Inverse problems are a very important topic in data and image processing and, more generally, in information theory. For this reason they are of interest in very different domains of applied science, such as medical imaging, microscopy, astronomy,
seismology, nondestructive evaluation etc. As it is well known, inverse problems are ill-posed in the sense of Hadamard. As a
consequence, the discretization of the equations relating the unknown solution to the data leads to problems affected by numerical
instability.
Therefore the formulation of inverse problems requires an accurate mathematical modeling, including a statistical model
of the noise affecting the data and additional constraints on the solution. As a result the solution of inverse problems is in general
reduced to the constrained minimization of suitable functionals.
The specific objective of the project is to provide methods and software for the solution of problems of image reconstruction and
denoising in medicine and astronomy. Such a result, that will necessarily require a multidisciplinary activity, will be obtained by
coordinating five groups from different Italian Universities, with different and complementary expertises, all with a consolidated
experience in their specific research area.
The reason for considering in the same project these very different domains of application is twofold. First, both in medicine and
astronomy data are typically acquired by means of devices such that the noise is mainly due to photon counting and is modeled by a
Poisson process. A maximum likelihood approach leads to the minimization of a non-quadratic convex functional known as Csiszar
divergence. Therefore, physical properties common to these problems lead to similar mathematical models.
The second reason is that we consider applications, both in Medicine and Astronomy, where different images of the same object can
be acquired with different orientations of the instrument with respect to the object. Therefore we consider problems of reconstructing
the same object from multiple images, in a situation similar to that encountered in tomography, namely object reconstruction from
projections. It must also be pointed out that all these problems are charaterized by a huge amount of data, i. e. they are large-scale
problems, so that computational issues are quite important.
The project team consists of five Units: three of them (Genova, Milano, Verona) are directly involved in applications of inverse
problems to Astronomy and Medicine; about the two others (Modena and Napoli), one has a long term experience in numerical
methods of optimization, the other in the numerical solution of inverse problems described by integral and differential operators.
Both have expertises in high end computing. Therefore the project can provide an important cross-fertilization of the activities of
these so different groups.
The problems coming from astronomical applications range from the reconstruction of the images of the Large Binocular Telescope,
an interferometer of new conception, to the reconstruction of X-ray images provided by the NASA satellite RHESSI, dedicated to the
observation of solar flares. On the other hand the medical applications include volumetric reconstruction from digital radiographs,
witth application to dental imaging, magnetoencephalography and ultrasound imaging. Efficient optimization methods and
algorithms, as well as edge-preserving methods for image reconstruction and denoising will be investigated and applied to the
problems whose solution is the objective of the project.