Spatiospectral Concentration on a Sphere

Abstract

We pose and solve the analogue of Slepian's time-frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the sphere, or, alternatively, of strictly spacelimited functions that are optimally concentrated within the spherical harmonic domain. Such a basis of simultaneously spatially and spectrally concentrated functions should be a useful data analysis and representation tool in a variety of geophysical and planetary applications, as well as in medical imaging, computer science, cosmology and numerical analysis. The spherical Slepian functions can be found either by solving an algebraic eigenvalue problem in the spectral domain or by solving a Fredholm integral equation in the spatial domain. The associated eigenvalues are a measure of the spatiospectral concentration. When the concentration region is an axisymmetric polar cap the spatiospectral projection operator commutes with a Sturm-Liouville operator; this enables the eigenfunctions to be computed extremely accurately and efficiently, even when their area-bandwidth product, or Shannon number, is large. In the asymptotic limit of a small concentration region and a large spherical harmonic bandwidth the spherical concentration problem approaches its planar equivalent, which exhibits self-similarity when the Shannon number is kept invariant.

Mapping Greenland’s mass loss in space and time

“…We truncate the expansion at the effective dimension of the combined spatiospectral space (Greenland in space, band-limited spectrally), known as the Shannon number (23,33). This truncation leaves only 20 target functions, each of which is an eigenmap that has its energy highly concentrated over Greenland.…”

“…Our inversion method relies on a spherical basis of spatiospectrally concentrated Slepian functions (23,24). We show its ability to resolve unprecedented geographical and temporal detail in the mass flux using gravity data alone.…”

On the response of the Earth to a fault system: its evaluation beyond the epicentral reference frame

“…Particularly, it simplifies the implementation of anisotropic filters for dealing with the peculiar noise at equatorial latitudes (the so-called 'stripes') of the Stokes coefficients from GRACE and the spatial localization of seismic source response in a polar cup including the near field of the earthquake using the Slepian functions. Indeed, both the anisotropic filters and the spatial localization are usually defined in terms of linear combinations of spherical harmonic coefficients of the perturbation in a fixed reference frame: the geographic reference frame for anisotropic filtering (Sweanson and Wahr, 2006;Kusche et al, 2007Kusche et al, , 2009) and the spherical reference frame with the polar axis along the center of the cup for the spatial localization (Simons, 2006;Wang et al, 2012;Cambiotti and Sabaini, 2012; APPENDIX A: SPHERICAL HARMONICS…”

Eigendecomposition of Images Correlated on $S^{1}$, $S^{2}$, and $SO(3)$ Using Spectral Theory

“…They have been used for solving PDE's in spherical geometry for weather and climate models [28], geophysics [29], [30], quantum mechanics [31], [32], as well as a host of other related applications [33]. Over the last decade, spherical harmonics have been gaining popularity in the computer vision and computer graphics arena.…”