DISCRETE AND GEOMETRIC TOMOGRAPHY PAOLO DULIO CATTHEORY Theoretical

CAT-THEORY Theoretical model FBP (continuous operators) E

APPLICATIVE APPROACH poor quality of reconstruction

LINEAR SYSTEM OF EQUATIONS Numerical approach required Image size Number of unknowns 10 x

RECONSTRUCTION ALGORITHMS INPUT: some image x 0 ITERATIVE METHODS ( D. A. R. T.

UNIQUENESS RESULTS Incorporation of possible available information reduces the number of allowed solutions. Uniqueness

DIFFERENT INVERSE PROBLEMS Reconstruction from collected data extends to different research areas In addition

TOPICS FOR POSSIBLE THESES…AND BEYOND General reconstruction theorems Uniqueness results under prior knowledge Modelization

WORKING GROUP THE NETHERLAND S GERMAN Y HUNGARY -Debrecen -Szeged U. S. A. BELGIUM

CONTACTS Email paolo. dulio@polimi. it Telephone 02 -2399 4577 Office NAVE – 4° FLOOR

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DISCRETE AND GEOMETRIC TOMOGRAPHY PAOLO DULIO

CAT-THEORY Theoretical model FBP (continuous operators) E

CAT-THEORY E

APPLICATIVE APPROACH poor quality of reconstruction

DISCRETIZATION

LINEAR SYSTEM OF EQUATIONS

LINEAR SYSTEM OF EQUATIONS Numerical approach required Image size Number of unknowns 10 x 10 100 128 x 128 16384 256 x 256 65536 512 x 512 262144

GHOSTS

INVISIBLE PROJECTIONS 1 -1 0 -1 1 0 0 0

0 0

RECONSTRUCTION ALGORITHMS INPUT: some image x 0 ITERATIVE METHODS ( D. A. R. T. ) OUTPUT: a (good approximation of a) solution

UNIQUENESS RESULTS Incorporation of possible available information reduces the number of allowed solutions. Uniqueness theorems sometimes can be proved. Usually exploited knowledge • Geometric features, e. g. convexity • Size of a confining grid • Number of gray levels, e. g. binary tomography • Directions of projections

DIFFERENT INVERSE PROBLEMS Reconstruction from collected data extends to different research areas In addition to Tomographic problems (X-ray data), our research also focuses on modeling the functional and the structural connectivity in the brain network (EEG, f. MRI, DTI, PET data) Graph Theory THE MODEL

TOPICS FOR POSSIBLE THESES…AND BEYOND General reconstruction theorems Uniqueness results under prior knowledge Modelization of noisy projections Reconstruction algorithms with MATLAB Combinatorial characterization of regions of uniqueness Explicit reconstruction of special geometric sets Theoretical investigation of ghosts Algebraic approach for special geometric classes Generalization of special geometric constructions Modelization of the brain neural network

WORKING GROUP THE NETHERLAND S GERMAN Y HUNGARY -Debrecen -Szeged U. S. A. BELGIUM AUSTRALIA FRANCE -Grenoble Chambery -Nantes POLITECNICO ITALY -Firenze -Siena -Piacenza -Trieste

CONTACTS Email paolo. dulio@polimi. it Telephone 02 -2399 4577 Office NAVE – 4° FLOOR Availability 10 a. m-6 p. m. every working day, by appointment is preferable