Alberto Sorrentino

Sara Sommariva, Alberto Sorrentino, Michele Piana et al.

In this work we use numerical simulation to investigate how the temporal length of the data affects the reliability of the estimates of brain connectivity from EEG time--series. We assume that the neural sources follow a stable MultiVariate AutoRegressive model, and consider three connectivity metrics: Imaginary part of Coherency (IC), generalized Partial Directed Coherence (gPDC) and frequency--domain Granger Causality (fGC). In order to assess the statistical significance of the estimated values, we use the surrogate data test by generating phase--randomized and autoregressive surrogate data. We first consider the ideal case where we know the source time courses exactly. Here we show how, expectedly, even exact knowledge of the source time courses is not sufficient to provide reliable estimates of the connectivity when the number of samples gets small; however, while gPDC and fGC tend to provide a larger number of false positives, the IC becomes less sensitive to the presence of connectivity. Then we proceed with more realistic simulations, where the source time courses are estimated using eLORETA, and the EEG signal is affected by biological noise of increasing intensity. Using the ideal case as a reference, we show that the impact of biological noise on IC estimates is qualitatively different from the impact on gPDC and fGC.

Alberto Sorrentino, Michele Piana

We provide an overview of the state-of-the-art for mathematical methods that are used to reconstruct brain activity from neurophysiological data. After a brief introduction on the mathematics of the forward problem, we discuss standard and recently proposed regularization methods, as well as Monte Carlo techniques for Bayesian inference. We classify the inverse methods based on the underlying source model, and discuss advantages and disadvantages. Finally we describe an application to the pre-surgical evaluation of epileptic patients.

Sara Sommariva, Alberto Sorrentino

We discuss the use of a recent class of sequential Monte Carlo methods for solving inverse problems characterized by a semi-linear structure, i.e. where the data depend linearly on a subset of variables and nonlinearly on the remaining ones. In this type of problems, under proper Gaussian assumptions one can marginalize the linear variables. This means that the Monte Carlo procedure needs only to be applied to the nonlinear variables, while the linear ones can be treated analytically; as a result, the Monte Carlo variance and/or the computational cost decrease. We use this approach to solve the inverse problem of magnetoencephalography, with a multi-dipole model for the sources. Here, data depend nonlinearly on the number of sources and their locations, and depend linearly on their current vectors. The semi-analytic approach enables us to estimate the number of dipoles and their location from a whole time-series, rather than a single time point, while keeping a low computational cost.

Valentina Vivaldi, Alberto Sorrentino

We describe a novel method for dynamic estimation of multi-dipole states from Magneto/Electro-encephalography (M/EEG) time series. The new approach builds on the recent development of particle filters for M/EEG; these algorithms approximate, with samples and weights, the posterior distribution of the neural sources at time t given the data up to time t. However, for off-line inference purposes it is preferable to work with the smoothing distribution, i.e. the distribution for the neural sources at time t conditioned on the whole time series. In this study, we use a Monte Carlo algorithm to approximate the smoothing distribution for a time-varying set of current dipoles. We show, using numerical simulations, that the estimates provided by the smoothing distribution are more accurate than those provided by the filtering distribution, particularly at the appearance of the source. We validate the proposed algorithm using an experimental dataset recorded from an epileptic patient. Improved localization of the source onset can be particularly relevant in source modeling of epileptic patients, where the source onset brings information on the epileptogenic zone.

Federica Sciacchitano, Silvio Lugaro, Alberto Sorrentino

We consider imaging of solar flares from NASA RHESSI data as a parametric imaging problem, where flares are represented as a finite collection of geometric shapes. We set up a Bayesian model in which the number of objects forming the image is a priori unknown, as well as their shapes. We use a Sequential Monte Carlo algorithm to explore the corresponding posterior distribution. We apply the method to synthetic and experimental data, largely known in the RHESSI community. The method reconstructs improved images of solar flares, with the additional advantage of providing uncertainty quantification of the estimated parameters.

Anna Maria Massone, Federica Sciacchitano, Michele Piana et al.

We describe two inversion methods for the reconstruction of hard X-ray solar images. The methods are tested against experimental visibilities recorded by the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) and synthetic visibilities based on the design of the Spectrometer/Telescope for Imaging X-rays (STIX).

Giulia Denevi, Sara Garbarino, Alberto Sorrentino

We consider the inverse problem of retrieving aerosol extinction coefficients from Raman lidar measurements. In this problem the unknown and the data are related through the exponential of a linear operator, the unknown is non-negative and the data follow the Poisson distribution. Standard methods work on the log-transformed data and solve the resulting linear inverse problem, but neglect to take into account the noise statistics. In this study we show that proper modelling of the noise distribution can improve substantially the quality of the reconstructed extinction profiles. To achieve this goal, we consider the non-linear inverse problem with non-negativity constraint, and propose two iterative algorithms derived using the Karush-Kuhn-Tucker conditions. We validate the algorithms with synthetic and experimental data. As expected, the proposed algorithms outperform standard methods in terms of sensitivity to noise and reliability of the estimated profile.

Sara Garbarino, Alberto Sorrentino, Anna Maria Massone et al.

We consider the problem of retrieving the aerosol extinction coefficient from Raman lidar measurements. This is an ill--posed inverse problem that needs regularization, and we propose to use the Expectation--Maximization (EM) algorithm to provide stable solutions. Indeed, EM is an iterative algorithm that imposes a positivity constraint on the solution, and provides regularization if iterations are stopped early enough. We describe the algorithm and propose a stopping criterion inspired by a statistical principle. We then discuss its properties concerning the spatial resolution. Finally, we validate the proposed approach by using both synthetic data and experimental measurements; we compare the reconstructions obtained by EM with those obtained by the Tikhonov method, by the Levenberg-Marquardt method, as well as those obtained by combining data smoothing and numerical derivation.