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 Trouve (1995) in which two images 10, 1, are given and connected via the diffeomorphic change of coordinates I-0 o phi(-1) = I, where p = 01 is the end point at t = 1 of curve phi(t), t is an element of [0, 1] satisfying (phi)over dot(t) = v(t)(phi(t)), t is an element of [0, 1] with phi(0) = id. The variational problem takes the form [GRAPHICS] where parallel tov(t)parallel to(V) is an appropriate Sobolev norm on the velocity field v(t)(.), and the second term enforces matching of the images with parallel to.parallel to(L2) representing the squared-error norm. In this paper we derive the Euler-Lagrange equations characterizing the minimizing vector fields vt(,) t is an element of [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 integral(0)(1) parallel tov(t)parallel to(v)dt on the geodesic shortest paths.