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.

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.

November 14, 2016

COMPUTERIZED AXIAL TOMOGRAPHY (CT)

General principles.

The Radon Transform.

Main theoretical reconstruction model.

Examples and applications.

November 15, 2016

DISCRETE TOMOGRAPHY (DT)

Discrete models

Ghosts and switching components.

Ryser algorithm.

Algebraic approach.

November 16, 2016

Uniqueness models.

Uniqueness and additivity.

Some reconstruction algorithms

Examples and applications.

November 17, 2016

GEOMETRIC TOMOGRAPHY (GT)

Hammer’s problem.

Parallel and point X-rays

Tomography for special geometric objects.

Examples and applications

November 18, 2016

Survey on the presented results.

Detailed proofs of some theorems.

Description of a few open problems.

Discussing with stutends possible research projects and supplementary material