# 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} }

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…

## 90 Citations

### Stochastic metamorphosis with template uncertainties

- MathematicsLecture Notes Series, Institute for Mathematical Sciences, National University of Singapore
- 2019

This paper investigates two stochastic perturbations of the metamorphosis equations of image analysis, in the geometrical context of the Euler-Poincar\'e theory, and applies this general geometric theory to several classical examples, including landmarks, images, and closed curves.

### Stochastic Metamorphosis in Imaging Science

- Geology
- 2017

In the pattern matching approach to imaging science, the process of \emph{metamorphosis} in template matching with dynamical templates was introduced in \cite{ty05b}. In \cite{HoTrYo2009} the…

### Computing metamorphoses between discrete measures

- Computer Science
- 2013

It is shown that, when matching sums of Dirac measures, minimizing evolutions can include other singular distributions, which complicates the numerical approximation of such solutions.

### Space-time metamorphosis

- MathematicsArXiv
- 2020

A gradient descent method is proposed on a primal conforming finite element method for the matching parameterized by a space-time velocity field and several promising numerical results are shown.

### Selective metamorphosis for growth modelling with applications to landmarks

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We present a framework for shape matching in computational anatomy allowing users control of the degree to which the matching is diffeomorphic. This control is given as a function defined over the…

### Metamorphosis of images in reproducing kernel Hilbert spaces

- Computer ScienceAdv. Comput. Math.
- 2016

This paper derives the relevant shooting equations from a Lagrangian frame of reference, presents the details of the numerical approach, and illustrates the method through morphing of some simple images.

### Metamorphosis of images in reproducing kernel Hilbert spaces

- Computer ScienceAdvances in Computational Mathematics
- 2015

This paper derives the relevant shooting equations from a Lagrangian frame of reference, presents the details of the numerical approach, and illustrates the method through morphing of some simple images.

### Metamorphoses of Functional Shapes in Sobolev Spaces

- MathematicsFound. Comput. Math.
- 2018

A model of geometric-functional variability between fshape spaces with adequate geodesic distances, encompassing in one common framework usual shape comparison and image metamorphoses, and a formulation of matching between any two fshapes from the optimal control perspective is proposed.

### Geometric dynamics of optimization

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A family of dynamical systems arising from an evolutionary re-interpretation of certain optimal control and optimization problems is investigated, particularly on the application in image registration of the theory of metamorphosis.

### A class of fast geodesic shooting algorithms for template matching and its applications via the $N$-particle system of the Euler-Poincar\'e equations

- Mathematics
- 2015

The Euler-Poincar\'e (EP) equations describe the geodesic motion on the diffeomorphism group. For template matching (template deformation), the Euler-Lagrangian equation, arising from minimizing an…

## References

SHOWING 1-10 OF 38 REFERENCES

### Metamorphoses Through Lie Group Action

- Computer ScienceFound. 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.

### Euler-Poincare Dynamics of Perfect Complex Fluids

- Mathematics, Physics
- 2002

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)…

### Diffeomorphisms Groups and Pattern Matching in Image Analysis

- MathematicsInternational 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.

### Local Geometry of Deformable Templates

- MathematicsSIAM 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.

### The Euler–Poincaré Equations and Semidirect Products with Applications to Continuum Theories

- 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…

### On a Camassa-Holm type equation with two dependent variables

- Mathematics
- 2005

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…

### The Kelvin-Helmholtz Instability of Momentum Sheets in the Euler Equations for Planar Diffeomorphisms

- Physics, MathematicsSIAM 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.

### Landmark Matching via Large Deformation Diffeomorphisms on the Sphere

- MathematicsJournal 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.

### On a Class of Diffeomorphic Matching Problems in One Dimension

- MathematicsSIAM 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.