Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms

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 Trouve (1995) in which two images I 0, I 1 are given and connected via the diffeomorphic change of coordinates I 0???1=I 1 where ?=?1 is the end point at t= 1 of curve ? t , t?[0, 1] satisfying .? t =v t (? t ), t? [0,1] with ?0=id. The variational problem takes the form $$\mathop {\arg {\text{m}}in}\limits_{\upsilon :\dot \phi _t = \upsilon _t \left( {\dot \phi } \right)} \left( {\int_0^1 {\left\| {\upsilon _t } \right\|} ^2 {\text{d}}t + \left\| {I_0 \circ \phi _1^{ - 1} - I_1 } \right\|_{L^2 }^2 } \right),$$ 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.