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.

48 pages, 17 figures. Submitted to SIAM Review, August 24th, 2004

References in corpus (1)

• Minimum bias multiple taper spectral estimation

Cited by in corpus (44)

• A novel sampling theorem on the sphere
• Spherical Slepian functions and the polar gap in geodesy
• Spectral estimation on a sphere in geophysics and cosmology
• Slepian functions and their use in signal estimation and spectral analysis
• Fast directional continuous spherical wavelet transform algorithms
• Spatiospectral concentration in the Cartesian plane
• CMB power spectrum estimation using wavelets
• Distributed soil moisture from crosshole ground-penetrating radar travel times using stochastic inversion
• An Optimal-Dimensionality Sampling Scheme on the Sphere with Fast Spherical Harmonic Transforms
• Super-resolution on the Sphere using Convex Optimization
• Lithospheric Structure of Venusian Crustal Plateaus
• Efficient analysis and representation of geophysical processes using localized spherical basis functions
• Wavelets and wavelet-like transforms on the sphere and their application to geophysical data inversion
• Sparse image reconstruction on the sphere: implications of a new sampling theorem
• When Slepian Meets Fiedler: Putting a Focus on the Graph Spectrum
• Internal and external potential-field estimation from regional vector data at varying satellite altitude
• Time and Band Limiting for Matrix Valued Functions, an Example
• Efficient Computation of Slepian Functions for Arbitrary Regions on the Sphere
• Spherical Multi-Modal Place Recognition for Heterogeneous Sensor Systems
• Determining the depth of Jupiter's Great Red Spot with Juno: a Slepian approach
• Harmonic Singular Integrals and Steerable Wavelets in $L_2({\mathbb{R}}^d)$
• A spatiospectral localization approach to estimating potential fields on the surface of a sphere from noisy, incomplete data taken at satellite altitudes
• Reflective prolate-spheroidal operators and the KP/KdV equations
• Bispectrality and Time-Band-Limiting: Matrix valued polynomials
• Fast directional spatially localized spherical harmonic transform
• Slepian Spatial-Spectral Concentration Problem on the Sphere: Analytical Formulation for Limited Colatitude-Longitude Spatial Region
• A Combination of Downward Continuation and Local Approximation for Harmonic Potentials
• Data compression on the sphere
• Sifting Convolution on the Sphere
• Time and band limiting for matrix valued functions: an integral and a commuting differential operator
• Augmented Slepians: Bandlimited Functions that Counterbalance Energy in Selected Intervals
• Slepian Spatial-Spectral Concentration on the Ball
• A General Approach to Regularizing Inverse Problems with Regional Data using Slepian Wavelets
• Slepian Scale-Discretised Wavelets on the Sphere
• Some Noncommutative Matrix Algebras Arising in the Bispectral Problem
• Commuting integral and differential operators and the master symmetries of the Korteweg-de Vries equation
• Spatial-Slepian Transform on the Sphere
• Joint SO(3)-Spectral Domain Filtering of Spherical Signals in the Presence of Anisotropic Noise
• Vectorial Slepian Functions on the Ball
• High-dimensional experiments for the downward continuation using the LRFMP algorithm
• The CMV bispectral problem
• A phase-sensitive method for filtering on the sphere
• Tight bounds on the mutual coherence of sensing matrices for Wigner D-functions on regular grids
• SLEPLET: Slepian Scale-Discretised Wavelets in Python