5,343 research outputs found
An Optimal-Dimensionality Sampling for Spin- Functions on the Sphere
For the representation of spin- band-limited functions on the sphere, we
propose a sampling scheme with optimal number of samples equal to the number of
degrees of freedom of the function in harmonic space. In comparison to the
existing sampling designs, which require samples for the
representation of spin- functions band-limited at , the proposed scheme
requires samples for the accurate computation of the spin-
spherical harmonic transform~(-SHT). For the proposed sampling scheme, we
also develop a method to compute the -SHT. We place the samples in our
design scheme such that the matrices involved in the computation of -SHT are
well-conditioned. We also present a multi-pass -SHT to improve the accuracy
of the transform. We also show the proposed sampling design exhibits superior
geometrical properties compared to existing equiangular and Gauss-Legendre
sampling schemes, and enables accurate computation of the -SHT corroborated
through numerical experiments.Comment: 5 pages, 2 figure
Optimal-Dimensionality Sampling on the Sphere: Improvements and Variations
For the accurate representation and reconstruction of band-limited signals on
the sphere, an optimal-dimensionality sampling scheme has been recently
proposed which requires the optimal number of samples equal to the number of
degrees of freedom of the signal in the spectral (harmonic) domain. The
computation of the spherical harmonic transform (SHT) associated with the
optimal-dimensionality sampling requires the inversion of a series of linear
systems in an iterative manner. The stability of the inversion depends on the
placement of iso-latitude rings of samples along co-latitude. In this work, we
have developed a method to place these iso-latitude rings of samples with the
objective of improving the well-conditioning of the linear systems involved in
the computation of the SHT. We also propose a multi-pass SHT algorithm to
iteratively improve the accuracy of the SHT of band-limited signals.
Furthermore, we review the changes in the computational complexity and
improvement in accuracy of the SHT with the embedding of the proposed methods.
Through numerical experiments, we illustrate that the proposed variations and
improvements in the SHT algorithm corresponding to the optimal-dimensionality
sampling scheme significantly enhance the accuracy of the SHT.Comment: 5 Pages, 4 figure
A novel sampling theorem on the rotation group
We develop a novel sampling theorem for functions defined on the
three-dimensional rotation group SO(3) by connecting the rotation group to the
three-torus through a periodic extension. Our sampling theorem requires
samples to capture all of the information content of a signal band-limited at
, reducing the number of required samples by a factor of two compared to
other equiangular sampling theorems. We present fast algorithms to compute the
associated Fourier transform on the rotation group, the so-called Wigner
transform, which scale as , compared to the naive scaling of .
For the common case of a low directional band-limit , complexity is reduced
to . Our fast algorithms will be of direct use in speeding up the
computation of directional wavelet transforms on the sphere. We make our SO3
code implementing these algorithms publicly available.Comment: 5 pages, 2 figures, minor changes to match version accepted for
publication. Code available at http://www.sothree.or
Sparse image reconstruction on the sphere: analysis and synthesis
We develop techniques to solve ill-posed inverse problems on the sphere by
sparse regularisation, exploiting sparsity in both axisymmetric and directional
scale-discretised wavelet space. Denoising, inpainting, and deconvolution
problems, and combinations thereof, are considered as examples. Inverse
problems are solved in both the analysis and synthesis settings, with a number
of different sampling schemes. The most effective approach is that with the
most restricted solution-space, which depends on the interplay between the
adopted sampling scheme, the selection of the analysis/synthesis problem, and
any weighting of the l1 norm appearing in the regularisation problem. More
efficient sampling schemes on the sphere improve reconstruction fidelity by
restricting the solution-space and also by improving sparsity in wavelet space.
We apply the technique to denoise Planck 353 GHz observations, improving the
ability to extract the structure of Galactic dust emission, which is important
for studying Galactic magnetism.Comment: 11 pages, 6 Figure
Sparse image reconstruction on the sphere: implications of a new sampling theorem
We study the impact of sampling theorems on the fidelity of sparse image
reconstruction on the sphere. We discuss how a reduction in the number of
samples required to represent all information content of a band-limited signal
acts to improve the fidelity of sparse image reconstruction, through both the
dimensionality and sparsity of signals. To demonstrate this result we consider
a simple inpainting problem on the sphere and consider images sparse in the
magnitude of their gradient. We develop a framework for total variation (TV)
inpainting on the sphere, including fast methods to render the inpainting
problem computationally feasible at high-resolution. Recently a new sampling
theorem on the sphere was developed, reducing the required number of samples by
a factor of two for equiangular sampling schemes. Through numerical simulations
we verify the enhanced fidelity of sparse image reconstruction due to the more
efficient sampling of the sphere provided by the new sampling theorem.Comment: 11 pages, 5 figure
A sparse representation of gravitational waves from precessing compact binaries
Many relevant applications in gravitational wave physics share a significant
common problem: the seven-dimensional parameter space of gravitational
waveforms from precessing compact binary inspirals and coalescences is large
enough to prohibit covering the space of waveforms with sufficient density. We
find that by using the reduced basis method together with a parametrization of
waveforms based on their phase and precession, we can construct ultra-compact
yet high-accuracy representations of this large space. As a demonstration, we
show that less than judiciously chosen precessing inspiral waveforms are
needed for cycles, mass ratios from to and spin magnitudes . In fact, using only the first reduced basis waveforms yields a
maximum mismatch of over the whole range of considered parameters. We
test whether the parameters selected from the inspiral regime result in an
accurate reduced basis when including merger and ringdown; we find that this is
indeed the case in the context of a non-precessing effective-one-body model.
This evidence suggests that as few as numerical simulations of
binary black hole coalescences may accurately represent the seven-dimensional
parameter space of precession waveforms for the considered ranges.Comment: 5 pages, 3 figures. The parameters selected for the basis of
precessing waveforms can be found in the source file
Optimal-Dimensionality Sampling on the Sphere: Improvements and Variations
For the accurate representation and reconstruction of band-limited signals on the sphere, an optimal-dimensionality sampling scheme has been recently proposed which requires the optimal number of samples equal to the number of degrees of freedom of the signal in the spectral (harmonic) domain. The computation of the spherical harmonic transform (SHT) associated with the optimal-dimensionality sampling requires the inversion of a series of linear systems in an iterative manner. The stability of the inversion depends on the placement of iso-latitude rings of samples along co-latitude. In this work, we have developed a method to place these iso-latitude rings of samples with the objective of improving the well-conditioning of the linear systems involved in the computation of the SHT. We also propose a multi-pass SHT algorithm to iteratively improve the accuracy of the SHT of band-limited signals. Furthermore, we review the changes in the computational complexity and improvement in accuracy of the SHT with the embedding of the proposed methods. Through numerical experiments, we illustrate that the proposed variations and improvements in the SHT algorithm corresponding to the optimal-dimensionality sampling scheme significantly enhance the accuracy of the SHT
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