smooth velocity vector fields

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 ϕ̇t = vt(ϕt), t ∈ [0, 1] with ϕ0 = id. The variational problem takes the form argmin v:ϕ̇t=vt(ϕt) ( ∫0^1 ‖vt‖_V^2 dt + ‖I0 ∘ ϕ_1^−1 − I1‖_L^2^2 ), where ‖vt‖_V is an appropriate Sobolev norm on the velocity field vt(·), 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 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. We also compute the metric distance on several anatomical configurations as measured by ∫0^1 ‖vt‖_V dt on the geodesic shortest paths.