A new technique is presented to study radio propagation and rough surface scattering problems based on a reformulation of the Magnetic Field Integration Equation (MFIE) called the Single-Scatter Subtraction (S^3) method. This technique amounts to a physical preconditioning by separating the single- and multiple-scatter currents and removing the single-scattering contribution from the integral term that is present in the MFIE. This requires the calculation of a new quantity that is the kernel of the MFIE integral call the kernel integral or Gbar. In this work, 1-dimensional deterministically rough surfaces are simulated by surfaces consisting of single and multiple cosines. In order to truncate the problem domain, a beam illumination is used as the source term and it is shown that this also causes the kernel integral to have a finite support. Using the Single Scatter Subtraction method on these surfaces, closed-form expressions are found for the kernel integral and thus the single-scatter current for a well defined region of validity of surface parameters which may then be efficiently radiated into the far field numerically. Both the closed-form expressions, and the computed radiated fields are studied for their physical significance. This provides a clear physical intuition for the technique as an augmentation to existing ones as a bent-plane approximation as shown analytically and also validated by numeric results. Further analysis resolves a controversy on the nature of Bragg scatter which is found to be a multiple-scatter phenomenon. Error terms present in the kernel integral also raise new questions on the effect of truncation for any MFIE-based solution. Additionally, a dramatic enhancement of backscatter predicted by this new approach versus the Kirchhoff method is observed as the angle of incidence increases due to the error terms. / Doctor of Philosophy / A new technique is presented to study the interaction of electromagnetic waves with rough surfaces. Building on the technique called the Magnetic Field Integral Equation (MFIE) which allows the solution for the electromagnetic fields scattered from the surface by considering only the induced electric and magnetic currents on the surface, the Single-Scatter Substraction (S 3 ) method separates the surface currents into those that interact with the surface only once or single-scatter, and those that interact multiple times called multiple-scatter. Since this is the introduction of this technique, only the former is investigated. In this study, a new quantity which is an integral of one of the components of the standard MFIE is studied and closed-form approximations are presented along with bounds of validity. This provides closed form solutions for the single-scattering currents, from which the radiated fields may be efficiently found numerically. Since they are closed form, the expressions provide insight into the nature of the physical scattering process. Numerical results of these expressions are compared to the standard approximate technique as well as the ”exact” solution found by numerically solving the MFIE. Compared to the standard approximate technique which approximates the surface by a tangent plane at each point on the surface, the single-scatter currents approximate the surface with a bent-plane at each point. This shifts the scattered fields from certain directions to others, and highlights where single- and multiple-scattering have an effect.

Positron emission tomography (PET) is a nuclear medicine imaging techniquethat uses radiotracers to visualize processes like metabolism and perfusion. Theradiotracer emits positrons, which collide with shell electrons of the atomsthat make up the surrounding tissue. Such a collision produces two gammarayphotons, emitted roughly 180 degrees apart [1]. PET captures thesephotons using a cylindrical arrangement of detectors. When two photons aredetected simultaneously by different detectors, it registers as a line of response(LOR). These LORs are then pre-processed into a sinogram. A mathematicalreconstruction method is used to computationally recover the 3D distribution ofthe radiotracer (activity map) from the sinogram. However, genuine LORs can becorrupted by false LORs that come from scattering, random events, and spuriousevents. Mitigating these in reconstruction algorithms is essential for improvingPET imaging accuracy and reliability.This paper explores the theoretical foundation of the Time of Flight (TOF) SingleScatter Simulation (SSS) model by Watson (2007) [2]. It also includes a Pythonimplementation of the MATLAB code associated with [2]. The model modelsCompton scattering to accurately estimate scattered photons in PET.Incorporating TOF data into the SSS model improves estimation accuracy, albeitat the cost of increased computational time. To expedite computations, thealgorithm was simplified by restricting operations to a subset of rings anddetectors and by pre-processing images through cropping and downscaling.Interpolation fills in missing data, ensuring complete estimation.The outcome of this project is a Python implementation that exhibited a strongcorrelation with the estimates obtained using the MATLAB implementation. Anotable issue arose during the comparison between the main components ofthe SSS algorithm in Python and MATLAB. The Euclidean norm between theresults from these two implementations was significant, indicating that they wereon different scales. Nevertheless, both implementations accurately predictedthe scatter in the same locations and relative magnitudes, despite the scalediscrepancy. Investigation into the discrepancy’s cause is ongoing, but theproject demonstrates the feasibility of implementing the TOF SSS algorithm inPython.