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Biometrika 2000 87(3):527-544; doi:10.1093/biomet/87.3.527
© 2000 by Biometrika Trust
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Using circulant symmetry to model featureless objects

JT KentZ, IL DrydenZZ and CR AndersonY

Z Department of Statistics, University of Leeds, Leeds LS2 9JT, UK E-mail: j.t.kent@leeds.ac.uk ZZ School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK E-mail: ild@maths.nott.ac.uk Y Garvin and Company, Redditch B97 4BS, UK

Grenander & Miller (1994) describe a model for representing amorphous two-dimensional objects with no obvious landmark. Each object is represented by n vertices around its perimeter, and is described by deforming an n-sided regular polygon using edge transformations. A multivariate normal distribution with a block circulant covariance matrix is used to model these edge transformations. The purpose of this paper is to describe in detail the statistical properties of this multivariate model and the eigenstructure of the covariance matrix. Various special cases of the model are considered, including articulated models and conditional Markov random field models. We consider maximum likelihood based inference and the model is applied to some datasets to explore shape variability.

Key Words: articulated; circulant matrix; circulant symmetry; complex symmetry; deformation; edge transformation; Markov random field; outline; particles; reflection; shape


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