3,502 research outputs found
Complex data processing: fast wavelet analysis on the sphere
In the general context of complex data processing, this paper reviews a
recent practical approach to the continuous wavelet formalism on the sphere.
This formalism notably yields a correspondence principle which relates wavelets
on the plane and on the sphere. Two fast algorithms are also presented for the
analysis of signals on the sphere with steerable wavelets.Comment: 20 pages, 5 figures, JFAA style, paper invited to J. Fourier Anal.
and Appli
On the computation of directional scale-discretized wavelet transforms on the sphere
We review scale-discretized wavelets on the sphere, which are directional and
allow one to probe oriented structure in data defined on the sphere.
Furthermore, scale-discretized wavelets allow in practice the exact synthesis
of a signal from its wavelet coefficients. We present exact and efficient
algorithms to compute the scale-discretized wavelet transform of band-limited
signals on the sphere. These algorithms are implemented in the publicly
available S2DW code. We release a new version of S2DW that is parallelized and
contains additional code optimizations. Note that scale-discretized wavelets
can be viewed as a directional generalization of needlets. Finally, we outline
future improvements to the algorithms presented, which can be achieved by
exploiting a new sampling theorem on the sphere developed recently by some of
the authors.Comment: 13 pages, 3 figures, Proceedings of Wavelets and Sparsity XV, SPIE
Optics and Photonics 2013, Code is publicly available at http://www.s2dw.org
S2LET: A code to perform fast wavelet analysis on the sphere
We describe S2LET, a fast and robust implementation of the scale-discretised
wavelet transform on the sphere. Wavelets are constructed through a tiling of
the harmonic line and can be used to probe spatially localised, scale-depended
features of signals on the sphere. The scale-discretised wavelet transform was
developed previously and reduces to the needlet transform in the axisymmetric
case. The reconstruction of a signal from its wavelets coefficients is made
exact here through the use of a sampling theorem on the sphere. Moreover, a
multiresolution algorithm is presented to capture all information of each
wavelet scale in the minimal number of samples on the sphere. In addition S2LET
supports the HEALPix pixelisation scheme, in which case the transform is not
exact but nevertheless achieves good numerical accuracy. The core routines of
S2LET are written in C and have interfaces in Matlab, IDL and Java. Real
signals can be written to and read from FITS files and plotted as Mollweide
projections. The S2LET code is made publicly available, is extensively
documented, and ships with several examples in the four languages supported. At
present the code is restricted to axisymmetric wavelets but will be extended to
directional, steerable wavelets in a future release.Comment: 8 pages, 6 figures, version accepted for publication in A&A. Code is
publicly available from http://www.s2let.or
Fast directional continuous spherical wavelet transform algorithms
We describe the construction of a spherical wavelet analysis through the
inverse stereographic projection of the Euclidean planar wavelet framework,
introduced originally by Antoine and Vandergheynst and developed further by
Wiaux et al. Fast algorithms for performing the directional continuous wavelet
analysis on the unit sphere are presented. The fast directional algorithm,
based on the fast spherical convolution algorithm developed by Wandelt and
Gorski, provides a saving of O(sqrt(Npix)) over a direct quadrature
implementation for Npix pixels on the sphere, and allows one to perform a
directional spherical wavelet analysis of a 10^6 pixel map on a personal
computer.Comment: 10 pages, 3 figures, replaced to match version accepted by IEEE
Trans. Sig. Pro
Spherical Needlets for CMB Data Analysis
We discuss Spherical Needlets and their properties. Needlets are a form of
spherical wavelets which do not rely on any kind of tangent plane approximation
and enjoy good localization properties in both pixel and harmonic space;
moreover needlets coefficients are asymptotically uncorrelated at any fixed
angular distance, which makes their use in statistical procedures very
promising. In view of these properties, we believe needlets may turn out to be
especially useful in the analysis of Cosmic Microwave Background (CMB) data on
the incomplete sky, as well as of other cosmological observations. As a final
advantage, we stress that the implementation of needlets is computationally
very convenient and may rely completely on standard data analysis packages such
as HEALPix.Comment: 7 pages, 7 figure
An Optimal Dimensionality Multi-shell Sampling Scheme with Accurate and Efficient Transforms for Diffusion MRI
This paper proposes a multi-shell sampling scheme and corresponding
transforms for the accurate reconstruction of the diffusion signal in diffusion
MRI by expansion in the spherical polar Fourier (SPF) basis. The sampling
scheme uses an optimal number of samples, equal to the degrees of freedom of
the band-limited diffusion signal in the SPF domain, and allows for
computationally efficient reconstruction. We use synthetic data sets to
demonstrate that the proposed scheme allows for greater reconstruction accuracy
of the diffusion signal than the multi-shell sampling schemes obtained using
the generalised electrostatic energy minimisation (gEEM) method used in the
Human Connectome Project. We also demonstrate that the proposed sampling scheme
allows for increased angular discrimination and improved rotational invariance
of reconstruction accuracy than the gEEM schemes.Comment: 4 pages, 4 figures presented at ISBI 201
Markov chain Monte Carlo analysis of Bianchi VII_h models
We have extended the analysis of Jaffe et al. to a complete Markov chain
Monte Carlo (MCMC) study of the Bianchi type models including a
dark energy density, using 1-year and 3-year Wilkinson Microwave Anisotropy
Probe (WMAP) cosmic microwave background (CMB) data. Since we perform the
analysis in a Bayesian framework our entire inference is contained in the
multidimensional posterior distribution from which we can extract marginalised
parameter constraints and the comparative Bayesian evidence. Treating the
left-handed Bianchi CMB anisotropy as a template centred upon the `cold-spot'
in the southern hemisphere, the parameter estimates derived for the total
energy density, `tightness' and vorticity from 3-year data are found to be:
, , with orientation ). This template is preferred by a factor of roughly
unity in log-evidence over a concordance cosmology alone. A Bianchi type
template is supported by the data only if its position on the sky is heavily
restricted. The low total energy density of the preferred template, implies a
geometry that is incompatible with cosmologies inferred from recent CMB
observations. Jaffe et al. found that extending the Bianchi model to include a
term in creates a degeneracy in the plane. We explore this region fully by MCMC and find that the
degenerate likelihood contours do not intersect areas of parameter space that 1
or 3 year WMAP data would prefer at any significance above . Thus we
can confirm that a physical Bianchi model is not responsible for
this signature.Comment: 8 pages, 10 figures, significant update to include more accurate
results and conclusions to match version accepted by MNRA
Strength in diversity: enhancing learning in vocationally-orientated, master's level courses
Postgraduate education in geography, especially at the Master’s level, is undergoing significant changes in the developed world. There is an expansion of vocationally-oriented degree programmes, increasing recruitment of international students, integration of work place skills, and the engagement of non-traditional postgraduate students as departments respond to policies for a more ‘inclusive’ higher education. This paper sets the context by outlining some programmatic changes in selected countries (Australia, the UK, and the USA). We briefly reflect on how postgraduate ‘bars’ or ‘levels’ are defined and explore in detail what ‘diversity’ or ‘heterogeneity’ means in these new postgraduate settings. The paper then explores some examples of practice drawn from our own experiences, whilst recognising that relevance will vary in other contexts. Finally we consider how diversity can be harnessed as a strength that has potential to enhance taught elements of contemporary postgraduate education in and beyond the discipline
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