The concept of tomography was first introduced around 1979 by the principal investigator, K.T. Smith. It was introduced formally in the mathematical literature in 1985 by F. Keinert and K.T. Smith and independently (with a different implementation) by E.I. Vainberg and M.L. Faingois. Local tomography allows the reconstruction of a function closely related to the attenuation coefficient of an object, within a region, from the attenuation along lines meeting that region, while standard tomography requires the attenuation along all lines meeting the full object. This has significant implications in regard to x-ray dose, resolution, the required dynamic range of the x-ray sensors, demands on the x-ray tube, scatter, compression distortions, scanner size, the size of the object to which tomography can be applied, the number of numerical computations, and the volume of data that must be collected, stored, and processed. While local tomography has been performed very successfully in a number of interesting cases, many problems remain in the area of a proper mathematical understanding of how it should be implemented. The objective of this project is to acquire that understanding, to write sound and effective numerical algorithms for local tomography, and to make them available to practitioners and researchers.
