Coarse-to-Fine Hamiltonian Dynamics of Hierarchical Flows in Computational Anatomy

We present here the Hamiltonian control equations for hierarchical diffeomorphic flows of particles. We define the controls to be a series of multi-scale vector fields, each with their own reproducing kernel Hilbert space norm. The hierarchical control is connected across scale through successive refinements that refine as they ascend the hierarchy with commensurately higher bandwidth Green's kernels. Interestingly the geodesic equations do not separate, with fine scale motions determined by all of the particle information simultaneously, from coarse to fine. Additionally, the hierarchical conservation law is derived, defining the geodesics and demonstrating the constancy of the Hamiltonian. We show results on one simulated example and one example from histological images of an Alzheimer's disease brain. We introduce the varifold action to transport the weights of micro-scale particles for mapping to sub millimeter scale cortical folds.