Aliasing effects for random fields over spheres of arbitrary dimension

Abstract

In this paper, aliasing effects are investigated for random fields defined on the $d$-dimensional sphere $\mathbb{S}^{d}$ and reconstructed from discrete samples. First, we introduce the concept of an aliasing function on $\mathbb{S}^{d}$. The aliasing function allows one to identify explicitly the aliases of a given harmonic coefficient in the Fourier decomposition. Then, we exploit this tool to establish the aliases of the harmonic coefficients approximated by means of the quadrature procedure named spherical uniform sampling. Subsequently, we study the consequences of the aliasing errors in the approximation of the angular power spectrum of an isotropic random field, the harmonic decomposition of its covariance function. Finally, we show that band-limited random fields are aliases-free, under the assumption of a sufficiently large amount of nodes in the quadrature rule.