Algorithmic Advances in Riemannian Geometry and Applications

In this chapter, we present Bayesian models for diffeomorphic shape variability in populations of images. The first model is a probabilistic formulation of the image atlas construction problem, which seeks to compute an atlas image most representative of a set of input images. The second model adds diffeomorphic modes of shape variation, or principal geodesics. Both of these models represent shape variability as random variables on the manifold of diffeomorphic transformations. We define a Gaussian prior distribution for diffeomorphic transformations using the inner product in the tangent space to the diffeomorphism group. We develop a Monte Carlo Expectation Maximization (MCEM) algorithm for the Bayesian inference, due to the lack of closed-form solutions, where the expectation step is approximated via Hamiltonian Monte Carlo (HMC) sampling of diffeomorphisms. The resulting inference produces estimates of the image atlas, principal geodesic modes of variation, and model parameters. We show that the advantage of the Bayesian formulation is that it provides a principled way to estimate both the regularization parameter of the diffeomorphic transformations and the intrinsic dimensionality of the input data.