International Journal of Computer Vision

, Volume 61, Issue 2, pp 139–157

Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms

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

DOI: 10.1023/B:VISI.0000043755.93987.aa

Cite this article as:
Beg, M.F., Miller, M.I., Trouvé, A. et al. Int J Comput Vision (2005) 61: 139. doi:10.1023/B:VISI.0000043755.93987.aa


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 I0, I1 are given and connected via the diffeomorphic change of coordinates I0○ϕ−1=I1 where ϕ=Φ1 is the end point at t= 1 of curve Φt, t∈[0, 1] satisfying .Φt=vtt), t∈ [0,1] with Φ0=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 ‖vtV is an appropriate Sobolev norm on the velocity field vt(·), and the second term enforces matching of the images with ‖·‖L2 representing the squared-error norm.

In this paper we derive the Euler-Lagrange equations characterizing the minimizing vector fields vt, 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 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 ∫01vtVdt on the geodesic shortest paths.

Computational Anatomy Euler-Lagrange Equation Variational Optimization Deformable Template Metrics 

Copyright information

© Kluwer Academic Publishers 2005

Authors and Affiliations

  • M. Faisal Beg
    • 1
  • Michael I. Miller
    • 2
  • Alain Trouvé
    • 3
  • Laurent Younes
    • 4
  1. 1.Center for Imaging Science & Department of Biomedical EngineeringThe Johns Hopkins UniversityBaltimoreUSA
  2. 2.Center for Imaging Science, department of Biomedical Engineering, Department of Electrical and Computer Engineering and The Department of Computer Science, Whiting School of EngineeringThe Johns Hopkins UniversityBaltimoreUSA
  3. 3.LAGAUniversité ParisFrance
  4. 4.CMLAEcole Normale Supérieure de CachanCachan CEDEXFrance