Statistica Sinica 19 (2009), 83-102

ROBUST DESIGNS FOR 3D SHAPE ANALYSIS

WITH SPHERICAL HARMONIC DESCRIPTORS

Holger Dette and Douglas P. Wiens

Ruhr-Universität Bochum and University of Alberta

Abstract: Spherical harmonic descriptors are frequently used for describing three-dimensional shapes in terms of Fourier coefficients corresponding to an expansion of a function defined on the unit sphere. In a recent paper Dette, Melas and Pepelysheff (2005) determined optimal designs with respect to Kiefer's $\Phi_{p}$-criteria for regression models derived from a truncated Fourier series. In particular it was shown that the uniform distribution on the sphere is $\Phi_{p}$-optimal for spherical harmonic descriptors, for all $p>-1$. These designs minimize a function of the variance-covariance matrix of the least squares estimate but do not take into account the bias resulting from the truncation of the series.

In the present paper we demonstrate that the uniform distribution is also optimal with respect to a minimax criterion based on the mean square error, and as a consequence these designs are robust with respect to the truncation error. Moreover, we also consider heteroscedasticity and possible correlations in the construction of the optimal designs. These features appear naturally in 3D shape analysis, and the uniform design again turns out to be minimax robust against erroneous assumptions of homoscedasticity and independence.

Key words and phrases: 3D-image data, dependent data, mean square error, minimax optimal designs, optimal designs, robust designs, shape analysis, spherical harmonic descriptors.