MI (1385) - MODELLI E METODI MATEMATICI PER L'INGEGNERIA / MATHEMATICAL MODELS AND METHODS IN ENGINEERING

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053494 - DISCRETE AND GEOMETRIC TOMOGRAPHY

Programma dettagliato e risultati di apprendimento attesi

PROGRAM OF THE COURSE

Overview of the Course. A brief history of CAT. 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.

The origin of Geometric Tomography and of Discrete Tomography. Continuous and discrete parallel X-rays. Continuous and discrete point X-rays. Remarks and examples.

Projections of lattice sets with discrete parallel X-rays. Description of the main models for discrete tomography. Algebraic approach. The reconstruction problem in the grid model as a linear system of equations. Some remarks on Singular Value Decomposition and on stability of solutions.

Binary Tomography. Bad configurations, weakly bad configurations, switching components, ghosts. Ryser algorithm and a few extensions. Examples of binary reconstruction and characterization of the solutions.Ridge functions and additivity.

Algebraic approach in a finite lattice grid and polynomial characterization of switching components. Uniqueness models in discrete tomography. Uniqueness and additivity. Reconstruction with suitable sets of four directions. Characterization of regions of interest in a finite lattice grid. Remarks on possible applications and examples.

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. The results of Gardner and Gritzmann in the integer lattice. Projections of convex bodies with point X-rays. The theorem of Volcic in the Euclidean plane. P-polygons. Some results and examples in the lattice.

EXPECTED RESULTS

The aim of the course is to provide an introduction to Discrete and Geometric Tomography, and to some relatedresearch problems. Moving from Computerized Axial Tomography, the focus is naturally turned on the discretization process. Students are expected to learn the main theorems, and the usual approaches to the reconstruction problem from a finite number of projection.

Note Sulla Modalità di valutazione

For evaluation, students can choose one of the following options

Reading one of the research papers cited during the course, and reporting on the corresponding results. Answering possible related questions from the teacher.

Writing a Matlab code concerning a discussed reconstruction problem. Running the code on different phantom images and producing the corresponding reconstructions.

Presenting and discussing a possible research project concerning Discrete Tomography. The project should be based on some preliminary result or conjecture.

Answering questions concerning the topics treated during the course. Detailed proofs of the presented theorems are required.

Intervallo di svolgimento dell'attività didattica

Data inizio

Data termine

Calendario testuale dell'attività didattica

Monday Overview of the Course. Computerized Axial Tomography.

Tuesday The reconstruction problem in Discrete Tomography.

Wednesday Uniqueness models in discrete tomography.

Thursday Geometric Tomography.

Friday Point X-rays

Bibliografia

Richard Gardner, Geometric Tomography (second edition), Editore: Cambridge University Press, New York, Anno edizione: 2006
Collection of research papers, Advances in discrete tomography and its applications, Editore: Gabor T. Herman and Attila Kuba Eds., BirkhÂuser Boston, Inc., Boston, MA., Anno edizione: 2007
Collection of papers, The Radon Transform, Inverse Problems, and Tomography, Editore: G. ¿lafsson, E. Todd Quinto (Eds.), American Mathematical Society, Boston, MA, USA., Anno edizione: 2006
Avinash C. Kak and Malcolm Slaney, Principles of Computerized Tomographic Imaginghttp://epubs.siam.org/doi/book/10.1137/1.9780898719277E. Barcucci, A. Del Lungo, M.Nivat, R. Pinzani, X-rays characterizing some classes of discrete sets Note:

Linear Algebra and its Applications 339 (2001) 3-21

S. Brunetti, P. Dulio, C. Peri, Discrete tomography determination of bounded lattice sets from four X-rays Note: