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

@article{Micheli2012SectionalCI, 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.}, year={2012}, volume={5}, pages={394-433} }

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…

## Figures from this paper

## 32 Citations

Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds

- Mathematics
- 2013

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

- MathematicsArXiv
- 2019

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

- Mathematics
- 2012

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

- Mathematics
- 2016

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

- MathematicsJ. Nonlinear Sci.
- 2014

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

- MathematicsJournal of Mathematical Imaging and Vision
- 2013

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

- MathematicsESAIM: Control, Optimisation and Calculus of Variations
- 2019

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

- Mathematics
- 2012

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

- Mathematics
- 2017

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

- Computer Science
- 2013

A systematic study and classification of non-scalar kernels for Reproducing Kernel Hilbert Spaces (RKHS), to be used in the analysis of deformation in shape spaces endowed with metrics induced by the action of groups of diffeomorphisms.

## References

SHOWING 1-10 OF 49 REFERENCES

Riemannian Geometries on Spaces of Plane Curves

- Mathematics
- 2003

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

- Mathematics
- 2006

A Metric on Shape Space with Explicit Geodesics

- Mathematics
- 2007

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

- MathematicsNature
- 1927

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

- Mathematics
- 1995

* 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

- Mathematics
- 2005

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…

SHAPE MANIFOLDS, PROCRUSTEAN METRICS, AND COMPLEX PROJECTIVE SPACES

- Mathematics
- 1984

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

- MathematicsCVPR
- 2004

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

- MathematicsIEEE Trans. Image Process.
- 2000

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

- MathematicsInternational 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.