Landmark matching via large deformation diffeomorphisms

This paper describes the generation of large deformation diffeomorphisms φ:Ω=[0,1]3&rlhar2;Ω for landmark matching generated as solutions to the transport equation dφ(x,t)/dt=ν(φ(x,t),t),t∈[0,1] and φ(x,0)=x, with the image map defined as φ(·,1) and therefore controlled via the velocity field ν(·,t),t∈[0,1]. Imagery are assumed characterized via sets of landmarks {xn, yn, n=1, 2, ..., N}. The optimal diffeomorphic match is constructed to minimize a running smoothness cost ||Lν||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 xn matched to yn measured with error covariances Σn, then the matching problem is solved generating the optimal diffeomorphism φˆ(x,1)=∫01 νˆ(φˆ(x,t),t)dt+x where νˆ(·)argminν(·)∫1 1∫Ω||Lν(x,t)||2dxdt +Σn=1N[yn-φ(xn,1)] TΣn-1[yn-φ(xn ,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 macaque monkey