We have been studying the well-established diffeomorphic landmark matching problem central to the emerging field of computational anatomy.

Computational anatomy models landmark evolution as a dynamical system governed by a smooth velocity field. Fundamental principles of mechanics allow the interpretation of this framework from a Hamiltonian point of viewin which the landmarks evolve as particles under specific equations of motion. This leads to a result stating that the landmark trajectories are entirely specified from only the initial conditions(position and momentum).

We exploit this analysis by designing landmark matching algorithms which optimize the trajectories with respect to the initial momentum, given a fixed initial position (template). Performing Flow of diffeomorphic transfunction of template (profile one) to target (profile six).this optimization in momentum space has the advantage of using a template-based, finitedimensional system of coordinates which contains all necessary information for the reconstruction of a dense diffeomorphism of the background space, and which provides the required landmark matching interpolation.

Shown in the figure are results from 3D hippocampus examples from the BIRN Morphometry database. The left and right panels show the template and target configurations. The flow sequence is shown in the center panels.

A by-product of this optimization scheme is to readily provide the momentum coordinates of a target configuration of landmarks relative to a given template. These coordinates can be linearly combined to provide new objects, and used to design statistical models of deformations.