Landmark matching via large deformation diffeomorphisms

Abstract

This paper describes the generation of large deformation diffeomorphisms /spl phi/:/spl Omega/=[0,1]/sup 3//spl rlhar2//spl Omega/ for landmark matching generated as solutions to the transport equation d/spl phi/(x,t)/dt=/spl nu/(/spl phi/(x,t),t),t/spl isin/[0,1] and /spl phi/(x,0)=x, with the image map defined as /spl phi/(/spl middot/,1) and therefore controlled via the velocity field /spl nu/(/spl middot/,t),t/spl isin/[0,1]. Imagery are assumed characterized via sets of landmarks {x/sub n/, y/sub n/, n=1, 2, ..., N}. The optimal diffeomorphic match is constructed to minimize a running smoothness cost /spl par/L/spl nu//spl par//sup 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/sub n/ matched to y/sub n/ measured with error covariances /spl Sigma//sub n/, then the matching problem is solved generating the optimal diffeomorphism /spl phi//spl circ/(x,1)=/spl int//sub 0//sup 1//spl nu//spl circ/(/spl phi//spl circ/(x,t),t)dt+x where /spl nu//spl circ/(/spl middot/)argmin/sub /spl nu/(/spl middot/)//spl int//sub 1//sup 1//spl int//sub /spl Omega///spl par/L/spl nu/(x,t)/spl par//sup 2/dxdt +/spl Sigma//sub n=1//sup N/[y/sub n/-/spl phi/(x/sub n/,1)]/sup T//spl Sigma//sub n//sup -1/[y/sub n/-/spl phi/(x/sub 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.