  This paper proposes a novel framework for multi-group shape analysis relying
on a hierarchical graphical statistical model on shapes within a population.The
framework represents individual shapes as point setsmodulo translation,
rotation, and scale, following the notion in Kendall shape space.While
individual shapes are derived from their group shape model, each group shape
model is derived from a single population shape model. The hierarchical model
follows the natural organization of population data and the top level in the
hierarchy provides a common frame of reference for multigroup shape analysis,
e.g. classification and hypothesis testing. Unlike typical shape-modeling
approaches, the proposed model is a generative model that defines a joint
distribution of object-boundary data and the shape-model variables.
Furthermore, it naturally enforces optimal correspondences during the process
of model fitting and thereby subsumes the so-called correspondence problem. The
proposed inference scheme employs an expectation maximization (EM) algorithm
that treats the individual and group shape variables as hidden random variables
and integrates them out before estimating the parameters (population mean and
variance and the group variances). The underpinning of the EM algorithm is the
sampling of pointsets, in Kendall shape space, from their posterior
distribution, for which we exploit a highly-efficient scheme based on
Hamiltonian Monte Carlo simulation. Experiments in this paper use the fitted
hierarchical model to perform (1) hypothesis testing for comparison between
pairs of groups using permutation testing and (2) classification for image
retrieval. The paper validates the proposed framework on simulated data and
demonstrates results on real data.
