Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms
 M. Faisal Beg,
 Michael I. Miller,
 Alain Trouvé,
 Laurent Younes
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Abstract
This paper examine the EulerLagrange equations for the solution of the large deformation diffeomorphic metric mapping problem studied in Dupuis et al. (1998) and Trouvé (1995) in which two images I _{0}, I _{1} are given and connected via the diffeomorphic change of coordinates I _{0}○ϕ^{−1}=I _{1} where ϕ=Φ_{1} is the end point at t= 1 of curve Φ_{ t }, t∈[0, 1] satisfying ^{.}Φ_{ t }=v _{t} (Φ_{ t }), t∈ [0,1] with Φ_{0}=id. The variational problem takes the form
$$\mathop {\arg {\text{m}}in}\limits_{\upsilon :\dot \phi _t = \upsilon _t \left( {\dot \phi } \right)} \left( {\int_0^1 {\left\ {\upsilon _t } \right\} ^2 {\text{d}}t + \left\ {I_0 \circ \phi _1^{  1}  I_1 } \right\_{L^2 }^2 } \right),$$
where ‖v _{t}‖_{ V } is an appropriate Sobolev norm on the velocity field v _{t}(·), and the second term enforces matching of the images with ‖·‖_{L} ^{2} representing the squarederror norm.
In this paper we derive the EulerLagrange equations characterizing the minimizing vector fields v _{t}, t∈[0, 1] assuming sufficient smoothness of the norm to guarantee existence of solutions in the space of diffeomorphisms. We describe the implementation of the Euler equations using semilagrangian method of computing particle flows and show the solutions for various examples. As well, we compute the metric distance on several anatomical configurations as measured by ∫_{0} ^{1}‖v _{t}‖_{ V }dt on the geodesic shortest paths.
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 Title
 Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms
 Journal

International Journal of Computer Vision
Volume 61, Issue 2 , pp 139157
 Cover Date
 20050201
 DOI
 10.1023/B:VISI.0000043755.93987.aa
 Print ISSN
 09205691
 Online ISSN
 15731405
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 Computational Anatomy
 EulerLagrange Equation
 Variational Optimization
 Deformable Template
 Metrics
 Industry Sectors
 Authors

 M. Faisal Beg ^{(1)}
 Michael I. Miller ^{(2)}
 Alain Trouvé ^{(3)}
 Laurent Younes ^{(4)}
 Author Affiliations

 1. Center for Imaging Science & Department of Biomedical Engineering, The Johns Hopkins University, 301 Clark Hall, Baltimore, MD, 21218, USA
 2. Center for Imaging Science, department of Biomedical Engineering, Department of Electrical and Computer Engineering and The Department of Computer Science, Whiting School of Engineering, The Johns Hopkins University, 301 Clark Hall, Baltimore, MD, 21218, USA
 3. LAGA, Université Paris, France
 4. CMLA, Ecole Normale Supérieure de Cachan, 61, Avenue du President Wilson, F94 235, Cachan CEDEX, France