### Abstract:

The Radon transform (first considered by J. Radon in 1917) is an integral transform achieved by integrating a function over a set of lines on the domain of the function, and is used in x-ray tomography to produce pictures of unseen objects. In the Radon transform, one may think of the domain as the body of a person and the function as a density function on the body. In this case, the Radon transform produces an average density along various lines through the body. Physically, these averages may be obtained by measuring the decrease in intensity of a beam of radiation passed through the body along various lines, so one may compute the image of the Radon transform without knowing the original density function explicitly. In x-ray tomography, one produces a picture of the body by reconstructing the original density function. The question "When it possible to reconstruct a picture of the body?" which is of great importance in x-ray tomography, translates into the mathematical question "When is it possible to invert the Radon transform?" In a variety of settings, this latter mathematical question remains unsolved. In order to shed light on these inversion questions, we will investigate the finite Radon transform. In this case, the body consists of finitely many points and is crossed by finitely many lines (each containing the same finite number of points). In this project, we will explore two instances of the finite Radon transform: the complete graph on n vertices (the vertices are the `body' and the edges are the 'lines') and finite dimensional vector spaces over finite fields (points in the vector space form the 'body' and one-dimensional subspaces, along with their translations, form the 'lines').