Irina Gorodnitsky

Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm

We present a nonparametric algorithm for finding localized energy solutions from limited data. The problem we address is underdetermined, and no prior knowledge of the shape of the region on which the solution is nonzero is assumed. Termed the FOcal Underdetermined System Solver (FOCUSS), the algorithm has two integral parts: a low-resolution initial estimate of the real signal and the iteration process that refines the initial estimate to the final localized energy solution. The iterations are based on weighted norm minimization of the dependent variable with the weights being a function of the preceding iterative solutions. The algorithm is presented as a general estimation tool usable across different applications. A detailed analysis laying the theoretical foundation for the algorithm is given and includes proofs of global and local convergence and a derivation of the rate of convergence. A view of the algorithm as a novel optimization method which combines desirable characteristics of both classical optimization and learning-based algorithms is provided. Mathematical results on conditions for uniqueness of sparse solutions are also given. Applications of the algorithm are illustrated on problems in direction-of-arrival (DOA) estimation and neuromagnetic imaging.

Neuromagnetic source imaging with FOCUSS: a recursive weighted minimum norm algorithm

The paper describes a new algorithm for tomographic source reconstruction in neural electromagnetic inverse problems. Termed FOCUSS (FOCal Underdetermined System Solution), this algorithm combines the desired features of the two major approaches to electromagnetic inverse procedures. Like multiple current dipole modeling methods, FOCUSS produces high resolution solutions appropriate for the highly localized sources often encountered in electromagnetic imaging. Like linear estimation methods, FOCUSS allows current sources to assume arbitrary shapes and it preserves the generality and ease of application characteristic of this group of methods. It stands apart from standard signal processing techniques because, as an initialization-dependent algorithm, it accommodates the non-unique set of feasible solutions that arise from the neuroelectric source constraints. FOCUSS is based on recursive, weighted norm minimization. The consequence of the repeated weighting procedure is, in effect, to concentrate the solution in the minimal active regions that are essential for accurately reproducing the measurements. The FOCUSS algorithm is introduced and its properties are illustrated in the context of a number of simulations, first using exact measurements in 2- and 3-D problems, and then in the presence of noise and modeling errors. The results suggest that FOCUSS is a powerful algorithm with considerable utility for tomographic current estimation.

EEG mu component responses to viewing emotional faces

Simulation theories for the perceptual processing of emotional faces assert that observers recruit the neural circuitry involved in creating their own emotional facial expressions in order to recognize the emotions and infer the feelings of others. The EEG mu rhythm is a sensorimotor oscillation hypothesized to index simulation of some actions during perceptual processing of these actions. The purpose of this research was to extend the study of mu rhythm simulation responses during perceptual tasks to the domain of emotional face perception. Subjects viewed happy and disgusted face photos with empathy and non-empathy task instructions while EEG responses were measured. EEG components were isolated and analyzed using a blind source separation (BSS) method. Mu components were found to respond to the perception of happy and disgusted faces during both empathy and non-empathy tasks with an event-related desynchronization (ERD), activation that is consistent with face simulation. Significant differences were found between responses to happy and to disgusted faces across the right hemisphere mu components beginning about 500ms after stimulus presentation. These findings support a simulation account of perceptual face processing based on a sensorimotor mirroring mechanism, and are the first report of distinct EEG mu responses to observation of positively and negatively valenced emotional faces.

Tracking eye fixations with electroocular and electroencephalographic recordings

We describe a method, based on recordings of the electroencephalogram (EEG) and eye movement potentials (electrooculogram), to track where on a screen (x,y coordinates) an individual is fixating. The method makes use of an empirically derived beam-forming filter (derived from a sequence of calibrated eye movements) to isolate eye motion from other electrophysiological and ambient electrical signals. Electrophysiological researchers may find this method a simple and inexpensive means of tracking eye movements and a useful complement to scalp recordings in studies of cognitive phenomena. The resolution is comparable to that of many commercial systems; the method can be implemented with as few as four electrodes around the eyes to complement the EEG electrodes already in use. This method may also find some specialized applications such as studying eye movements during sleep and in human-machine interfaces that make use of gaze information.

Global modeling of the Rössler system from the z-variable

Obtaining a global model from the z-variable of the Rossler system is considered to be difficult because of its spiky structure. In this Letter, a 3D global model from the z-variable is derived in a space spanned by the state variable of the time-series itself and generic functions of the other two state variables. We term this space the Ansatz Space. The procedure consists of two steps. First, models built in the derivative coordinates are obtained. Second, we use the analytical form of the map ϕ between systems in the original state space and in the differential space to find a class of models in the Ansatz Space. We find eight models in this class which we show to be dynamically equivalent to the original Rossler system. The important attribute of this approach is that we do not need to use any prior knowledge of the dynamical system other than the measured time series data in order to obtain global models from a single time series.  2003 Elsevier B.V. All rights reserved.

Signal processing for NQR discrimination of buried land mines

Nuclear quadrupole resonance (NQR) is a technique that discriminates mines from clutter by exploiting unique properties of explosives, rather than the attributes of the mine that exist in many forms of anthropic clutter. After exciting the explosive with a properly designed electromagnetic-induction (EMI) system, one attempts to sense late-time spin echoes, which are characterized by radiation at particular frequencies. It is this narrow-band radiation that indicates the presence of explosives, since this effect is not seen in most clutter, both natural and anthropic. However, this problem is complicated by several issues. First, the late-time radiation if often very weak, particularly for TNT, and therefore the signal-to-noise ratio must be high for extracting the NQR response. Further, the frequency at which the explosive radiates is often a strong function of the background environment, and therefore in practice the NQR radiation frequency is not known a priori. Finally, at the frequencies of interest, there is a significant amount of background radiation, which induces radio frequency interference (RFI). In this paper we discuss several signal processing tools we have developed to enhance the utility of NQR explosives detection. In particular, with regard to the RFI, we exposure least-mean-squares algorithms which have proven well suited to extracting background interference. Algorithm performance is assessed through consideration of actual measured data. With regard to the detection of the NQR electromagnetic echo, we consider a Bayesian discrimination algorithm. The performance of the Bayesian algorithm is presented, again using measured NQR data.

A recursive weighted minimum norm algorithm: Analysis and applications

Reliable estimation of a highly localized signal from insufficient data is a difficult problem common to many fields. The authors present an iterative weighted norm minimization method that utilizes a posteriori constraints for the estimation of such signals. Analysis is performed to determine the convergence properties and characterize the solutions of the algorithm. The advantages and applications of the approach and the new algorithm are demonstrated using electromagnetic imaging and direction-of-arrival (DOA) estimation problems.<<ETX>>

Source localization in magnetoencephalography using an iterative weighted minimum norm algorithm

Imaging of brain activity based on magnetoencephalography (MEG) requires high-resolution estimates that closely approximate the spatial distribution of the underlying currents. The authors examine the physics of the MEG problem to give the motivation for developing a new algorithm that meets its unique requirements. The technique is a nonparametric, iterative, weighted norm minimization procedure with a posteriori constraints. The authors develop the algorithm and determine the necessary requirements for convergence. Issues of initialization and bias equalization for MEG reconstruction, and techniques for analysis of noisy data are discussed.<<ETX>>

A new iterative weighted norm minimization algorithm and its applications

A general class of linear inverse problems in which the solutions are sparse and localized is considered. The proposed algorithm is a nonparametric approach that finds sparse and localized solutions without prior information on the constraints. Each step of the iterative procedure consists in solving a weighted least squares problem wherein the weighting matrix is determined by the solution from the previous iteration. Some properties of the algorithm along with its applications to problems in direction of arrival and spectrum estimation are presented.<<ETX>>

Affine scaling transformation based methods for computing low complexity sparse solutions

This paper presents affine scaling transformation based methods for finding low complexity sparse solutions to optimization problems. The methods achieve sparse solutions in a more general context, and generalize our earlier work on FOCUSS developed to deal with the underdetermined linear inverse problem. The key result is a theorem which shows a simple condition that a sequence has to satisfy for it to converge to a sparse limiting solution. Three approaches to incorporate this condition into optimization problems are presented. These consist of either imposing the condition as an additional optimization constraint, or suitably modifying the cost function, or using a combination of the two. The benefits of the methodology when applied to the linear inverse problem are twofold. Firstly, it allows for the treatment of the overdetermined problem in addition to the underdetermined problem, and secondly it enables establishing sufficient conditions under which regularized versions of FOCUSS are assured of convergence to sparse solutions.