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December 2005 Optimal designs for three-dimensional shape analysis with spherical harmonic descriptors
Holger Dette, Viatcheslav B. Melas, Andrey Pepelyshev
Ann. Statist. 33(6): 2758-2788 (December 2005). DOI: 10.1214/009053605000000552

Abstract

We determine optimal designs for some regression models which are frequently used for describing three-dimensional shapes. These models are based on a Fourier expansion of a function defined on the unit sphere in terms of spherical harmonic basis functions. In particular, it is demonstrated that the uniform distribution on the sphere is optimal with respect to all Φp criteria proposed by Kiefer in 1974 and also optimal with respect to a criterion which maximizes a p mean of the r smallest eigenvalues of the variance–covariance matrix. This criterion is related to principal component analysis, which is the common tool for analyzing this type of image data. Moreover, discrete designs on the sphere are derived, which yield the same information matrix in the spherical harmonic regression model as the uniform distribution and are therefore directly implementable in practice. It is demonstrated that the new designs are substantially more efficient than the commonly used designs in three-dimensional shape analysis.

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Holger Dette. Viatcheslav B. Melas. Andrey Pepelyshev. "Optimal designs for three-dimensional shape analysis with spherical harmonic descriptors." Ann. Statist. 33 (6) 2758 - 2788, December 2005. https://doi.org/10.1214/009053605000000552

Information

Published: December 2005
First available in Project Euclid: 17 February 2006

zbMATH: 1084.62065
MathSciNet: MR2253101
Digital Object Identifier: 10.1214/009053605000000552

Subjects:
Primary: 62K05 , 65D32

Keywords: 3D image data , Optimal designs , Principal Component Analysis , quadrature formulas , shape analysis , spherical harmonic descriptors

Rights: Copyright © 2005 Institute of Mathematical Statistics

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Vol.33 • No. 6 • December 2005
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