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Transport of Relational Structures in Groups of Diffeomorphisms

Laurent Younes¹, Anqi Qiu¹, Raimond L. Winslow² and Michael I. Miller¹

(1)	Center for Imaging Science, Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218, USA

(2)	Department of Biomedical Engineering, Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218, USA

Received: 22 May 2007 Accepted: 21 February 2008 Published online: 8 March 2008

Abstract This paper focuses on the issue of translating the relative variation of one shape with respect to another in a template centered representation. The context is the theory of Diffeomorphic Pattern Matching which provides a representation of the space of shapes of objects, including images and point sets, as an infinite dimensional Riemannian manifold which is acted upon by groups of diffeomorphisms. We discuss two main options for achieving our goal; the first one is the parallel translation, based on the Riemannian metric; the second one, based on the group action, is the coadjoint transport. These methods are illustrated with 3D experiments.

Keywords Groups of diffeomorphisms - Jacobi fields - Image registration - Shape analysis - Deformable templates

This work is partially supported by NSF DMS-0456253, NIH R01-EB000975, NIH P41-RR15241, NIH R01-MH064838, NIH 1R24-HL08534301A1 and the D.W. Reynolds Foundation.

Laurent Younes
Email: laurent.younes@jhu.edu

Laurent Younes

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is the former student of the Ecole Normale Supérieure in Paris, Laurent Younes was awarded the Ph.D. from the University Paris Sud in 1989, and the thesis advisor certification from the same university in 1995. He was a junior, then senior researcher at CNRS (French National Research Center) from 1991 to 2003. He is now professor in the Department of Applied Mathematics and Statistics at Johns Hopkins University. Here is a core faculty member of the Center for Imaging Science and of the Institute for Computational Medicine at JHU. His main research interests are on statistical analyses of images and shapes, and on mathematical models for shape deformations and shape spaces.

Anqi Qiu

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received her Ph.D. degree in Electrical and Computer Engineering from Johns Hopkins University in 2006. She also holds M.S. degrees in Biomedical Engineering and Applied Mathematics and Statistics. From 2007, she joined National University of Singapore as assistant professor. Her research interests focus on medical imaging analysis.

Raimond L. Winslow

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is Professor of Biomedical Engineering at Johns Hopkins University School of Medicine. He also directs the Institute for Computational Medicine at Johns Hopkins University. His interests are in integrative modeling of heart function in health and disease. This includes mathematical and computational modeling of signal transduction, metabolism, ion channels, intracellular calcium cycling, and force generation within the cardiac myocyte. His lab has also measured and modeled the structure of the cardiac ventricles using diffusion tensor MR imaging and finite-element modeling methods.

Michael I. Miller

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is the Herschel and Ruth Seder Professor of Biomedical Engineering, Professor of Electrical and Computer Engineering and Director of Center for Imaging Science at Johns Hopkins University, Baltimore, MD. Throughout his career his research interests have focused on image understanding and computer vision; medical imaging and computational anatomy.

Previous to his current position, Michael Miller was the Newton R. and Sarah Louisa Glasgow Professor in Biomedical Engineering at the Washington University Department of Electrical Engineering. From 1995–2000 he was the director of the Army Center for Imaging Science, an Academic Center of Excellence consisting of 7 universities dedicated to the development of new methods for image understanding and automated target recognition. He has been a visiting professor at several over several years at the Ecole Normale in France and a visiting professor at Brown University.

Michael Miller is a Fellow of the American Institute for Medical and Biological Engineering. He is a Presidential Young Investigator Award Winner and is the recipient of International Order of Merit; International Man of the Year, Warrant of Proclamation and Medal of Recognition from the International Biographical Centre, Cambridge England; and Universal Award of Accomplishment. He has been an invited and plenary speaker at numerous national and international meetings. In 2002 he was recognized by in-cites as having the greatest increase in citation rates in the field of Engineering for his work in Computational Anatomy.

Michael Miller received a B.S.E.E., Electrical Engineering, State University of New York at Stony Brook, an M.S. in Electrical Engineering from Johns Hopkins University and a Ph.D. in Biomedical Engineering also from Johns Hopkins University, Baltimore, MD.

Transport of Relational Structures in Groups of Diffeomorphisms

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