Landmark matching via large deformation diffeomorphisms

This paper describes the generation of large deformation diffeomorphisms phi : Omega = [0, 1](3) <-> for landmark matching generated as solutions to the transport equation d phi(x,t)/dt = nu(phi(x,t),t),t epsilon [0,1] and phi(x,0) = x, with the image map defined as phi(, 1) and therefore controlled via the velocity field nu(., t),t epsilon [0, 1], Imagery are assumed characterized via sets of landmarks {x(n), y(n), n = 1, 2,..., N}. The optimal diffeomorphic match is constructed to minimize a running smoothness cost parallel to L nu parallel to(2) associated with a linear differential operator L on the velocity field generating the diffeomorphism while simultaneously minimizing the matching end point condition of the landmarks, Both inexact and exact Landmark matching is studied here. Given noisy landmarks x(n) matched to y(n) measured with error covariances Sigma(n) then the matching problem is solved generating the optimal diffeomorphism <(phi)over cap>(x, 1) = integral(0)(1) <(nu)over cap>(<(phi)over cap>(x,t), t) dt + x where <(nu)over cap>(.) = arg min(nu(.)) integral(0)(1) integral(Omega) parallel to L nu(x,t)parallel to(2) dx dt + Sigma(n=1)(N) [yn-phi(xn,1)](T)Sigma(n)(-1)[yn-phi(x(n),1)]. (1) Conditions for the existence of solutions in the space of diffeomorphisms are established, with a gradient algorithm provided for generating the optimal flow solving the minimum problem. Results on matching two-dimensional (2-D) and three-dimensional (3-D) imagery are presented in the the macaque monkey.