Course code POLI8
Course title Discrete and Geometric Tomography
Institution Politecnico di Milano
Course address Department of Mathematics, (Building NAVE). Via Bonardi 9 20133 Milano
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 +39 02 2399 4577
Fax
Email paolo.dulio@polimi.it
Participating professors
Number of places Minimum: 5, Maximum: 10, Reserved for local students:
Objectives
COMPUTERIZED AXIAL TOMOGRAPHY (CT)
Principles and main mathematical reconstruction model.
Examples and applications.
 
DISCRETE TOMOGRAPHY (DT)
Ghosts and switching components.
Ryser algorithm.
Algebraic approach.
Uniqueness models.
Uniqueness and Additivity.
Examples and Applications.
  
GEOMETRIC TOMOGRAPHY (GT)
Hammer’s problem.
Parallel and point X-rays
Tomography for special geometric objects.
 
 

MAIN REFERENCES

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.

 

Further references will be given during the course, and cited papers will be supplied to all interested students

Programme to be followed

MONDAY, 16

Morning 9:15-13:15

Overview of the Course. A brief history of CAT. Qualitative description of the Radon transform and its inversion for X-ray image reconstruction. Applications and related problems. The origin of Geometric Tomography and of Discrete Tomography. Continuous and discrete parallel X-rays. Continuous and discrete point X-rays. Remarks and examples.

Afternoon 2:30-4:30 Discussion and exercise section.

 

TUESDAY, 17

Morning 9:15-13:15

Projections of lattice sets with discrete parallel X-rays. The reconstruction problem in Discrete Tomography. Ryser algorithm. Bad configurations, weakly bad configurations, switching components, ghosts. Ridge functions and additivity.

Afternoon 2:30-4:30 Discussion and exercise section.

 

WEDNESDAY, 18

Morning 9:15-13:15

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

Afternoon 2:30-4:30 Discussion and exercise section.

 

THURSDAY, 19

Morning 9:15-13:15

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.

Afternoon 2:30-4:30 Discussion and exercise section.

 

FRIDAY, 20

Morning 9:15-12:15 Exam Section.

Correction, marking and discussion.

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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