Abstract
In this paper, aliasing effects are investigated for random fields defined on the d-dimensional sphere and reconstructed from discrete samples. First, we introduce the concept of an aliasing function on the sphere. 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.
Highlights
We are concerned with the study of the aliasing effects for the harmonic expansion of a random field defined on the d-dimensional sphere Sd
The analysis of spherical random fields over Sd is strongly motivated by a growing set of applications in several scientific disciplines, such as Cosmology and Astrophysics for d = 2, as well as in Medical Image Analysis ([HCW+13, HCK+15]), Material Physics ([MS08]), and Nuclear Physics ([AA18]) for d > 2
The so-called 4D-hyperspherical harmonic representation of surface anatomy aims to solve this issue by means of a stereographic projection of an entire collection of disjoint 3-dimensional objects onto the hypersphere of dimension 4
Summary
It is there proved that band-limited random fields over S2, which can be roughly viewed as linear combinations of finitely many spherical harmonics, can be uniquely reconstructed with a sufficiently large sample size. An explicit definition of the aliasing function, a crucial tool to identify the aliases of a given harmonic coefficient, is developed when the sampling is based on the combination of a Gauss-Legendre quadrature formula and a trapezoidal rule (see Section 4 for further details).
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