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 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

$$\mathop {\arg {\text{m}}in}\limits_{\upsilon :\dot \phi _t = \upsilon _t \left( {\dot \phi } \right)} \left( {\int_0^1 {\left\| {\upsilon _t } \right\|} ^2 {\text{d}}t + \left\| {I_0 \circ \phi _1^{ - 1} - I_1 } \right\|_{L^2 }^2 } \right),$$

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.