Statistical shape analysis: clustering, learning, and testing

TLDR

The authors develop a differential‑geometric framework for planar shapes that enables hierarchical clustering, learning of probability models for shape clusters, and hypothesis testing of new shapes. Clustering is performed with a minimum‑variance Markov criterion, iteratively using cluster means to build a shape hierarchy, while probability models are imposed on finite‑dimensional tangent approximations at sample means for random sampling and classification. The integrated clustering and hypothesis‑testing approach yields an efficient shape‑retrieval system, as illustrated on datasets from ETH, Surrey, and AMCOM.

Abstract

Using a differential-geometric treatment of planar shapes, we present tools for: 1) hierarchical clustering of imaged objects according to the shapes of their boundaries, 2) learning of probability models for clusters of shapes, and 3) testing of newly observed shapes under competing probability models. Clustering at any level of hierarchy is performed using a mimimum variance type criterion criterion and a Markov process. Statistical means of clusters provide shapes to be clustered at the next higher level, thus building a hierarchy of shapes. Using finite-dimensional approximations of spaces tangent to the shape space at sample means, we (implicitly) impose probability models on the shape space, and results are illustrated via random sampling and classification (hypothesis testing). Together, hierarchical clustering and hypothesis testing provide an efficient framework for shape retrieval. Examples are presented using shapes and images from ETH, Surrey, and AMCOM databases.