Analytical reconstruction formula for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" display="inline" overflow="scroll"><mml:mi>n</mml:mi></mml:math>-dimensional conical Radon transform

Gouia‐Zarrad, Rim

doi:10.1016/j.camwa.2014.04.019

Cited by 22 publications

(21 citation statements)

References 17 publications

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“…We guess that some artifacts evident in the figures primarily arises from the jump discontinuity of the function. They are typical in numerical simulation of the V-line transform (eg, see previous studies 3,18,23 ). In fact, they appear when the V-line is tangentially touching the support of the function.…”

Section: Resultsmentioning

confidence: 93%

“…The following Theorem 2 is the general version of one derived in Gouia-Zarrad. 18 This means that can be zero unlike Theorem 1.…”

Section: Inversion Formulasmentioning

confidence: 98%

“…The reconstruction problem for C 0 f was solved in R n+1 in Gouia-Zarrad. 18 This paper is dedicated to the generalization of the conical Radon transform by adding a weight function e − l (u,v) to each cone where ≥ 0 is a constant known inside the domain.…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…To prove the theorem, we follow a similar method to the one used in Gouia-Zarrad. 18 Proof. Taking the n-dimensional Fourier transform of C f with respect to u yields…”

Section: Inversion Formulasmentioning

confidence: 99%

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Inversion of the attenuated conical Radon transform with a fixed opening angle

Gouia‐Zarrad¹,

Moon

2017

Math Methods in App Sciences

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Since Compton cameras were introduced in the use of single photon emission computed tomography, various types of conical Radon transforms, which integrate the emission distribution over circular cones, have been studied. Most of previous works did not address the attenuation factor, which may lead to significant degradation of image quality. In this paper, we consider the problem of recovering an unknown function from conical projections affected by a known constant attenuation coefficient called an attenuated conical Radon transform.In the case of a fixed opening angle and vertical central axis, new explicit inversion formula is derived. Two-dimensional numerical simulations were performed to demonstrate the efficiency of the suggested algorithm.

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Section: Resultsmentioning

confidence: 93%

“…The following Theorem 2 is the general version of one derived in Gouia-Zarrad. 18 This means that can be zero unlike Theorem 1.…”

Section: Inversion Formulasmentioning

confidence: 98%

Section: Formulation Of the Problemmentioning

confidence: 99%

“…To prove the theorem, we follow a similar method to the one used in Gouia-Zarrad. 18 Proof. Taking the n-dimensional Fourier transform of C f with respect to u yields…”

Section: Inversion Formulasmentioning

confidence: 99%

See 2 more Smart Citations

Inversion of the attenuated conical Radon transform with a fixed opening angle

Gouia‐Zarrad¹,

Moon

2017

Math Methods in App Sciences

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“…The VLT in other 2D geometries was studied in [2,3,4,25,26]. Inversion formulae for a generalization of the VLT to higher dimensions, namely the Conical Radon Transform mapping a function to a family of circular cones with vertices inside the image domain, were presented in [13,14,38].…”

Section: Introductionmentioning

confidence: 99%

The V-line transform with some generalizations and cone differentiation

Ambartsoumian

Jebelli

2019

Inverse Problems

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The paper studies various properties of the V-line transform (VLT) in the plane and the conical Radon transform (CRT) in R n . The VLT maps a function to a family of its integrals along trajectories made of two rays emanating from a common point. The CRT considered in this paper maps a function to a set of its integrals over surfaces of polyhedral cones. These types of operators appear in mathematical models of single scattering optical tomography, Compton camera imaging and other applications. We derive new explicit inversion formulae for the VLT and the CRT, as well as proving some previously known results using more intuitive geometric ideas. Using our inversion formula for the VLT, we describe the range of that transformation when applied to a fairly broad class of functions and prove some support theorems. The efficiency of our method is demonstrated on several numerical examples. As an auxiliary result that plays a big role in this article, we derive a generalization of the Fundamental Theorem of Calculus, which we call the Cone Differentiation Theorem.

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