Inversion formulas for cone transforms arising in application of Compton cameras

Jung, Chang-Yeol; Moon, Sunghwan

doi:10.1088/0266-5611/31/1/015006

Cited by 42 publications

(53 citation statements)

References 23 publications

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“…where in the last line, we used the identity (6). Changing the variables z → r − v and doing some manipulations, we have…”

Section: Inversion Formulasmentioning

confidence: 99%

“…During the last decade, the interest towards such transforms was triggered by the connection between the conical Radon transform and the mathematical models of many novel imaging modalities. [1][2][3][4][5][6][7][8][9][10][11][12][13][14] In all the above works, the attenuation phenomena was neglected. However, in many medical imaging techniques, ignoring the effect of the attenuation of photon can significantly degrade the quality of the reconstruction image.…”

Section: Introductionmentioning

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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“…where in the last line, we used the identity (6). Changing the variables z → r − v and doing some manipulations, we have…”

Section: Inversion Formulasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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“…Also, we assume that the density of the photons decreases at the 1:1 rate of distance from the source to detectors in the case of C 1 f . When the density decreases as a different power of distance, we need a different power of r (see, e.g., [16]). For simplicity, set s = cos ψ ∈ [−1, 1], and define the cone transform C k of a function f ∈ C(R 3 ) with compact support by…”

Section: The 3-dimensional Cone Transformmentioning

confidence: 99%

“…Several inversion formulas for various types of cone transforms were derived in [2,5,6,11,12,13,16,19,21,24,25,28,31,34]. The cone transform with planar vertex positions and a fixed central axis was studied in [5,6,16,21,24,25,34].…”

Section: Introductionmentioning

confidence: 99%

“…The cone transform with planar vertex positions and a fixed central axis was studied in [5,6,16,21,24,25,34]. Haltme ier derived an exact backprojection-type inversion formula for the n-dimensional cone transform which integrates a given n-dimensional function over all conical surfaces having vertices on a hyperplane and a central axis orthogonal to this hyperplane in [13].…”

Section: Introductionmentioning

confidence: 99%

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Exact Inversion of the Cone Transform Arising in an Application of a Compton Camera Consisting of Line Detectors

Jung¹,

Moon²

2016

SIAM J. Imaging Sci.

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Abstract.A Compton camera has been suggested for use in single photon emission computed tomography because a conventional gamma camera has low efficiency. Here we consider a cone transform brought about by a Compton camera with line detectors. A cone transform takes a given function on the 3-dimensional space and assigns to it the surface integral of the function over cones determined by the 1-dimensional vertex space, the 1-dimensional central axis, and the 1-dimensional opening angle. We generalize this cone transform to n-dimensional space and provide an inversion formula. Also, numerical simulations are presented to demonstrate our suggested algorithm in three dimensions.Key words. Compton camera, SPECT, tomography, cone transform, inversion, gamma camera AMS subject classifications. 44A12, 65R10, 92C55DOI. 10.1137/15M10336171. Introduction. A Radon-type transform that assigns to a given function its surface integral over various sets of cones has been studied, since it is known that this kind of a transform relates to a Compton camera. A Compton camera, also called an electronically collimated γ-camera, was introduced for use in single photon emission computed tomography (SPECT) because of the low efficiency of a conventional γ-camera [26,32]. A Compton camera has very high sensitivity and flexibility of geometrical design, so it has attracted a lot of interest and applications in many areas including monitoring nuclear power plants and astronomy [1,3].A standard Compton camera consists of two planar detectors: a scatter detector and an absorption detector, positioned one behind the other. A photon emitted from a radioactive source toward the camera undergoes Compton scattering in the scatter detector, and is absorbed in the absorption detector positioned behind (see Figure 1). In each detector, the position of the hit and the energy of the photon are measured. The scattering angle or the

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Inversion formula for the conical Radon transform arising in a single first semicircle second Compton camera with rotation

Moon

Kwon

2019

Japan J. Indust. Appl. Math.

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Inversion formulas for cone transforms arising in application of Compton cameras

Cited by 42 publications

References 23 publications

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

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

Exact Inversion of the Cone Transform Arising in an Application of a Compton Camera Consisting of Line Detectors

Inversion formula for the conical Radon transform arising in a single first semicircle second Compton camera with rotation

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