Graduate Thesis Or Dissertation

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  • Single-scattering tomography describes a model of photon transfer through a object in which photons are assumed to scatter at most once. The Broken Ray transform arises from this model, and was first investigated by Lucia Florescu, Vadim A. Markel, and John C. Schotland, [2], in 2010, followed by an inversion, [1], in the case of fixed initial and terminal directions. Later, in 2013, Katsevich and Krylov, [6], investigated settings where terminal rays were permitted to vary, either heading towards or away from a focal point, providing inversion formulas in two- and three-detector settings. In this thesis, we will explore these transforms, give them distributional meaning, and analyze how these transforms propagate singularities. This requires analysis of the relationships between a function and its antiderivative obtained from integration over a ray. This thesis also introduces the Polar Broken Ray transform, in which the source position is fixed, initial direction varies, and the scattering angle is held constant. We will discover that the Polar Broken Ray transform is injective on spaces of functions supported in an annulus that is bounded away from the origin, and also derive distributional meaning to the Polar Broken Ray transform, and derive a relationship between a wavefront set of a function and that of its Polar Broken Ray transform. Provided in the appendix are implementations of numerical inversions of the Broken Ray transform with fixed initial and terminal directions, as well as the Polar Broken Ray transform.
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