Radon and Abel Transform equivalence in atmospheric radio occultation

Thomson, F.; Tyler, G. L.

doi:10.1029/2006rs003583

Cited by 7 publications

(6 citation statements)

References 9 publications

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“…), then its Fourier coefficients γn satisfy the following bound, which follows immediately from (37):…”

Section: Stability Estimates In the Presence Of Noise And Regularizationmentioning

confidence: 98%

“…Abel inversion is widely used in plasma physics to obtain the electronic density from phase-shift maps obtained by laser interferometry [35] or radial emission patterns from observed plasma radiances [21,32]. Photoion and photoelectron imaging in molecular dynamics [17], evaluation of mass density and velocity laws of stellar winds in astrophysics [13,30], and atmospheric radio occultation signal analysis [28,37] are additional fields which frequently require the numerical solution of Abel's equations of type (1).…”

Section: Introductionmentioning

confidence: 99%

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A fast algorithm for the inversion of Abel's transform

De Micheli

2022

Preprint

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We present a new algorithm for the computation of the inverse Abel transform, a problem which emerges in many areas of physics and engineering. We prove that the Legendre coefficients of a given function coincide with the Fourier coefficients of a suitable periodic function associated with its Abel transform. This allows us to compute the Legendre coefficients of the inverse Abel transform in an easy, fast and accurate way by means of a single Fast Fourier Transform. The algorithm is thus appropriate also for the inversion of Abel integrals given in terms of samples representing noisy measurements. Rigorous stability estimates are proved and the accuracy of the algorithm is illustrated also by some numerical experiments.

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“…), then its Fourier coefficients γn satisfy the following bound, which follows immediately from (37):…”

Section: Stability Estimates In the Presence Of Noise And Regularizationmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

A fast algorithm for the inversion of Abel's transform

De Micheli

2022

Preprint

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“…For example: Thomson (2007) proposed in his research that radio occultation measurements made with a receiver inside the earth's atmosphere can be inverted with an Abel transform to provide an estimate of atmospheric refractive index profile. Healy et al (2002), also proposed in their study that the reformulation of the radon transform as path integral for the case of a radio ray refracting in a spherical symmetric atmosphere can be change to Abel transformation as both are equivalent in atmospheric radio occultation and I agree with their proposed theory.…”

Section: Discussionmentioning

confidence: 99%

Proof of Analytic Extension Theorem for Zeta Function Using Abel Transformation and Euler Product

Zacharie¹

2010

Journal of Mathematics and Statistics

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Problem statement:In the prime number the Riemann zeta function is unquestionable and undisputable one of the most important questions in mathematics whose many researchers are still trying to find answer to some unsolved problems such as Riemann Hypothesis. In this study we proposed a new method that proves the analytic extension theorem for zeta function. Approach: Abel transformation was used to prove that the extension theorem is true for the real part of the complex variable that is strictly greater than one and consequently provides the required analytic extension of the zeta function to the real part greater than zero and Euler product was used to prove the real part of the complex that are less than zero and greater or equal to one. Results: From this proposed study we noted that the real values of the complex variable are lying between zero and one which may help to understand the relation between zeta function and its properties and consequently can pay the way to solve some complex arithmetic problems including the Riemann Hypothesis. Conclusion: The combination of Abel transformation and Euler product is a powerful tool for proving theorems and functions related to Zeta function including other subjects such as radio atmospheric occultation.

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“…reconstruct the 3D attenuation map slice by slice, which ignores the longitudinal tilt of x-ray paths. For each slice reconstruction, one needs to solve the Abel transform [1,10]. The reconstruction map by FDK method often suffers from strong artefacts for long objects.…”

Section: Introductionmentioning

confidence: 99%

Data-driven Method for 3D Axis-symmetric Object Reconstruction from Single Cone-beam Projection Data

Luo

Meng

Wei

et al. 2019

Proceedings of the Third International Symposium on Image Computing and Digital Medicine

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In this paper we consider 3D axis-symmetric (AS) object reconstruction from single cone-beam x-ray projection data. Traditional x-ray CT fails to capture fleeting state of material due to the long time for data acquisition at all angles. Therefore, AS object is devised to investigate the instant deformation of material under pulse change of environment because single projection data is enough to reconstruct its inner structure. Previous reconstruction methods are layer by layer, and ignore the longitudinal tilt of x-ray paths. We propose a regularization method using adaptive tight frame to reconstruct the 3D AS object structure simultaneously. Alternating direction method is adopted to solve the proposed model. More importantly, a numerical algorithm is developed to compute imaging matrix. Experiments on simulation data verify the effectiveness of our method.

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Radon and Abel Transform equivalence in atmospheric radio occultation

Cited by 7 publications

References 9 publications

A fast algorithm for the inversion of Abel's transform

A fast algorithm for the inversion of Abel's transform

Proof of Analytic Extension Theorem for Zeta Function Using Abel Transformation and Euler Product

Data-driven Method for 3D Axis-symmetric Object Reconstruction from Single Cone-beam Projection Data

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