# 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…

## 4 Citations

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.

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.

Hypoelliptic Diffusion Maps I: Tangent Bundles

- Mathematics, Computer Science
- 2015

This paper analyzes HDM on tangent bundles, revealing its intimate connection with sub-Riemannian geometry and a family of hypoelliptic differential operators and in a later paper, it shall consider more general fibre bundles.

Hypoelliptic Diffusion Maps and Their Applications in Automated Geometric Morphometrics

- Computer Science
- 2015

A correspondence-based, landmark-free approach is proposed that automates this process while maintaining morphological interpretability of Hypoelliptic Diffusion Maps and their applications in Automated Geometric Morphometrics.

## References

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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.

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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…

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2D-Shape Analysis Using Conformal Mapping

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- 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.

2D-Shape Analysis Using Conformal Mapping

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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.

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The existence of the Euler–Poincaré equation for diffeomorphisms of S 1 is proved and bounds on its long-term behavior are found, showing that it is asymptotic to a one-parameter subgroup in Diff(S 1).