  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 $\Phi_p$
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.
