# Computable Elastic Distances Between Shapes

@article{Younes1998ComputableED, title={Computable Elastic Distances Between Shapes}, author={Laurent Younes}, journal={SIAM J. Appl. Math.}, year={1998}, volume={58}, pages={565-586} }

We dene distances between geometric curves by the square root of the minimal energy required to transform one curve into the other. The energy is formally dened from a left invariant Riemannian distance on an innite dimensional group acting on the curves, which can be explicitly computed. The obtained distance boils down to a variational problem for which an optimal matching between the curves has to be computed. An analysis of the distance when the curves are polygonal leads to a numericalâ€¦Â

## 362 Citations

A distance for elastic matching in object recognition

- Mathematics, Computer ScienceProceedings of 13th International Conference on Pattern Recognition
- 1996

An analysis of the distance when the curves are polygonal leads to a numerical procedure for the solution of the variational problem, which can efficiently be implemented, as illustrated by experiments.

Similarity Metric for Curved Shapes in Euclidean Space

- Mathematics2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)
- 2016

The proposed method represents a given curve as a point in the deformation space, the direct product of rigid transformation matrices, such that the successive action of the matrices on a fixed starting point reconstructs the full curve.

Simplifying Transforms for General Elastic Metrics on the Space of Plane Curves

- MathematicsSIAM J. Imaging Sci.
- 2020

This paper extends the transformations appearing in the existing literature to a family of isometries, which take any elastic metric to the flat L 2 metric, and extends the transforms to treat piecewise linear curves and demonstrates the existence of optimal matchings over the diffeomorphism group in this setting.

A Discrete Framework to Find the Optimal Matching Between Manifold-Valued Curves

- MathematicsJournal of Mathematical Imaging and Vision
- 2018

A simple algorithm is introduced that finds an optimal matching between two curves by computing the geodesic of the infinite-dimensional manifold of curves that is at all time horizontal to the fibers of the shape bundle, and comparison with dynamic programming is established.

Matching Shapes Using the Current Distance

- Computer ScienceArXiv
- 2010

An interesting aspect of this work is that it can compute the current distance between curves, surfaces, and higher-order manifolds via a simple reduction to instances of weighted point sets, thus obviating the need for different kind of algorithms for different kinds of shapes.

Fast Approximation of Distance Between Elastic Curves using Kernels

- Computer Science, MathematicsBMVC
- 2013

This paper proposes a new procedure for metric approximation using the framework of kernel functions and demonstrates that this provides a fast approximation of the metric while preserving its invariance properties.

A New Variational Model for Shape Graph Registration with Partial Matching Constraints

- MathematicsSIAM Journal on Imaging Sciences
- 2022

This paper introduces a new extension of Riemannian elastic curve matching to a general class of geometric structures, which we call (weighted) shape graphs, that allows for shape registration withâ€¦

Metric registration of curves and surfaces using optimal control

- MathematicsHandbook of Numerical Analysis
- 2019

Elastic Geodesic Paths in Shape Space of Parameterized Surfaces

- MathematicsIEEE Transactions on Pattern Analysis and Machine Intelligence
- 2012

A novel Riemannian framework for shape analysis of parameterized surfaces provides efficient algorithms for computing geodesic paths which, in turn, are important for comparing, matching, and deforming surfaces.

Conformal metrics and true "gradient flows" for curves

- Computer Science, MathematicsTenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1
- 2005

This paper shows how this Riemannian metric arises as the unique metric which validates the common references to a wide variety of contour evolution models in the literature as "gradient flows" to various formulated energy functionals using conformal factors that depend upon a curve's total arclength.

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