Diagonality Measures of Hermitian Positive-Definite Matrices with Application to the Approximate Joint Diagonalization Problem

In this paper, we introduce properly-invariant diagonality measures of Hermitian positive-definite matrices. These diagonality measures are defined as distances or divergences between a given positive-definite matrix and its diagonal part. We then give closed-form expressions of these diagonality measures and discuss their invariance properties. The diagonality measure based on the log-determinant $\alpha$-divergence is general enough as it includes a diagonality criterion used by the signal processing community as a special case. These diagonality measures are then used to formulate minimization problems for finding the approximate joint diagonalizer of a given set of Hermitian positive-definite matrices. Numerical computations based on a modified Newton method are presented and commented

On the comparison to EEG Norms : a new method and a simulation study

Quantitative Electroencephalography (qEEG) as a tool for the diagnosis of neurological and psychiatric disorders is receiving an increased interest. While qEEG analysis is restricted to the scalp, the recent development of electromagnetic tomographies (ET) allows the study of the electrical activity of cortical structures. Electrical measures of a patient can be compared to a normative database derived on a large sample of healthy individuals. The deviance from the database\u27s norms provides a probabilistic measure of the likelihood that the patient\u27s electrical activity reflects normal brain functioning. The focus of this thesis is the method for estimating such deviance. The method currently employed estimates the mean and the standard deviation of the normative sample. The deviance is then expressed in terms of z-scores. This method is referred to as the parametric method. The accuracy of the parametric method relies on the assumption that the distribution of the normative sample is gaussian, but this assumption is not always fulfilled in real qEEG and especially ET data. A new method based on percentiles ( nonparametric ) is proposed. The parametric and the non-parametric methods are compared using simulated data. The accuracy of both methods is assessed as a function of normative sample size and gaussianity for three different alpha levels. Results suggest that the performance of the parametric method is unaffected by sample size (bigger than 100), but that non-gaussianity jeopardizes accuracy even if the normative distribution is close to gaussianity. On the contrary the performance of the non-parametric method is unaffected by non-gaussianity, but is a function of sample size only. It is shown that, with n\u3e160, the non-parametric method can be considered always preferable. Results are discussed taking into consideration technical issues related to the peculiar nature of qEEG and ET data. It is suggested that the sample size is the only constant across EEG frequency bands, measurement locations, and kind of quantitative measures. As a consequence, for a given database, the error rate of the non-parametric database is homogeneous, however the same is not true for the parametric method

Tomographic neurofeedback : a new technique for the self-regulation of brain electrical activity

A major limitation of the current neurofeedback paradigm is the limited information provided by a single or a small number of electrodes placed on the scalp. A considerable improvement of the neurofeedback efficacy and specificity could be obtained feeding back brain activity of delimited structures. While traditional EEG information reflects the superposition of the electrical activity of a large number of neurons, by means of inverse solutions such as the Low-Resolution Electromagnetic Tomography (LORETA) spatially delimited brain activity can be evaluated in neocortical tissue. In this Dissertation we implement LORETA neurofeedback, we introduce a new feedback function ( 1 ) sensitive to dynamic change over time, and we clarify several issues related to the learning process observable with neurofeedback. The reported set of three experiments is the first attempt I am aware of to prove learning of brain current density activity. Three individuals were trained to improve brain activation (suppress low Alpha (8-10 Hz) and enhance low Beta (16 -20 Hz) current density) in the anterior cingulate gyros cognitive division (ACcd). Participants took part of six experimental sessions, each lasting approximately 30 minutes. Randomization-Permutation ANCOVA tests were conducted on recordings of the neurofeedback training. In addition a randomized trial was performed at the end of the treatment. During eight two-minutes periods (trials) participants were asked to try to obtain as many rewards as they could ( 4 1 trials) or as few rewards as they could ( 4 0 trials). The order of trials was decided at random. The hypothesis under testing was that participants acquired volitional control over their brain activity so to be able to obtain more rewards during the plus condition as compared to the minus condition. We found evidence of volitional control for two subjects (p=0.043 and p=O.l) and no evidence of volitional control for one of them (p=0.27 1). The combination of the three p-values provided an overall probability value for this experiment of 0.012 with the additive method and 0.035 with the multiplicative method. These results strongly support the hypothesis of volitional control across the experimental group. Trends of the Beta/ Alpha power ratio in the ACcd were in the expected direction for all the three subjects, however the combined p-values did not reach significance. With as few as six training sessions, typically insufficient to produce any form of learning with scalp neurofeedback, the experiment showed overall signs of volitional control of the electrical activity of the ACcd. Possible applications of the technique are important and include the treatment of epileptic foci, the treatment of specific brain regions damaged as a consequence of traumatic brain injury, and in general of any specific cortical electrical activity

Copernicus high-resolution layers for land cover classification in Italy

The high-resolution layers (HRLs) are land cover maps produced for the entire Italian territory (approximately 30 million hectares) in 2012 by the European Environment Agency, aimed at monitoring soil imperviousness and natural cover, such as forest, grassland, wetland, and water surface, with a high spatial resolution of 20 m. This study presents the methodologies developed for the production, verification, and enhancement of the HRLs in Italy. The innovative approach is mainly based on (a) the use of available reference data for the enhancement process, (b) the reduction of the manual work of operators by using a semi-automatic approach, and (c) the overall increase in the cost-efficiency in relation to the production and updating of land cover maps. The results show the reliability of these methodologies in assessing and enhancing the quality of the HRLs. Finally, an integration of the individual layers, represented by the HRLs, was performed in order to produce a National High-Resolution Land Cover ma

Least-Squares Joint Diagonalization of a matrix set by a congruence transformation

The approximate joint diagonalization (AJD) is an important analytic tool at the base of numerous independent component analysis (ICA) and other blind source separation (BSS) methods, thus finding more and more applications in medical imaging analysis. In this work we present a new AJD algorithm named SDIAG (Spheric Diagonalization). It imposes no constraint either on the input matrices or on the joint diagonalizer to be estimated, thus it is very general. Whereas it is well grounded on the classical leastsquares criterion, a new normalization reveals a very simple form of the solution matrix. Numerical simulations shown that the algorithm, named SDIAG (spheric diagonalization), behaves well as compared to state-of-the art AJD algorithms.Comment: 2nd Singaporean-French IPAL Symposium, Singapour : Singapour (2009

Single-Trial Classification of Multi-User P300-Based Brain-Computer Interface Using Riemannian Geometry

International audienceThe classification of electroencephalographic (EEG) data recorded from multiple users simultaneously is an important challenge in the field of Brain-Computer Interface (BCI). In this paper we compare different approaches for classification of single-trials Event-Related Potential (ERP) on two subjects playing a collaborative BCI game. The minimum distance to mean (MDM) classifier in a Riemannian framework is extended to use the diversity of the inter-subjects spatio-temporal statistics (MDM-hyper) or to merge multiple classifiers (MDM-multi). We show that both these classifiers outperform significantly the mean performance of the two users and analogous classifiers based on the step-wise linear discriminant analysis. More importantly, the MDM-multi outperforms the performance of the best player within the pair

A Fixed-Point Algorithm for Estimating Power Means of Positive Definite Matrices

International audienceThe estimation of means of data points lying on the Riemannian manifold of symmetric positive-definite (SPD) matrices is of great utility in classification problems and is currently heavily studied. The power means of SPD matrices with exponent p in the interval [-1, 1] interpolate in between the Harmonic (p =-1) and the Arithmetic mean (p = 1), while the Geometric (Karcher) mean corresponds to their limit evaluated at 0. In this article we present a simple fixed point algorithm for estimating means along this whole continuum. The convergence rate of the proposed algorithm for p = ±0.5 deteriorates very little with the number and dimension of points given as input. Along the whole continuum it is also robust with respect to the dispersion of the points on the manifold. Thus, the proposed algorithm allows the efficient estimation of the whole family of power means, including the geometric mean