm provided for generating the optimal flow solving the minimum problem.
Group Actions, Homeomorphisms, and Matching: A General Framework
  • M. Miller, L. Younes
  • Mathematics, Computer Science
    International Journal of Computer Vision
  • 2004
Left-invariant metrics are defined on the product G × I thus allowing the generation of transformations of the background geometry as well as the image values, and structural generation in which image values are changed supporting notions such as tissue creation in carrying one image to another.