Sur la transformation de Radon de la sphère $S\sp d$

Abouelaz, Ahmed; Daher, Radouan

doi:10.24033/bsmf.2212

Cited by 25 publications

(33 citation statements)

References 5 publications

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“…Namely, 1) up to a constant multiple depending on u. For sufficiently nice wavelet function u, formula (1.1) can be extended analytically to all a √ ‫.ރ‬ The case a Å 0 in (1.1) corresponds to the well-known Calderón reproducing formula [10].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…We also give the characterization of the range R(L p ‫ޓ(‬ n )). In connection with the subject of the article one should mention the papers [1,34,35] in which the notion of the spherical Radon transform has another meaning. Note also that the methodology developed below is applicable to explicit inversion (in the wavelet form) of the totally geodesic Radon transform on the Lobachevsky space [5].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Spherical Radon Transform and Related Wavelet Transforms

Rubin

1998

Applied and Computational Harmonic Analysis

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Spherical Radon Transform and Related Wavelet Transforms

Rubin

1998

Applied and Computational Harmonic Analysis

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“…The case a = 0 for F a gives the classical Funk transform. In the case, when a = 1, n = 2, and f is smooth, these operators were introduced by Abouelaz and Daher [1] and studied by Helgason [7, p. 145]. The case a = 1 for all n ≥ 2 was studied in [27,Section 7.2], [28,Section 5] in the general context of Lebesgue integrable functions.…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of functions on the sphere from their integrals over hyperplane sections

Rubin

2019

Anal.Math.Phys.

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We obtain new inversion formulas for the Funk type transforms of two kinds associated to spherical sections by hyperplanes passing through a common point A which lies inside the n-dimensional unit sphere or on the sphere itself. Transforms of the first kind are defined by integration over complete subspheres and can be reduced to the classical Funk transform. Transforms of the second kind perform integration over truncated subspheres, like spherical caps or bowls, and can be reduced to the hyperplane Radon transform. The main tools are analytic families of λ-cosine transforms, Semyanisyi's integrals, and modified stereorgraphic projection with the pole at A. Assumptions for functions are close to minimal.2010 Mathematics Subject Classification. Primary 44A12; Secondary 44A15. Key words and phrases. Radon transform and Funk transform and cosine transform and Semyanistyi integrals and inversion formulas.

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“…Thus, this formal calculations show that formula (7) corresponds to formula (2) and serves as its regularization…”

Section: Spherical Convolution Operatorsmentioning

confidence: 63%

Funk–Minkowski transform and spherical convolution of Hilbert type in reconstructing functions on the sphere

Kazantsev¹

2018

Sib. Elektron. Mat. Izv.

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The Funk-Minkowski transform F associates a function f on the sphere S 2 with its mean values (integrals) along all great circles of the sphere. Thepresented analytical inversion formula reconstruct the unknown function f completely if two Funk-Minkowski transforms, Ff and F∇f , are known. Another result of this article is related to the problem of Helmholtz-Hodge decomposition for tangent vector field on the sphere S 2 . We proposed solution for this problem which is used the Funk-Minkowski transform F and Hilbert type spherical convolution S.Keywords: Funk-Minkowski transform, Funk-Radon transform, spherical convolution of Hilbert type, Fourier multiplier operators, inverse operator, scalar and vector spherical harmonics, surface gradient, tangential spherical vector fields, Helmholtz-Hodge decomposition.Kazantsev, S. G., Funk-Minkowski transform and spherical convolution of Hilbert type in reconstructing functions on the sphere.

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Sur la transformation de Radon de la sphère $S\sp d$

Cited by 25 publications

References 5 publications

Spherical Radon Transform and Related Wavelet Transforms

Spherical Radon Transform and Related Wavelet Transforms

Reconstruction of functions on the sphere from their integrals over hyperplane sections

Funk–Minkowski transform and spherical convolution of Hilbert type in reconstructing functions on the sphere

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