Using circulant symmetry to model featureless objects

Grenander & Miller (1994) describe a model for representing amorphous two-d imensional objects with no obvious landmark. Each object is represented by n vertices around its perimeter, and is described by deforming an n-sided r egular polygon using edge transformations. A multivariate normal distributi on with a block circulant covariance matrix is used to model these edge tra nsformations. The purpose of this paper is to describe in detail the statis tical properties of this multivariate model and the eigenstructure of the c ovariance matrix. Various special cases of the model are considered, includ ing articulated models and conditional Markov random field models. We consi der maximum likelihood based inference and the model is applied to some dat asets to explore shape variability.