Robust Designs for 3d Shape Analysis with Spherical Harmonic Descriptors

Spherical harmonic descriptors are frequently used for describing threedimensional 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 Φp-criteria for regression models derived from a truncated Fourier series. In particular it was shown that the uniform distribution on the sphere is Φp-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.