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 {x_n, y_n, 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 x_n matched to y_n measured with error covariances Σ_n, then the matching problem is solved generating the optimal diffeomorphism φˆ(x,1)=∫₀¹ νˆ(φˆ(x,t),t)dt+x where νˆ(·)argmin_ν(·)∫₁ ¹∫_Ω||Lν(x,t)||²dxdt +Σ_n=1^N[y_n-φ(x_n,1)] ^TΣ_n^-1[y_n-φ(x_n ,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