A Geometric Framework for Stochastic Shape Analysis

We introduce a stochastic model of diffeomorphisms, whose action on a variety of data types descends to stochastic evolution of shapes, images and landmarks. The stochasticity is introduced in the vector field which transports the data in the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework for shape analysis and image registration. The stochasticity thereby models errors or uncertainties of the flow in following the prescribed deformation velocity. The approach is illustrated in the example of finite dimensional landmark manifolds, whose stochastic evolution is studied both via the Fokker-Planck equation and by numerical simulations. We derive two approaches for inferring parameters of the stochastic model from landmark configurations observed at discrete time points. The first of the two approaches matches moments of the Fokker-Planck equation to sample moments of the data, while the second approach employs an Expectation-Maximisation based algorithm using a Monte Carlo bridge sampling scheme to optimise the data likelihood. We derive and numerically test the ability of the two approaches to infer the spatial correlation length of the underlying noise.