97 research outputs found
Spatiospectral concentration of vector fields on a sphere
We construct spherical vector bases that are bandlimited and spatially
concentrated, or, alternatively, spacelimited and spectrally concentrated,
suitable for the analysis and representation of real-valued vector fields on
the surface of the unit sphere, as arises in the natural and biomedical
sciences, and engineering. Building on the original approach of Slepian,
Landau, and Pollak we concentrate the energy of our function bases into
arbitrarily shaped regions of interest on the sphere, and within certain
bandlimits in the vector spherical-harmonic domain. As with the concentration
problem for scalar functions on the sphere, which has been treated in detail
elsewhere, a Slepian vector basis can be constructed by solving a
finite-dimensional algebraic eigenvalue problem. The eigenvalue problem
decouples into separate problems for the radial and tangential components. For
regions with advanced symmetry such as polar caps, the spectral concentration
kernel matrix is very easily calculated and block-diagonal, lending itself to
efficient diagonalization. The number of spatiospectrally well-concentrated
vector fields is well estimated by a Shannon number that only depends on the
area of the target region and the maximal spherical-harmonic degree or
bandwidth. The spherical Slepian vector basis is doubly orthogonal, both over
the entire sphere and over the geographic target region. Like its scalar
counterparts it should be a powerful tool in the inversion, approximation and
extension of bandlimited fields on the sphere: vector fields such as gravity
and magnetism in the earth and planetary sciences, or electromagnetic fields in
optics, antenna theory and medical imaging.Comment: Submitted to Applied and Computational Harmonic 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
Efficient analysis and representation of geophysical processes using localized spherical basis functions
While many geological and geophysical processes such as the melting of
icecaps, the magnetic expression of bodies emplaced in the Earth's crust, or
the surface displacement remaining after large earthquakes are spatially
localized, many of these naturally admit spectral representations, or they may
need to be extracted from data collected globally, e.g. by satellites that
circumnavigate the Earth. Wavelets are often used to study such nonstationary
processes. On the sphere, however, many of the known constructions are somewhat
limited. And in particular, the notion of `dilation' is hard to reconcile with
the concept of a geological region with fixed boundaries being responsible for
generating the signals to be analyzed. Here, we build on our previous work on
localized spherical analysis using an approach that is firmly rooted in
spherical harmonics. We construct, by quadratic optimization, a set of
bandlimited functions that have the majority of their energy concentrated in an
arbitrary subdomain of the unit sphere. The `spherical Slepian basis' that
results provides a convenient way for the analysis and representation of
geophysical signals, as we show by example. We highlight the connections to
sparsity by showing that many geophysical processes are sparse in the Slepian
basis.Comment: To appear in the Proceedings of the SPIE, as part of the Wavelets
XIII conference in San Diego, August 200
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