Stochastically evolving geometric systems are studied in shape analysis and computational anatomy for modeling random evolutions of human organ shapes. The notion of geodesic paths between shapes is central to shape analysis and has a natural generalization as diffusion bridges in a stochastic setting. Simulation of such bridges is key to solving inference and registration problems in shape analysis. We demonstrate how to apply state-of-the-art diffusion bridge simulation methods to recently introduced stochastic shape deformation models, thereby substantially expanding the appli-cability of such models. We exemplify these methods by estimating template shapes from observed shape configurations while simultaneously learning model parameters.

Suppose X is a multivariate diffusion process that is observed discretely in time. At each observation time, a transformation of the state of the process is observed with noise. The smoothing problem consists of recovering the path of the process, consistent with the observations. We derive a novel Markov Chain Monte Carlo algorithm to sample from the exact smoothing distribution. The resulting algorithm is called the Backward Filtering Forward Guiding (BFFG) algorithm. We extend the algorithm to include parameter estimation. The proposed method relies on guided proposals introduced in [53]. We illustrate its efficiency in a number of challenging problems.