73 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
Implications for compressed sensing of a new sampling theorem on the sphere
A sampling theorem on the sphere has been developed recently, requiring half
as many samples as alternative equiangular sampling theorems on the sphere. A
reduction by a factor of two in the number of samples required to represent a
band-limited signal on the sphere exactly has important implications for
compressed sensing, both in terms of the dimensionality and sparsity of
signals. We illustrate the impact of this property with an inpainting problem
on the sphere, where we show the superior reconstruction performance when
adopting the new sampling theorem compared to the alternative.Comment: 1 page, 2 figures, Signal Processing with Adaptive Sparse Structured
Representations (SPARS) 201
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
Iterative Residual Fitting for Spherical Harmonic Transform of Band-Limited Signals on the Sphere: Generalization and Analysis
We present the generalized iterative residual fitting (IRF) for the
computation of the spherical harmonic transform (SHT) of band-limited signals
on the sphere. The proposed method is based on the partitioning of the subspace
of band-limited signals into orthogonal subspaces. There exist sampling schemes
on the sphere which support accurate computation of SHT. However, there are
applications where samples~(or measurements) are not taken over the predefined
grid due to nature of the signal and/or acquisition set-up. To support such
applications, the proposed IRF method enables accurate computation of SHTs of
signals with randomly distributed sufficient number of samples. In order to
improve the accuracy of the computation of the SHT, we also present the
so-called multi-pass IRF which adds multiple iterative passes to the IRF. We
analyse the multi-pass IRF for different sampling schemes and for different
size partitions. Furthermore, we conduct numerical experiments to illustrate
that the multi-pass IRF allows sufficiently accurate computation of SHTs.Comment: 5 Pages, 7 Figure
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
An Optimal Dimensionality Sampling Scheme on the Sphere for Antipodal Signals In Diffusion Magnetic Resonance Imaging
We propose a sampling scheme on the sphere and develop a corresponding
spherical harmonic transform (SHT) for the accurate reconstruction of the
diffusion signal in diffusion magnetic resonance imaging (dMRI). By exploiting
the antipodal symmetry, we design a sampling scheme that requires the optimal
number of samples on the sphere, equal to the degrees of freedom required to
represent the antipodally symmetric band-limited diffusion signal in the
spectral (spherical harmonic) domain. Compared with existing sampling schemes
on the sphere that allow for the accurate reconstruction of the diffusion
signal, the proposed sampling scheme reduces the number of samples required by
a factor of two or more. We analyse the numerical accuracy of the proposed SHT
and show through experiments that the proposed sampling allows for the accurate
and rotationally invariant computation of the SHT to near machine precision
accuracy.Comment: Will be published in the proceedings of the International Conference
Acoustics, Speech and Signal Processing 2015 (ICASSP'2015
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