@article {M.:February 2005:0920-5691:139, author = "M. Faisal Beg", author = "Michael I. Miller", author = "Alain Trouve", author = "Laurent Younes", title = "Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms", journal = "International Journal of Computer Vision", volume = "61", year = "February 2005", 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 I0, I1 are given and connected via the diffeomorphic change of coordinates I0○PHgr-1 = I1 where PHgr = phiv1 is the end point at t = 1 of curve phivt, tisin[0, 1] satisfying ·phivt = vt (phivt), t isin [0,1] with phiv0 = id. The variational problem takes the form

\argmin_{v: \dot \phi_t = v_t (\phi_t)} \Bigg ( \int_0ˆ1 \| v_t\|ˆ2_V \dt+\big \| I_0 \circ \phi_1ˆ{-1} - I_1\big \|_{Lˆ2}ˆ2 \Bigg),

where VerbarvtVerbarV is an appropriate Sobolev norm on the velocity field vt(·), and the second term enforces matching of the images with Verbar·VerbarL2 representing the squared-error norm.

In this paper we derive the Euler-Lagrange equations characterizing the minimizing vector fields vt, tisin[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 int01 VerbarvtVerbarVdt on the geodesic shortest paths.", pages = "139-157(19)", url = "http://www.ingentaconnect.com/content/klu/visi/2005/00000061/00000002/05384378" doi = "doi:10.1023/B:VISI.0000043755.93987.aa" }