Slepian Scale-Discretised Wavelets on the Sphere

Abstract

Wavelets are widely used in various disciplines to analyse signals both in space and scale. Whilst many fields measure data on manifolds (i.e., the sphere), often data are only observed on a partial region of the manifold. Wavelets are a typical approach to data of this form, but the wavelet coefficients that overlap with the boundary become contaminated and must be removed for accurate analysis. Another approach is to estimate the region of missing data and to use existing whole-manifold methods for analysis. However, both approaches introduce uncertainty into any analysis. Slepian wavelets enable one to work directly with only the data present, thus avoiding the problems discussed above. Applications of Slepian wavelets to areas of research measuring data on the partial sphere include gravitational/magnetic fields in geodesy, ground-based measurements in astronomy, measurements of whole-planet properties in planetary science, geomagnetism of the Earth, and cosmic microwave background analyses.

Slepian Scale-Discretised Wavelets on Manifolds

“…Many fields measure data that are intrinsically non-Euclidean in structure and are better modelled by manifolds or graphs. One manifold, in particular, which is commonplace in science and engineering is the sphere; such as in: cosmology [1], geophysics [2], planetary science [3], computer graphics [4], and signal processing [5], [6]. Often data are not observed in some regions of the manifold, and hence methods which work over the whole manifold may not be appropriate.…”

“…Contamination of the wavelet coefficients at the boundaries of the region, however, still presents a problem. To overcome this, in [6], Slepian wavelets are constructed within a region of the sphere. Here, in this work, Slepian wavelets are generalised to the manifold setting following an analogous construction.…”

“…[23]. This work generalises Slepian scale-discretised wavelets on the sphere [6] to Riemannian manifolds. These wavelets constitute a basis designed for incomplete data on manifolds or graphs.…”

“…Slepian scale-discretised wavelets on the sphere were introduced before in [6]. This work generalises these wavelets to Riemannian manifolds; however, wavelets in the Slepian domain have been considered before.…”

“…The Laplace-Beltrami operator and the graph/mesh Laplacian are also introduced. A brief review of Slepian scale-discretised wavelets on the sphere [6] follows, which this work builds upon. In Section III the Slepian concentration problem is adapted to the manifold setting, which results in the basis functions of a region of the manifold, i.e.…”