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○ϕ−1=I 1 where ϕ=Φ1 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 Φ0=id. The variational problem takes the form
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 squared-error norm.
In this paper we derive the Euler-Lagrange 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 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 ∫0 1‖v t‖ V dt on the geodesic shortest paths.
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Beg, M.F., Miller, M.I., Trouvé, A. et al. Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms. Int J Comput Vision 61, 139–157 (2005). https://doi.org/10.1023/B:VISI.0000043755.93987.aa
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DOI: https://doi.org/10.1023/B:VISI.0000043755.93987.aa