Efficient Spherical Designs with Good Geometric Properties

Womersley, Robert S.

doi:10.1007/978-3-319-72456-0_57

Cited by 67 publications

(80 citation statements)

References 65 publications

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“…Another sampling pattern on the sphere is the so-called t-design [52]. Unfortunately, the spherical t-design does not exist for an arbitrary pair m, t as given in [53], which also restricts the flexibility to choose an arbitrary number of samples. To the best of our knowledge there is nothing related to spherical designs on the rotation group.…”

Section: Sampling Pattern Design Using Coherence Minimizationmentioning

confidence: 99%

Sensing Matrix Design and Sparse Recovery on the Sphere and the Rotation Group

Bangun

Behboodi

Mathar

2020

IEEE Trans. Signal Process.

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In this paper, the goal is to design deterministic sampling patterns on the sphere and the rotation group and, thereby, construct sensing matrices for sparse recovery of band-limited functions. It is first shown that random sensing matrices, which consists of random samples of Wigner D-functions, satisfy the Restricted Isometry Property (RIP) with proper preconditioning and can be used for sparse recovery on the rotation group. The mutual coherence, however, is used to assess the performance of deterministic and regular sensing matrices. We show that many of widely used regular sampling patterns yield sensing matrices with the worst possible mutual coherence, and therefore are undesirable for sparse recovery. Using tools from angular momentum analysis in quantum mechanics, we provide a new expression for the mutual coherence, which encourages the use of regular elevation samples. We construct low coherence deterministic matrices by fixing the regular samples on the elevation and minimizing the mutual coherence over the azimuth-polarization choice. It is shown that once the elevation sampling is fixed, the mutual coherence has a lower bound that depends only on the elevation samples. This lower bound, however, can be achieved for spherical harmonics, which leads to new sensing matrices with better coherence than other representative regular sampling patterns. This is reflected as well in our numerical experiments where our proposed sampling patterns perfectly match the phase transition of random sampling patterns.that are obtained from sampling functions in finite-dimensional function spaces. The sensing matrix entries in these applications are samples of orthonormal basis functions of the ambient space. Fourier matrices [5], matrices from trigonometric polynomials [6], orthogonal polynomials [7,8] and spherical harmonics [9,10] are some examples of these matrices. Fortunately, when the orthonormal functions are uniformly bounded, also called Bounded Orthonormal Systems (BOSs), a similar recovery guarantee can be obtained. If the samples are taken randomly from a certain probability measure, BOS matrices are proven to satisfy the RIP property [4, Chapter 12]. If the orthonormal functions are uniformly bounded by K, the required number of measurements scales with K 2 .This randomness in the measurement process, however, is inadmissible in many applications, for instance when the measurement process involves movements of mechanical devices. Random measurements require arbitrary movements that are possibly harmful to the measurement device. In these applications, the measurement process should be designed by considering the physical characteristics of the measurement device. An example, which is the main motivation of the current work, is the antenna measurement application. The samples in antenna measurements are taken using a robotic arm or which samples of a smooth trajectory are preferred over random samples. Therefore, regular sampling patterns like equiangular patterns are widely used for the measurement proces...

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Section: Sampling Pattern Design Using Coherence Minimizationmentioning

confidence: 99%

Sensing Matrix Design and Sparse Recovery on the Sphere and the Rotation Group

Bangun

Behboodi

Mathar

2020

IEEE Trans. Signal Process.

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“…For the sampling points in the radial direction, we used the zeros of Chebyshev polynomials of the first kind on [−1, 1] mapped to [r in , r out ]. For the quadrature points and weights in the angular direction, we use the spherical t-designs (rules with equal weights) taken fromWomersley (2016). As functions to approximate, we take these choices: f 1 (r, θ, ϕ) = e r −1 0.001 cos 4 θ ,Fig.…”

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confidence: 99%

A fully discretised polynomial approximation on spherical shells

Kazashi

2016

Int J Geomath

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A fully discretised polynomial approximation on spherical shells is considered. The radial direction and the angular direction of the spherical shells are treated differently: for the radial direction, polynomial interpolation is employed, whereas for the angular direction, hyperinterpolation is employed. The error in terms of a weighted L 2 -norm is bounded by the sum of two terms, one for the radial, one for the angular direction. Numerical results support our result.

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“…The main difficulty of numerically solving the multi-species Boltzmann equation (33) lies in the collision operator (34). In this section, we introduce a fast Fourier spectral method (in the velocity space) to approximate this operator.…”

Section: A Fast Fourier Spectral Methods For the Multi-species Boltzmamentioning

confidence: 99%

“…where for the radial direction, we use the Gauss-Legendre quadrature with N ρ = O(N) points (since the integral oscillates roughly on O(N)); for the integral over the sphere, we use the M-point spherical design quadrature [32,33] (usually M N 2 ).…”

Section: A Fast Fourier Spectral Methods For the Multi-species Boltzmamentioning

confidence: 99%