15 research outputs found
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
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
Scale-discretised ridgelet transform on the sphere
We revisit the spherical Radon transform, also called the Funk-Radon
transform, viewing it as an axisymmetric convolution on the sphere. Viewing the
spherical Radon transform in this manner leads to a straightforward derivation
of its spherical harmonic representation, from which we show the spherical
Radon transform can be inverted exactly for signals exhibiting antipodal
symmetry. We then construct a spherical ridgelet transform by composing the
spherical Radon and scale-discretised wavelet transforms on the sphere. The
resulting spherical ridgelet transform also admits exact inversion for
antipodal signals. The restriction to antipodal signals is expected since the
spherical Radon and ridgelet transforms themselves result in signals that
exhibit antipodal symmetry. Our ridgelet transform is defined natively on the
sphere, probes signal content globally along great circles, does not exhibit
blocking artefacts, supports spin signals and exhibits an exact and explicit
inverse transform. No alternative ridgelet construction on the sphere satisfies
all of these properties. Our implementation of the spherical Radon and ridgelet
transforms is made publicly available. Finally, we illustrate the effectiveness
of spherical ridgelets for diffusion magnetic resonance imaging of white matter
fibers in the brain.Comment: 5 pages, 4 figures, matches version accepted by EUSIPCO, code
available at http://www.s2let.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
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
Localisation of directional scale-discretised wavelets on the sphere
Scale-discretised wavelets yield a directional wavelet framework on the
sphere where a signal can be probed not only in scale and position but also in
orientation. Furthermore, a signal can be synthesised from its wavelet
coefficients exactly, in theory and practice (to machine precision).
Scale-discretised wavelets are closely related to spherical needlets (both were
developed independently at about the same time) but relax the axisymmetric
property of needlets so that directional signal content can be probed. Needlets
have been shown to satisfy important quasi-exponential localisation and
asymptotic uncorrelation properties. We show that these properties also hold
for directional scale-discretised wavelets on the sphere and derive similar
localisation and uncorrelation bounds in both the scalar and spin settings.
Scale-discretised wavelets can thus be considered as directional needlets.Comment: 28 pages, 8 figures, minor changes to match version accepted for
publication by ACH
Spherical continuous wavelet transforms arising from sections of the Lorentz group
We consider the conformal group of the unit sphere Sn−1, the so-called proper Lorentz group Spin+(1,n), for the study of spherical continuous wavelet transforms. Our approach is based on the method for construction of general coherent states associated to square integrable group representations over homogeneous spaces. The underlying homogeneous space is an extension to the whole of the group Spin+(1,n) of the factorization of the
gyrogroup of the unit ball by an appropriate gyro-subgroup. Sections on it are constituted by rotations of the subgroup Spin(n) and Möbius transformations of the type ϕa(x) = (x −a)(1 + ax)−1, where a belongs to a given section on a quotient space of the unit ball. This extends in a natural way the work of Antoine and Vandergheynst to anisotropic conformal dilations on the unit sphere.info:eu-repo/semantics/publishedVersio