DISCRETE AND GEOMETRIC TOMOGRAPHY

Image reconstruction from projections is one of the main inverse problems which appears in several applications. The image is usually represented by an unknown real valued function $f(x,y)$, with bounded support. The values of $f$ are related to physical properties of a two-dimensional section of the object under investigation. Projections are taken with the help of some kind of rays. For instance, in Computerized Tomography (CT), a portion of a human body is reconstructed by measuring the coefficient of linear attenuation of each beam of the $X$-ray traveling  along a line crossing the body. The radiation is produced by photons, issued from a source and collected by a detector, both translating and rotating around the body. The differences between issued and collected photons measure the absorption of radiation by different tissues.

For our purposes, the usual density functions appearing in CT are replaced by geometric objects, and  one of the main  goal is to find  conditions which guarantee a faithful reconstruction, possibly unique, within a given geometric class of geometric subsets.

NEUROMATHEMATICS

The main goal of the Neuromathematics research Group is to understand integrative and segregative aspects of the brain. A  special interest concerns the structural and functional connectivity which are responsible for the brain dynamics. Their investigation could be of great help for understanding human (and animal) cognition and behavior.

Our main approach is to view the brain as a complex network, that can be considered as the support to process and integrate the information among the nodes of the net. The nodes represent neural elements, such as neurons, groups of neurons or even brain regions.

In particular we are interested in understanding how the changes in topological properties of the graphs representing the complex brain networks could lead to model neurological diseases and, possibly, to forecast their onset. In addition we wish to detect the individual differences in brain networks across healthy populations, as well as disturbances in brain injury and disease (e.g. Epilepsy and Schizophrenia).

Our ongoing projects involve collaborations with University Medical Centers specialized in the study of brain and its degenerations. These collaborations give us the opportunity to approach time series analysis, complexity and information theory.