Hierarchical Graphical Models for Multigroup Shape Analysis using Expectation Maximization with Sampling in Kendall's Shape Space

TL;DR

This paper introduces a hierarchical generative shape model in Kendall's shape space, employing EM with sampling to analyze multiple shape groups, addressing correspondence and enabling classification and hypothesis testing.

Contribution

It presents a novel hierarchical graphical model for multigroup shape analysis that naturally enforces correspondences and uses EM with Hamiltonian Monte Carlo sampling for inference.

Findings

Effective hypothesis testing between shape groups.

Accurate classification in shape retrieval tasks.

Validated on simulated and real datasets.

Abstract

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.