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Inversion of the Broken Ray Transform

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Date Issued:
2014
Abstract/Description:
The broken ray transform (BRT) is an integral of a functionalong a union of two rays with a common vertex.Consider an X-ray beam scanning an object of interest.The ray undergoes attenuation and scatters in all directions inside the object.This phenomena may happen repeatedly until the photons either exit the object or are completely absorbed.In our work we assume the single scattering approximation when the intensity of the raysscattered more than once is negligibly small.Among all paths that the scattered rays travel inside the object we pick the one that isa union of two segments with one common scattering point.The intensity of the ray which traveled this path and exited the object can be measured by a collimated detector.The collimated detector is able to measure the intensity of X-rays from the selected direction.The logarithm of such a measurement is the broken ray transform of the attenuation coefficientplus the logarithm of the scattering coefficient at the scattering point (vertex)and a known function of the scattering angle.In this work we consider the reconstruction of X-ray attenuation coefficient distributionin a plane from the measurements on two or three collimated detector arrays.We derive an exact local reconstruction formula for three flat collimated detectorsor three curved or pin-hole collimated detectors.We obtain a range condition for the case of three curved or pin-hole detectors and provide a special caseof the range condition for three flat detectors.We generalize the reconstruction formula to four and more detectors and find anoptimal set of parameters that minimize noise in the reconstruction.We introduce a more accurate scattering model which takes into accountenergy shifts due to the Compton effect, derive an exact reconstruction formula and develop an iterativereconstruction method for the energy-dependent case.To solve the problem we assume that the radiation source is monoenergeticand the dependence of the attenuation coefficient on energy is linearon an energy interval from the minimal to the maximal scattered energy. %initial radiation energy.We find the parameters of the linear dependence of the attenuation on energy as a function of a pointin the reconstruction plane.
Title: Inversion of the Broken Ray Transform.
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Name(s): Krylov, Roman, Author
Katsevich, Alexander, Committee Chair
Tamasan, Alexandru, Committee Member
Nashed, M, Committee Member
Zeldovich, Boris, Committee Member
University of Central Florida, Degree Grantor
Type of Resource: text
Date Issued: 2014
Publisher: University of Central Florida
Language(s): English
Abstract/Description: The broken ray transform (BRT) is an integral of a functionalong a union of two rays with a common vertex.Consider an X-ray beam scanning an object of interest.The ray undergoes attenuation and scatters in all directions inside the object.This phenomena may happen repeatedly until the photons either exit the object or are completely absorbed.In our work we assume the single scattering approximation when the intensity of the raysscattered more than once is negligibly small.Among all paths that the scattered rays travel inside the object we pick the one that isa union of two segments with one common scattering point.The intensity of the ray which traveled this path and exited the object can be measured by a collimated detector.The collimated detector is able to measure the intensity of X-rays from the selected direction.The logarithm of such a measurement is the broken ray transform of the attenuation coefficientplus the logarithm of the scattering coefficient at the scattering point (vertex)and a known function of the scattering angle.In this work we consider the reconstruction of X-ray attenuation coefficient distributionin a plane from the measurements on two or three collimated detector arrays.We derive an exact local reconstruction formula for three flat collimated detectorsor three curved or pin-hole collimated detectors.We obtain a range condition for the case of three curved or pin-hole detectors and provide a special caseof the range condition for three flat detectors.We generalize the reconstruction formula to four and more detectors and find anoptimal set of parameters that minimize noise in the reconstruction.We introduce a more accurate scattering model which takes into accountenergy shifts due to the Compton effect, derive an exact reconstruction formula and develop an iterativereconstruction method for the energy-dependent case.To solve the problem we assume that the radiation source is monoenergeticand the dependence of the attenuation coefficient on energy is linearon an energy interval from the minimal to the maximal scattered energy. %initial radiation energy.We find the parameters of the linear dependence of the attenuation on energy as a function of a pointin the reconstruction plane.
Identifier: CFE0005514 (IID), ucf:50324 (fedora)
Note(s): 2014-12-01
Ph.D.
Sciences, Mathematics
Doctoral
This record was generated from author submitted information.
Subject(s): mathematics -- X-ray -- tomography -- medical imaging -- broken ray transform -- brt
Persistent Link to This Record: http://purl.flvc.org/ucf/fd/CFE0005514
Restrictions on Access: public 2014-12-15
Host Institution: UCF

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