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Scheda Riassuntiva
Anno Accademico 2018/2019
Tipo incarico Dottorato
Insegnamento 053494 - DISCRETE AND GEOMETRIC TOMOGRAPHY
Docente Dulio Paolo
Cfu 5.00 Tipo insegnamento Monodisciplinare

Corso di Dottorato Da (compreso) A (escluso) Insegnamento
MI (1385) - MODELLI E METODI MATEMATICI PER L'INGEGNERIA / MATHEMATICAL MODELS AND METHODS IN ENGINEERINGAZZZZ053494 - 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 related research 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
Risorsa bibliografica facoltativaRichard Gardner, Geometric Tomography (second edition), Editore: Cambridge University Press, New York, Anno edizione: 2006
Risorsa bibliografica facoltativaCollection 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
Risorsa bibliografica facoltativaCollection 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
Risorsa bibliografica facoltativaAvinash C. Kak and Malcolm Slaney, Principles of Computerized Tomographic Imaging http://epubs.siam.org/doi/book/10.1137/1.9780898719277
Risorsa bibliografica obbligatoriaE. 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

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

Discrete Applied Mathematics 161, (2013), 2281-2292

Risorsa bibliografica obbligatoriaP. Dulio, R. J. Gardner, and C. Peri, Discrete Point X-Rays
Note:

Siam J. Discrete Math. Vol. 20, No. 1, Pp. 171-188

Risorsa bibliografica obbligatoriaP. Dulio and C. Peri, On the geometric structure of lattice U-polygons
Note:

Discrete Mathematics 307, (2007), 2330-2340

Risorsa bibliografica obbligatoriaR.J. Gardner and P. Gritzmann, Discrete tomography: Determination of finite sets by X-rays
Note:

Trans. Amer. Math. Soc., 349 (6), (1997), 2271-2295

Risorsa bibliografica obbligatoriaL. Hajdu and R. Tijdeman, Algebraic aspects of discrete tomography
Note:

J. reine angew. Math. 534 (2001), 119-128


Mix Forme Didattiche
Tipo Forma Didattica Ore didattiche
lezione
30.0
esercitazione
0.0
laboratorio informatico
0.0
laboratorio sperimentale
0.0
progetto
0.0
laboratorio di progetto
0.0

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Note Docente
schedaincarico v. 1.6.1 / 1.6.1
Area Servizi ICT
17/02/2020