An Optimal-Dimensionality Sampling Scheme on the Sphere With Fast Spherical Harmonic Transforms

@article{Khalid2014AnOS,
  title={An Optimal-Dimensionality Sampling Scheme on the Sphere With Fast Spherical Harmonic Transforms},
  author={Zubair Khalid and Rodney A. Kennedy and Jason D. McEwen},
  journal={IEEE Transactions on Signal Processing},
  year={2014},
  volume={62},
  pages={4597-4610}
}
We develop a sampling scheme on the sphere that permits accurate computation of the spherical harmonic transform and its inverse for signals band-limited at L using only L2 samples. We obtain the optimal number of samples given by the degrees of freedom of the signal in harmonic space. The number of samples required in our scheme is a factor of two or four fewer than existing techniques, which require either 2L2 or 4L2 samples. We note, however, that we do not recover a sampling theorem on the… 

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  • Mathematics
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  • 2015
TLDR
It is demonstrated, through numerical experiments, that the proposed spherical harmonic transform is sufficiently accurate for band-limits of interest in diffusion magnetic resonance imaging and for a regular grid with equiangular sampling.

Optimal-dimensionality sampling on the sphere: Improvements and variations

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This work has developed a method to place iso-latitude rings of samples with the objective of improving the well-conditioning of the linear systems involved in the computation of the spherical harmonic transform (SHT).

An Antipodally Symmetric Optimal Dimensionality Sampling on the Sphere

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  • Mathematics
    ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
  • 2019
TLDR
The antipodal symmetry of the sampling points is exploited to separate the signal into antipodally symmetric and asymmetric signals due to which the signal splits in harmonic space into the signals of even and odd spherical harmonic degrees.

An Optimal Dimensionality Sampling Scheme on the Sphere with Accurate and Efficient Spherical Harmonic Transform for Diffusion MRI

TLDR
This work designs a sampling scheme on the sphere and a corresponding spherical harmonic transform (SHT) for the measurement and reconstruction of the diffusion signal in diffusion magnetic resonance imaging (dMRI) and demonstrates that the proposed scheme enables more accurate computation of the SHT, and this accuracy is practically rotationally invariant.

Improving the spatial dimensionality of Gauss-Legendre and equiangular sampling schemes on the sphere

TLDR
An efficient GL sampling scheme with spatial dimensionality equal to that of equiangular scheme is proposed and it is demonstrated that the accuracy of the SHT is not affected with the proposed reduction in the spatialdimensionality.

Comparative analysis of geometrical properties of sampling schemes on the sphere

  • U. ElahiZ. KhalidR. Kennedy
  • Computer Science, Mathematics
    2016 10th International Conference on Signal Processing and Communication Systems (ICSPCS)
  • 2016
TLDR
This work illustrates that the optimal dimensionality, extremal system and spherical design sampling schemes exhibit desirable geometrical properties, and proposes to use optimal dimensional sampling scheme for moderate to large band-limits as it exhibits desirable geometric properties.

Iterative residual fitting for spherical harmonic transform of band-limited signals on the sphere: Generalization and analysis

TLDR
The generalized iterative residual fitting (IRF) for the computation of the spherical harmonic transform (SHT) of band-limited signals on the sphere is presented and the so-called multi-pass IRF which adds multiple iterative passes to the IRF is presented.

An optimal dimensionality sampling scheme on the sphere for antipodal signals in diffusion magnetic resonance imaging

TLDR
This work proposes a sampling scheme on the sphere and develops a corresponding spherical harmonic transform (SHT) for the accurate reconstruction of the diffusion signal in diffusion magnetic resonance imaging (dMRI) to near machine precision accuracy.

A Novel Sampling Theorem on the Rotation Group

TLDR
A novel sampling theorem for functions defined on the three-dimensional rotation group SO(3) is developed by connecting the rotation group to theThree-torus through a periodic extension, and fast algorithms to compute the associated Fourier transform are presented.

Efficient Sampling on HEALPix Grid

TLDR
The proposed sampling scheme is designed as a variant of the widely used Hierarchical Equal Area iso-Latitude Pixelization (HEALPix) scheme on the sphere and the spherical harmonic transform is formulated.

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