August 2009
Statistical analysis on landmark-based shape spaces has diverse applications in morphometrics, medical diagnostics, machine vision, robotics and other areas. These shape spaces are non-Euclidean quotient manifolds, often the quotient of the unit sphere under a group of transformations. To conduct nonparametric inferences, one may define notions of center and spread of a probability distribution on an arbitrary manifold and work with their estimates. There has been a significant amount of work done in this direction. However, it is useful to consider full likelihood-based methods, which allow nonparametric estimation of the probability density. This article proposes a class of mixture models constructed using suitable kernels on a general compact non-Euclidean manifold and then on the planar shape space in particular. Following a Bayesian approach with a nonparametric prior on the mixing distribution, conditions are obtained under which the Kullback-Leibler property holds, implying large support and weak posterior consistency. Gibbs sampling methods are developed for posterior computation, and the methods are applied to problems in density estimation on shape space and classification with shape-based predictors.
Keywords: Non-Euclidean manifold; Planar shape space; Nonparametric Bayes; Dirichlet process mixture; KL property; Posterior consistency; Discriminant analysis.
The manuscript is available in PDF format.