Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms
 M. Faisal Beg,
 Michael I. Miller,
 Alain Trouvé,
 Laurent Younes
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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.
 Amit, Y. 1994. A nonlinear variational problem for image matching. SIAM Journal on Scientific Computing, 15(1):207224. CrossRef
 Bajcsy, R. and Broit, C. 1982. Matching of deformed images. In Proc. 6th Int. Joint Conf Patt. Recog., pp. 351353.
 Bajcsy, R., Lieberson, R., and Reivich, M. 1983. A computerized system for the elastic matching of deformed radiographic images to idealized atlas images. Journal of Computer Assisted Tomography, 7(4):618625.
 Broit, C. 1981. Optimal registration of deformed images. PhD thesis,University of Pennsylvania.
 Do Carmo, M.P 1976. Differential geometry of curves and surfaces. PrenticeHall Engineering/Science/Mathematics.
 Do Carmo, M.P 1993. Riemannian Geometry. Birkhauser.
 Christensen, G.E., Rabbitt, R.D., and Miller, M.I. 1996. Deformable templates using large deformation kinematics. IEEE Transactions on Image Processing, 5(10):14351447. CrossRef
 Christensen, G. 1994. Deformable shape models for anatomy. PhD Thesis, Dept. of Electrical Engineering, Sever Institute of Technology, Washington Univ., St. Louis, MO.
 Dupuis, P., Grenander, U., and Miller, M.I. 1998. Variational problems on flows of diffeomorphisms for image matching. Quarterly of Applied Mathematics, LVI:587600.
 Grenander, U. and Miller, M.I. 1998. Computational anatomy: An emerging discipline. Quarterly of Applied Mathematics, 56:617694.
 Miller, M.I., Trouv6, A., and Younes, L. 2002. On the metrics and EulerLagrange equations of computational anatomy. Annual Review of Biomedical Engineering, 4:375405. CrossRef
 Miller, M.I. and Younes, L. 2001. Group actions, homeomorphisms, and matching: A general framework. International Journal of Computer Vision, 41:6184. CrossRef
 Morton, K.W. and Mayers, D.E 1996. Numerical Solution of Partial Differential Equations. Cambridge University Press, University of Cambridge.
 Robb, R.A. 1999. Biomedical Imaging, Vizualization and Analysis. John Wiley and Sons, Inc., New York, NY.
 Staniforth, A. and C6t6, J. 1991. Semilagrangian integration schemes for atmospheric modelsa review. Monthly Weather Review, 119:22062223. CrossRef
 Thirion, J.P. 1998. Image matching as a diffusion process: An analogy with maxwell’s demons. Medical Image Analysis, 2(3):243260. CrossRef
 Trouvé, A. 1995. An infinite dimensional group approach for physics based models in patterns recognition. Preprint.
 Trouvé, A. 1998. Diffeomorphic groups and pattern matching in image analysis. Int. J. Computer Vision, 28:213221 CrossRef
 Younes, L. 1999. Optimal matching between shapes via elastic deformations. Image and Vision Computing, 17:381389. CrossRef
 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