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Inversion of the Broken Ray Transform
 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 Xray 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 Xrays 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 Xray 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 pinhole collimated detectors.We obtain a range condition for the case of three curved or pinhole 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 energydependent 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 Xray 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 Xrays 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 Xray 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 pinhole collimated detectors.We obtain a range condition for the case of three curved or pinhole 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 energydependent 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): 
20141201 Ph.D. Sciences, Mathematics Doctoral This record was generated from author submitted information. 

Subject(s):  mathematics  Xray  tomography  medical imaging  broken ray transform  brt  
Persistent Link to This Record:  http://purl.flvc.org/ucf/fd/CFE0005514  
Restrictions on Access:  public 20141215  
Host Institution:  UCF 