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  Quarterly of Applied Mathematics
Quarterly of Applied Mathematics
Online ISSN 1552-4485; Print ISSN 0033-569X

The Euler-Poincaré theory of metamorphosis

Authors: Darryl D. Holm, Alain Trouvé and Laurent Younes
Journal: Quart. Appl. Math. 67 (2009), 661-685
MSC (2000): Primary 58E50
Published electronically: September 2, 2009
MathSciNet review: 2588229
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Abstract: In the pattern matching approach to imaging science, the process of ``metamorphosis'' is template matching with dynamical templates (Trouvé and Younes, Found. Comp. Math., 2005). Here, we recast the metamorphosis equations of that paper into the Euler-Poincaré variational framework of Holm, Marsden, and Ratiu, Adv. in Math., 1998 and show that the metamorphosis equations contain the equations for a perfect complex fluid (Holm, Springer, 2002). This result connects the ideas underlying the process of metamorphosis in image matching to the physical concept of an order parameter in the theory of complex fluids. After developing the general theory, we reinterpret various examples, including point set, image and density metamorphosis. We finally discuss the issue of matching measures with metamorphosis, for which we provide existence theorems for the initial and boundary value problems.

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Additional Information

Darryl D. Holm
Affiliation: Department of Mathematics, Imperial College London SW7 2AZ, United Kingdom and Computer and Computational Science, Los Alamos National Laboratory, MS D413 Los Alamos, New Mexico 87545

Alain Trouvé
Affiliation: CMLA (CNRS, URA 1611), Ecole Normale Supérieure de Cachan, 61, avenue du Président Wilson, F-94 235 Cachan Cedex

Laurent Younes
Affiliation: Center for Imaging Science, The Johns Hopkins University, 3400 N-Charles Street, Baltimore, Maryland 21218-2686

PII: S 0033-569X(09)01134-2
Keywords: Groups of diffeomorphisms, EPDiff, image registration, shape analysis, deformable templates
Received by editor(s): April 27, 2008
Published electronically: September 2, 2009
Additional Notes: The work of D. D. Holm was partially supported by the US Department of Energy, Office of Science, Applied Mathematical Research, and the Royal Society of London Wolfson Research Merit Award. D. D. Holm is grateful for stimulating discussions with C. Tronci.
The work of Laurent Younes was partially supported by NSF DMS-0456253.
Article copyright: © Copyright 2009 Brown University

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