Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms

  • Authors:

  • M. Faisal Beg;Michael I. Miller;Alain Trouvé;Laurent Younes

  • Affiliations:

  • Center for Imaging Science & Department of Biomedical Engineering, The Johns Hopkins University, Baltimore, USA 21218;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 ...;LAGA, Université Paris, France;CMLA, Ecole Normale Supérieure de Cachan, Cachan CEDEX, France F-94 235

  • Venue:
  • International Journal of Computer Vision
  • Year:
  • 2005

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Abstract

This paper examine the Euler-Lagrange 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驴驴驴1=I 1 where 驴=驴1 is the end point at t= 1 of curve 驴 t , t驴[0, 1] satisfying .驴 t =v t (驴 t ), t驴 [0,1] with 驴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 驴v t驴 V is an appropriate Sobolev norm on the velocity field v t(·), and the second term enforces matching of the images with 驴·驴L 2 representing the squared-error norm.In this paper we derive the Euler-Lagrange equations characterizing the minimizing vector fields v t, t驴[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 semi-lagrangian 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 驴0 1驴v t驴 V dt on the geodesic shortest paths.