# On Shape of Plane Elastic Curves

@article{Mio2006OnSO, title={On Shape of Plane Elastic Curves}, author={Washington Mio and Anuj Srivastava and Shantanu H. Joshi}, journal={International Journal of Computer Vision}, year={2006}, volume={73}, pages={307-324} }

We study shapes of planar arcs and closed contours modeled on elastic curves obtained by bending, stretching or compressing line segments non-uniformly along their extensions. Shapes are represented as elements of a quotient space of curves obtained by identifying those that differ by shape-preserving transformations. The elastic properties of the curves are encoded in Riemannian metrics on these spaces. Geodesics in shape spaces are used to quantify shape divergence and to develop morphing…

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## References

SHOWING 1-10 OF 23 REFERENCES

Analysis of planar shapes using geodesic paths on shape spaces

- MathematicsIEEE Transactions on Pattern Analysis and Machine Intelligence
- 2004

This work uses a Fourier basis to represent tangents to the shape spaces and a gradient-based shooting method to solve for the tangent that connects any two shapes via a geodesic.

Non-Rigid Shape Comparison of Plane Curves in Images

- MathematicsJournal of Mathematical Imaging and Vision
- 2004

A mathematical theory for establishing correspondences between curves and for non-rigid shape comparison using bimorphisms is developed, which is more general than those obtained from one-to-one functions.

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,…

Computable Elastic Distances Between Shapes

- Computer ScienceSIAM J. Appl. Math.
- 1998

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.

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.

Statistical shape analysis: clustering, learning, and testing

- Computer ScienceIEEE Transactions on Pattern Analysis and Machine Intelligence
- 2005

This work presents tools for hierarchical clustering of imaged objects according to the shapes of their boundaries, learning of probability models for clusters of shapes, and testing of newly observed shapes under competing probability models.

Size and Shape Spaces for Landmark Data in Two Dimensions

- Mathematics
- 1986

Biometric studies of the forms of organisms usually consider size and shape variations in the geometric configuration of landmarks, points that correspond biologically from form to form. The size…

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 ~…