Course code POLI8
Course title Discrete and Geometric Tomography
Institution Politecnico di Milano
Course address Politecnico di Milano (Dipartimento di Matematica e laboratorio di Computer Vision), Piazza L.da Vinci,32, 20133 Milano, Italy
City Milan
Minimum year of study 3rd year
Minimum level of English Good
Minimum level of French None
Key words

Tomography, X-ray, projective geometry, reconstruction, uniqueness problem
Language English
Professor responsible Paolo Dulio
Telephone +390223994577
Fax +390223994629
Email paolo.dulio@polimi.it
Participating professors
Number of places Minimum: 5, Maximum: 10, Reserved for local students:
Objectives Discrete and Geometric Tomography represent the geometric approach to the inverse problem of Computerized Axial Tomography, concerning the reconstruction of a body by means of X-rays. The purpose of the course is to outline, from a theoretic and geometric point of view, some of the topics usually considered, such as reconstruction algorithms, uniqueness problems and stability of reconstructions.  
Programme to be followed

The course is organized on lectures in the morning and interactive sessions (such as exercises, answer to questions or seminars) in the afternoon. Useful references are

 

1)      Richard Gardner, Geometric Tomography, Cambridge University Press, New York, second edition, 2006.

2)      Gabor T. Herman and Attila Kuba Eds., Advances in discrete tomography and its applications, Applied and Numerical Harmonic Analysis. Birkhäuser Boston, Inc., Boston, MA, 2007.

 Outline and provisional schedule:

A brief history of CAT. Qualitative description of the Radon transform. The origin of Geometric Tomography. Hammer’s problem and related uniqueness problems. Discrete Tomography an related problems. Continuous and discrete parallel X-rays. Continuous and discrete point X-rays. An overview of geometric transformations in the plane. Projective transformations. Cross-ratio for collinear points and for line in a pencil.

Radiographies of lattice sets with discrete parallel X-rays The reconstruction problem in Discrete Tomography. Description of some algorithms and examples of applications. Switching components. Mid-point construction. U-polygons.

 

Stability of reconstruction and uniqueness problem. Uniqueness results by means of radiographies of convex bodies with continuous parallel X-rays.  The theorem of Gardner-McMullen in the Euclidean plane. Uniqueness results for classes of lattice sets by means of discrete parallel X-rays. The results of Gardner and Gritzmann in the integer lattic.

 

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

Final exam

 Corrections and valuations
Prerequisites Elementary geometry, trigonometry, geometric transformations,  linear algebra, analytic geometry, calculus.
Course exam The final exam is scheduled on Friday morning. It consists of a written test organized in a few questions with open answers. A possible additional oral examination could be considered to clarify some  works
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