Efficient Computation of Slepian Functions for Arbitrary Regions on the Sphere

In this paper, we develop a new method for the fast and memory-efficient computation of Slepian functions on the sphere. Slepian functions, which arise as the solution of the Slepian concentration problem on the sphere, have desirable properties for applications where measurements are only available within a spatially limited region on the sphere and/or a function is required to be analyzed over the spatially limited region. Slepian functions are currently not easily computed for large band-limits for an arbitrary spatial region due to high computational and large memory storage requirements. For the special case of a polar cap, the symmetry of the region enables the decomposition of the Slepian concentration problem into smaller subproblems and consequently the efficient computation of Slepian functions for large band-limits. By exploiting the efficient computation of Slepian functions for the polar cap region on the sphere, we develop a formulation, supported by a fast algorithm, for the approximate computation of Slepian functions for an arbitrary spatial region to enable the analysis of modern datasets that support large band-limits. For the proposed algorithm, we carry out accuracy analysis of the approximation, computational complexity analysis, and review of memory storage requirements. We illustrate, through numerical experiments, that the proposed method enables faster computation, and has smaller storage requirements, while allowing for sufficiently accurate computation of the Slepian functions.Alice P. Bates is supported by the Australian Research Council’s Discovery Projects funding scheme (Project no. DP150101011). Rodney A. Kennedy is supported by the Australian Research Council’s Discovery Projects funding scheme (Project no. DP170101897)

Fast directional spatially localized spherical harmonic transform

We propose a transform for signals defined on the sphere that reveals their localized directional content in the spatio-spectral domain when used in conjunction with an asymmetric window function. We call this transform the directional spatially localized spherical harmonic transform (directional SLSHT) which extends the SLSHT from the literature whose usefulness is limited to symmetric windows. We present an inversion relation to synthesize the original signal from its directional-SLSHT distribution for an arbitrary window function. As an example of an asymmetric window, the most concentrated band-limited eigenfunction in an elliptical region on the sphere is proposed for directional spatio-spectral analysis and its effectiveness is illustrated on the synthetic and Mars topographic data-sets. Finally, since such typical data-sets on the sphere are of considerable size and the directional SLSHT is intrinsically computationally demanding depending on the band-limits of the signal and window, a fast algorithm for the efficient computation of the transform is developed. The floating point precision numerical accuracy of the fast algorithm is demonstrated and a full numerical complexity analysis is presented.Comment: 12 pages, 5 figure

Scalar and vector Slepian functions, spherical signal estimation and spectral analysis

It is a well-known fact that mathematical functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation and modeling scientific data, and we often only have access to, or are only interested in, a study area that is temporally or spatially bounded. In the geosciences we may be interested in spectrally modeling a time series defined only on a certain interval, or we may want to characterize a specific geographical area observed using an effectively bandlimited measurement device. It is clear that analyzing and representing scientific data of this kind will be facilitated if a basis of functions can be found that are "spatiospectrally" concentrated, i.e. "localized" in both domains at the same time. Here, we give a theoretical overview of one particular approach to this "concentration" problem, as originally proposed for time series by Slepian and coworkers, in the 1960s. We show how this framework leads to practical algorithms and statistically performant methods for the analysis of signals and their power spectra in one and two dimensions, and, particularly for applications in the geosciences, for scalar and vectorial signals defined on the surface of a unit sphere.Comment: Submitted to the 2nd Edition of the Handbook of Geomathematics, edited by Willi Freeden, Zuhair M. Nashed and Thomas Sonar, and to be published by Springer Verlag. This is a slightly modified but expanded version of the paper arxiv:0909.5368 that appeared in the 1st Edition of the Handbook, when it was called: Slepian functions and their use in signal estimation and spectral analysi

Spatiospectral concentration on a sphere

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.Comment: 48 pages, 17 figures. Submitted to SIAM Review, August 24th, 200

Slepian Wavelets for the Analysis of Incomplete Data on Manifolds

Many fields in science and engineering measure data that inherently live on non-Euclidean geometries, such as the sphere. Techniques developed in the Euclidean setting must be extended to other geometries. Due to recent interest in geometric deep learning, analogues of Euclidean techniques must also handle general manifolds or graphs. Often, data are only observed over partial regions of manifolds, and thus standard whole-manifold techniques may not yield accurate predictions. In this thesis, a new wavelet basis is designed for datasets like these. Although many definitions of spherical convolutions exist, none fully emulate the Euclidean definition. A novel spherical convolution is developed, designed to tackle the shortcomings of existing methods. The so-called sifting convolution exploits the sifting property of the Dirac delta and follows by the inner product of a function with the translated version of another. This translation operator is analogous to the Euclidean translation in harmonic space and exhibits some useful properties. In particular, the sifting convolution supports directional kernels; has an output that remains on the sphere; and is efficient to compute. The convolution is entirely generic and thus may be used with any set of basis functions. An application of the sifting convolution with a topographic map of the Earth demonstrates that it supports directional kernels to perform anisotropic filtering. Slepian wavelets are built upon the eigenfunctions of the Slepian concentration problem of the manifold - a set of bandlimited functions which are maximally concentrated within a given region. Wavelets are constructed through a tiling of the Slepian harmonic line by leveraging the existing scale-discretised framework. A straightforward denoising formalism demonstrates a boost in signal-to-noise for both a spherical and general manifold example. Whilst these wavelets were inspired by spherical datasets, like in cosmology, the wavelet construction may be utilised for manifold or graph data

Spectral estimation on a sphere in geophysics and cosmology

We address the problem of estimating the spherical-harmonic power spectrum of a statistically isotropic scalar signal from noise-contaminated data on a region of the unit sphere. Three different methods of spectral estimation are considered: (i) the spherical analogue of the one-dimensional (1-D) periodogram, (ii) the maximum likelihood method, and (iii) a spherical analogue of the 1-D multitaper method. The periodogram exhibits strong spectral leakage, especially for small regions of area $A\ll 4\pi$, and is generally unsuitable for spherical spectral analysis applications, just as it is in 1-D. The maximum likelihood method is particularly useful in the case of nearly-whole-sphere coverage, $A\approx 4\pi$, and has been widely used in cosmology to estimate the spectrum of the cosmic microwave background radiation from spacecraft observations. The spherical multitaper method affords easy control over the fundamental trade-off between spectral resolution and variance, and is easily implemented regardless of the region size, requiring neither non-linear iteration nor large-scale matrix inversion. As a result, the method is ideally suited for most applications in geophysics, geodesy or planetary science, where the objective is to obtain a spatially localized estimate of the spectrum of a signal from noisy data within a pre-selected and typically small region.Comment: Submitted to the Geophysical Journal Internationa