Discrete tomography on modules [Elektronische Ressource] : decomposition, separation, and uniqueness / Barbara Langfeld

Technische Universit¨at Mu¨nchenZentrum MathematikDiscrete Tomography on Modules:Decomposition, Separation,and UniquenessBarbara LangfeldVollst¨andiger Abdruck der von der Fakult¨at fu¨r Mathematik der TechnischenUniversit¨at Mu¨nchen zur Erlangung des akademischen Grades einesDoktors der Naturwissenschaften (Dr.rer.nat.)genehmigten Dissertation.Vorsitzender: Univ.-Prof.Dr. Martin BrokatePru¨fer der Dissertation: 1. Univ.-Prof.Dr. Peter Gritzmann2. Univ.-Prof.Dr. Martin HenkOtto-von-Guericke-Universit¨at Magdeburg3. Prof. Gabor T. Herman, Ph.D.City University of New York, USADie Dissertation wurde am 28.08.2007 bei der Technischen Universit¨at Mu¨ncheneingereicht und durch die Fakult¨at fu¨r Mathematik am 10.12.2007 angenommen.Fu¨r Aiso,Milo,Ulrichund meine Eltern.iiAbstractWe study three basic questions of discrete tomography on modules: First, we charac-terize under which conditions the complete tomographicgriddecomposes intofinitelymany translates of the underlying module. Second, we deal with a geometric separa-tion problem that arises naturally in reconstructing quasicrystalline point sets fromX-ray data. We show how to solve the separation problem algorithmically in a semi-algebraic setting. Finally, we study the problem of finding the minimal number ofpoints in a tomographic grid that have to be prescribed as (non-)positions so as toguarantee a unique reconstruction of a given sample from the X-ray data.