Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms

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 Trouv´e (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 = vt (φt ), t ∈ [0, 1] with φ0 = id. The variational problem takes the form argmin v:˙φt=vt (φt ) 1 0 vt 2 V dt + I0 ◦ φ −1 1 − I1 2 L2 , where vt V 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.We also compute the metric distance on several anatomical configurations as measured by 1 0 vt V dt on the geodesic shortest paths.