The Euler-Poincare theory of Metamorphosis

@article{Holm2008TheET,
title={The Euler-Poincare theory of Metamorphosis},
author={Darryl D. Holm and Alain Trouv{\'e} and Laurent Younes},
journal={ArXiv},
year={2008},
volume={abs/0806.0870}
}
• Published 4 June 2008
• Mathematics
• ArXiv
In the pattern matching approach to imaging science, the process of metamorphosis'' is template matching with dynamical templates. Here, we recast the metamorphosis equations of into the Euler-Poincare variational framework of and show that the metamorphosis equations contain the equations for a perfect complex fluid \cite{Ho2002}. This result connects the ideas underlying the process of metamorphosis in image matching to the physical concept of order parameter in the theory of complex fluids…

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References

SHOWING 1-10 OF 38 REFERENCES

• Computer Science
Found. Comput. Math.
• 2005
A modification of the metric is introduced by partly expressing displacements on M as an effect of the action of some group element, called metamorphoses, which can and has been applied to image processing problems, providing in particular diffeomorphic matching algorithms for pattern recognition.
Lagrangian reduction by stages is used to derive the Euler-Poincare equations for the nondissipative coupled motion and micromotion of complex fluids. We mainly treat perfect complex fluids (PCFs)
• A. Trouvé
• Mathematics
International Journal of Computer Vision
• 2004
This paper constructs a distance between deformations defined through a metric given the cost of infinitesimal deformations, and proposes a numerical scheme to solve a variational problem involving this distance and leading to a sub-optimal gradient pattern matching.
• Mathematics
SIAM J. Math. Anal.
• 2005
This paper provides a rigorous and general construction of this infinite dimensional "shape manifold" on which a Riemannian metric is placed and uses this to provide a geometrically founded linear approximation of the deformations of shapes in the neighborhood of a given template.
• Mathematics
• 1998
We study Euler–Poincare systems (i.e., the Lagrangian analogue of Lie–Poisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler–Poincare
We consider a generalization of the Camassa–Holm (CH) equation with two dependent variables, called CH2, introduced in a paper by Liu and Zhang (Liu S-Q and Zhang Y 2005 J. Geom. Phys. 54 427–53). We
• Physics, Mathematics
SIAM J. Appl. Dyn. Syst.
• 2006
It is proved that straight sheets moving normally to themselves under an $H^1$ metric, corresponding to peakons for the one‐dimensional (1D) Camassa–Holm equation, are linearly stable in Eulerian coordinates, suffering only a weak instability of Lagrangian particle paths, while most other cases are unstable but well‐posed.
• Mathematics
Journal of Mathematical Imaging and Vision
• 2004
A methodology and algorithm for generating diffeomorphisms of the sphere onto itself, given the displacements of a finite set of template landmarks, has application in brain mapping, where surface data is typically mapped to the sphere as a common coordinate system.
• Mathematics
SIAM J. Control. Optim.
• 2000
This work provides sufficient conditions under which an optimal matching can be found between two mappings from a fixed interval to some "feature space", the optimal matching being a homeomorphism of the interval I.