Tesi di LAUREA Titolo Problemi di unicità e costruzione ambigue per raggi X sorgenti in tomografia geometrica Data 2008-07-22 Autore/i Carminati Jennifer Relatore Dulio, P. Full text non disponibile Abstract The main issues of this work are about Geometrical Tomography, a particular aspect of mathematics whose aim is to gain information about a geometrical object, having only few data of their own sections, own projections or both of them. In general, data gets the name of the X rays of the specific object. We define projection the shadow of an object in a subspace. Thanks to polar duality we have a link between projections and sections for a fixed point, usually called point X rays. When this point moves to infinity we talk about parallel X rays, that have a further quantity of information with respect to a simple projection. Generally, going up with the dimension, we don t have uniqueness results, except for some trivial cases concerning to the plane situation ([9], Chapter 2 and Chapter 5). However, in E2, there are open problems concerning points X rays. Working in Z2, those problems can be solved in a non-uniqueness way, that gives rise to ambiguous con¯gurations. In a more explicit way, we have couples of reticular convexes that have the same X rays, outgoing from a given set P of points. In the present work we have focused on those two problems. On one hand, we have obtained a n-dimensional extension of the uniqueness theorem, concerning convex body in the interior of triangle, that has its vertices in points X rays, generalizing it in ipertetrahedral configurations of finite subset P of Z2. On the other hand, we focused on work of configurations, that produce reticular ambiguous reconstructions, determined by three points, lined up or not. From a mathematical point of view, presence of those configurations, named P-polygons, involves questions which concern the theory of the numbers, the theory of the measure and the projective geometry, and many of the questions linked to them are still outstanding. Having disposal of programs which reconstruct reticular P-polygons could be of great help to develop conjectures relating these outstanding questions. In this manner we have translated in an algorithm way the theoretical results in [6] producing then the relative Matlab programs which provide the reconstruction.