## Keywords

Mathematics, x ray, tomography, medical imaging, broken ray transform, brt

## Abstract

The broken ray transform (BRT) is an integral of a function along 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 rays scattered more than once is negligibly small. Among all paths that the scattered rays travel inside the object we pick the one that is a 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 coefficient plus 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 distribution in a plane from the measurements on two or three collimated detector arrays. We derive an exact local reconstruction formula for three flat collimated detectors or 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 case of the range condition for three flat detectors. We generalize the reconstruction formula to four and more detectors and find an optimal set of parameters that minimize noise in the reconstruction. We introduce a more accurate scattering model which takes into account energy shifts due to the Compton effect, derive an exact reconstruction formula and develop an iterative reconstruction method for the energy-dependent case. To solve the problem we assume that the radiation source is monoenergetic and the dependence of the attenuation coefficient on energy is linear on 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 point in the reconstruction plane.

## Notes

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## Graduation Date

2014

## Semester

Fall

## Advisor

Katsevich, Alexander

## Degree

Doctor of Philosophy (Ph.D.)

## College

College of Sciences

## Department

Mathematics

## Degree Program

Mathematics

## Format

application/pdf

## Identifier

CFE0005514

## URL

http://purl.fcla.edu/fcla/etd/CFE0005514

## Language

English

## Release Date

December 2014

## Length of Campus-only Access

None

## Access Status

Doctoral Dissertation (Open Access)

## Subjects

Dissertations, Academic -- Sciences; Sciences -- Dissertations, Academic

## STARS Citation

Krylov, Roman, "Inversion of the Broken Ray Transform" (2014). *Electronic Theses and Dissertations*. 4610.

https://stars.library.ucf.edu/etd/4610