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Dettaglio Insegnamento Contesto Anno Accademico 2021/2022 Corso di Studi Dott. - MI (1385) Modelli e Metodi Matematici per l'Ingegneria / Mathematical Models and Methods in Engineering Anno di Corso 1 Scheda Insegnamento Codice Identificativo 053494 Denominazione Insegnamento DISCRETE AND GEOMETRIC TOMOGRAPHY Tipo Insegnamento MONODISCIPLINARE Crediti Formativi Universitari (CFU) 5.0 Programma sintetico Computerized Tomography and discretization The Radon transform and its inversion for X-ray image reconstruction. Applications and related problems. Discretization of the reconstruction process. The Matlab "radon" and "iradon" functions. Examples and applications. Reconstruction from a limited number of projections The problem of ghosts. Models for discrete tomography. The reconstruction problem in the grid model as a linear system of equations. Investigation of the space of ghosts. Algebraic approach in a finite lattice grid and polynomial characterization of switching components. Discrete tomography Uniqueness models. Uniqueness and additivity. A few notion of projective geometry. Cross-ratio and its invariance under projections and sections. The results of Gardner and Gritzmann in the integer lattice. A few remarks on polyominoes. Binary Tomography Bad configurations, weakly bad configurations, switching components, ghosts. Ryser algorithm and a few extensions. Examples of binary reconstruction and characterization of the set of solutions. A uniqueness theorem for binary tomography. Examples of reconstructions of differently shaped discrete binary objects in a finite grid, by exploiting suitable sets of four directions. Geometric Tomography Hammer's problem and related uniqueness problems. Mid-point construction. U-polygons and their properties. The theorem of Gardner-McMullen in the Euclidean plane. Projections of convex bodies with point X-rays. The theorem of Volcic in the Euclidean plane. P-polygons. Some results and examples Settori Scientifico Disciplinari (SSD) Codice SSD Descrizione SSD CFU MAT/03 GEOMETRIA 5.0 Dettaglio Scaglione Docente Programma dettagliato Da (compreso) A (escluso) A ZZZZ Dulio Paolo

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12/08/2022