Bridge Simulation and Metric Estimation on Landmark Manifolds
This work addresses a domain-specific challenge in computational anatomy for researchers dealing with nonlinear landmark manifolds, representing an incremental advancement in metric estimation techniques.
The authors tackled the problem of estimating the metric structure from landmark configurations under the LDDMM metric by developing an inference algorithm that approximates the transition density using Monte Carlo sampling of conditioned Brownian bridges, enabling parameter estimation via maximum likelihood.
We present an inference algorithm and connected Monte Carlo based estimation procedures for metric estimation from landmark configurations distributed according to the transition distribution of a Riemannian Brownian motion arising from the Large Deformation Diffeomorphic Metric Mapping (LDDMM) metric. The distribution possesses properties similar to the regular Euclidean normal distribution but its transition density is governed by a high-dimensional PDE with no closed-form solution in the nonlinear case. We show how the density can be numerically approximated by Monte Carlo sampling of conditioned Brownian bridges, and we use this to estimate parameters of the LDDMM kernel and thus the metric structure by maximum likelihood.