Statistical shape analysis: clustering, learning, and testing

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 minimum variance type 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.

1 Introduction

An important goal in image analysis is to classify and recognize objects of interest in given images. Imaged objects can be characterized in several ways, using their colors, textures, shapes, movements, and locations. The past decade has seen large efforts in modeling and analysis of pixel statistics in images to attain these goals albeit with limited success. An emerging opinion is that global features such as shapes be taken into account. Characterization of complex objects using their global shapes is fast becoming a major tool in computer vision and image understanding. Analysis of shapes, especially those of complex objects, is a challenging task and requires sophisticated mathematical tools. Applications of shape analysis include biomedical image analysis, morphometry, database retrieval, surveillance, biometrics, military target recognition, and general computer vision.