Inversion formulas for the broken-ray Radon transform

Florescu, Lucia; Markel, Vadim A.; Schotland, John C.

doi:10.1088/0266-5611/27/2/025002

Cited by 55 publications

(102 citation statements)

References 15 publications

(30 reference statements)

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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: 91%

“…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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Section: Resultsmentioning

confidence: 91%

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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“…These reconstructions are from full data and most of them assume specific boundaries at least for the reflection part, for example a flat one or a circle. It also should be mentioned that the broken ray transform or the V-line Radon transform sometimes refers to a different transform from the one we consider in this work, see [25,7,1,19]. In their settings, the V-line vertices are inside the object with a fixed axis direction.…”

Section: Introductionmentioning

confidence: 99%

Artifacts in the inversion of the broken ray transform in the plane

Zhang¹

2020

Inverse Problems &Amp; Imaging

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We study the integral transform over a general family of broken rays in R 2 . It is natural for broken rays to have conjugate points, for example, when they are reflected from a curved boundary. If there are conjugate points, we show that the singularities cannot be recovered from local data and therefore artifacts arise in the reconstruction. We apply these conclusions to two examples, the V-line Radon transform and the parallel ray transform. In each example, a detailed discussion of the local and global recovery of singularities is given and we perform numerical experiments to illustrate the results.

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“…Image reconstruction from such measurements is called single scattering optical tomography (SSOT). The measured data set in SSOT corresponds to a generalized Radon transform integrating the light attenuation coefficient along broken rays that coincide with the trajectories of scattered photons [9,10,11]. Hence, one of the crucial mathematical tasks in SSOT is the inversion of the BRT (or the VLT) in various geometric setups.…”

Section: Introductionmentioning

confidence: 99%

“…C and D), one obtains an integral of the light attenuation coefficient along a V-line (in our case CBD) in which integration along each ray is done with a different algebraic sign (see Figure 1). Such "signed" VLT's have been studied before in [10,25].…”

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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Inversion formulas for the broken-ray Radon transform

Cited by 55 publications

References 15 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

Artifacts in the inversion of the broken ray transform in the plane

The V-line transform with some generalizations and cone differentiation

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