A Mathematical View on Spin-Weighted Spherical Harmonics and Their Applications in Geodesy

Michel, Volker; Seibert, Katrin

doi:10.1007/978-3-662-55854-6_102

Cited by 4 publications

(7 citation statements)

References 29 publications

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“…Similar to the classical spherical harmonics, the spin-weighted spherical harmonics also satisfy recursion relations, a Christoffel-Darboux formula, and an addition theorem. These have been proven in [25,34] for the first time. In the following, we report the results and refer to [25] and [34] for the proofs.…”

Section: Spin-weighted Spherical Harmonicsmentioning

confidence: 74%

“…These have been proven in [25,34] for the first time. In the following, we report the results and refer to [25] and [34] for the proofs.…”

Section: Spin-weighted Spherical Harmonicsmentioning

confidence: 74%

“…In addition, commuting operators are known for the scalar and the vector Slepian case for spherical-cap regions ( [16,20]). The advantage of the spin-weighted spherical harmonics by Newman and Penrose [30] (which we introduce in this section) over the classical spherical harmonics is that the former allow us to (i) derive a unified approach to construct Slepian functions for arbitary tensor ranks and (ii) construct commuting operators for spherical-cap regions for arbitary tensor ranks (see also [25,34]). Definition 2.14.…”

Section: Spin-weighted Spherical Harmonicsmentioning

confidence: 99%

“…See [25,34] for a proof of Theorem 2.30. It can also be shown that the spin-weighted spherical harmonic functions form a complete orthonormal system for L 2 (Ω), see e.g.…”

Section: Spin-weighted Spherical Harmonicsmentioning

confidence: 99%

“…For this purpose, we utilize the spin-weighted spherical harmonics of Newman and Penrose [30] (see also [25,34]). As a consequence the eigenvalue problems for vectorial or higher-ranked Slepian function constructions, which are coupled when using the classical vector or tensor spherical harmonics, decouple.…”

Section: Introductionmentioning

confidence: 99%

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A Unified Approach to Scalar, Vector, and Tensor Slepian Functions on the Sphere and Their Construction by a Commuting Operator

Michel¹,

Plattner²,

Seibert³

2021

Preprint

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We present a unified approach for constructing Slepian functions -also known as prolate spheroidal wave functions -on the sphere for arbitrary tensor ranks including scalar, vectorial, and rank 2 tensorial Slepian functions, using spin-weighted spherical harmonics. For the special case of spherical cap regions, we derived commuting operators, allowing for a numerically stable and computationally efficient construction of the spin-weighted spherical-harmonic-based Slepian functions. Linear relationships between the spin-weighted and the classical scalar, vectorial, tensorial, and higher-rank spherical harmonics allow the construction of classical spherical-harmonic-based Slepian functions from their spin-weighted counterparts, effectively rendering the construction of spherical-cap Slepian functions for any tensorial rank a computationally fast and numerically stable task.

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Section: Spin-weighted Spherical Harmonicsmentioning

confidence: 74%

“…These have been proven in [25,34] for the first time. In the following, we report the results and refer to [25] and [34] for the proofs.…”

Section: Spin-weighted Spherical Harmonicsmentioning

confidence: 74%

Section: Spin-weighted Spherical Harmonicsmentioning

confidence: 99%

“…See [25,34] for a proof of Theorem 2.30. It can also be shown that the spin-weighted spherical harmonic functions form a complete orthonormal system for L 2 (Ω), see e.g.…”

Section: Spin-weighted Spherical Harmonicsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Unified Approach to Scalar, Vector, and Tensor Slepian Functions on the Sphere and Their Construction by a Commuting Operator

Michel¹,

Plattner²,

Seibert³

2021

Preprint

Self Cite

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Spherical harmonics to quantify cranial asymmetry in deformational plagiocephaly

Grieb

Barbero-García

Lerma

2022

Sci Rep

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Cranial deformation and deformational plagiocephaly (DP) in particular affect an important percentage of infants. The assessment and diagnosis of the deformation are commonly carried by manual measurements that provide low interuser accuracy. Another approach is the use of three-dimensional (3D) models. Nevertheless, in most cases, deformation measurements are carried out manually on the 3D model. It is necessary to develop methodologies for the detection of DP that are automatic, accurate and take profit on the high quantity of information of the 3D models. Spherical harmonics are proposed as a new methodology to identify DP from head 3D models. The ideal fitted ellipsoid for each head is computed and the orthogonal distances between head and ellipsoid are obtained. Finally, the distances are modelled using spherical harmonics. Spherical harmonic coefficients of degree 2 and order − 2 are identified as the correct ones to represent the asymmetry characteristic of DP. The obtained coefficient is compared to other anthropometric deformation indexes, such as Asymmetry Index, Oblique Cranial Length Ratio, Posterior Asymmetry Index and Anterior Asymmetry Index. The coefficient of degree 2 and order − 2 with a maximum degree of 4 is found to provide better results than the commonly computed anthropometric indexes in the detection of DP.

show abstract

A unified approach to scalar, vector, and tensor Slepian functions on the sphere and their construction by a commuting operator

2022

View full text Add to dashboard Cite

We present a unified approach for constructing Slepian functions — also known as prolate spheroidal wave functions — on the sphere for arbitrary tensor ranks including scalar, vectorial, and rank 2 tensorial Slepian functions, using spin-weighted spherical harmonics. For the special case of spherical cap regions, we derived commuting operators, allowing for a numerically stable and computationally efficient construction of the spin-weighted spherical-harmonic-based Slepian functions. Linear relationships between the spin-weighted and the classical scalar, vectorial, tensorial, and higher-rank spherical harmonics allow the construction of classical spherical-harmonic-based Slepian functions from their spin-weighted counterparts, effectively rendering the construction of spherical-cap Slepian functions for any tensorial rank a computationally fast and numerically stable task.

show abstract

A Mathematical View on Spin-Weighted Spherical Harmonics and Their Applications in Geodesy

Cited by 4 publications

References 29 publications

A Unified Approach to Scalar, Vector, and Tensor Slepian Functions on the Sphere and Their Construction by a Commuting Operator

A Unified Approach to Scalar, Vector, and Tensor Slepian Functions on the Sphere and Their Construction by a Commuting Operator

Spherical harmonics to quantify cranial asymmetry in deformational plagiocephaly

A unified approach to scalar, vector, and tensor Slepian functions on the sphere and their construction by a commuting operator

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