The geometry and curvature of shape spaces

@inproceedings{Mumford2012TheGA,
  title={The geometry and curvature of shape spaces},
  author={David Mumford},
  year={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 Riemann. We quote his quite amazing Habilitatsionschrift: There are, however, manifolds in which the fixing of position requires not a finite number but either an infinite series or a continuous manifold of determinations of quantity. Such manifolds are constituted for example by the possible shapes of a… 

Figures from this paper

Consistent Curvature Approximation on Riemannian Shape Spaces
TLDR
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.
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
Hypoelliptic Diffusion Maps I: Tangent Bundles
We introduce the concept of Hypoelliptic Diffusion Maps (HDM), a framework generalizing Diffusion Maps in the context of manifold learning and dimensionality reduction. Standard non-linear
Hypoelliptic Diffusion Maps and Their Applications in Automated Geometric Morphometrics
Hypoelliptic Diffusion Maps and Their Applications in Automated Geometric Morphometrics by Tingran Gao Department of Mathematics Duke University Date: Approved: Ingrid Daubechies, Supervisor Mauro

References

SHOWING 1-10 OF 19 REFERENCES
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,
VANISHING GEODESIC DISTANCE ON SPACES OF SUBMANIFOLDS AND DIFFEOMORPHISMS
The L 2 -metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type M in a Riemannian man- ifold (N;g) induces geodesic distance 0. We discuss another metric which
Infinite dimensional geodesic flows and the universal Teichmüller space
The subject of this thesis lies in the intersection of differential geometry and functional analysis, a domain usually called global analysis. A central object in this work is the group Ds(M) of all
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,
Sectional Curvature in Terms of the Cometric, with Applications to the Riemannian Manifolds of Landmarks
TLDR
This paper fully explore the case of geodesics on which only two points have nonzero momenta and compute the sectional curvatures of 2-planes spanned by the tangents to such geodesic, and gives insight into the geometry of the full manifolds of landmarks.
The differential geometry of landmark shape manifolds: metrics, geodesics, and curvature
The study of shapes and their similarities is central in computer vision, in that it allows to recognize and classify objects from their representation. One has the interest of defining a distance
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
2D-Shape Analysis Using Conformal Mapping
TLDR
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.
2D-Shape Analysis Using Conformal Mapping
  • E. Sharon, D. Mumford
  • Mathematics, Computer Science
    Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004.
  • 2004
TLDR
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.
Computable Elastic Distances Between Shapes
  • L. Younes
  • Computer Science, Mathematics
    SIAM J. Appl. Math.
  • 1998
TLDR
An elastic matching algorithm which is based on a true distance between intrinsic properties of the shapes, taking into account possible invariance to scaling or Euclidean transformations in the case they are required.
...
1
2
...