N. von Ellenrieder

Effects of geometric head model perturbations on the EEG forward and inverse problems

We study the effect of geometric head model perturbations on the electroencephalography (EEG) forward and inverse problems. Small magnitude perturbations of the shape of the head could represent uncertainties in the head model due to errors on images or techniques used to construct the model. They could also represent small scale details of the shape of the surfaces not described in a deterministic model, such as the sulci and fissures of the cortical layer. We perform a first-order perturbation analysis, using a meshless method for computing the sensitivity of the solution of the forward problem to the geometry of the head model. The effect on the forward problem solution is treated as noise in the EEG measurements and the Crame/spl acute/r-Rao bound is computed to quantify the effect on the inverse problem performance. Our results show that, for a dipolar source, the effect of the perturbations on the inverse problem performance is under the level of the uncertainties due to the spontaneous brain activity. Thus, the results suggest that an extremely detailed model of the head may be unnecessary when solving the EEG inverse problem.

A meshless method for solving the EEG forward problem

We present a numerical method to solve the quasi-static Maxwell equations and compute the electroencephalography (EEG) forward problem solution. More generally, we develop a computationally efficient method to obtain the electric potential distribution generated by a source of electric activity inside a three-dimensional body of arbitrary shape and layers of different electric conductivities. The method needs only a set of nodes on the surface and inside the head, but not a mesh connecting the nodes. This represents an advantage over traditional methods like boundary elements or finite elements since the generation of the mesh is typically computationally intensive. The performance of the proposed method is compared with the boundary element method (BEM) by numerically solving some EEG forward problems examples. For a large number of nodes and the same precision, our method has lower computational load than BEM due to a faster convergence rate and to the sparsity of the linear system to be solved.

Effect of Head Shape Variations Among Individuals on the EEG/MEG Forward and Inverse Problems

We study the effect of the head shape variations on the EEG/magnetoencephalography (MEG) forward and inverse problems. We build a random head model such that each sample represents the head shape of a different individual and solve the forward problem assuming this random head model, using a polynomial chaos expansion. The random solution of the forward problem is then used to quantify the effect of the geometry when the inverse problem is solved with a standard head model. The results derived with this approach are valid for a continuous family of head models, rather than just for a set of cases. The random model consists of three random surfaces that define layers of different electric conductivity, and we built an example based on a set of 30 deterministic models from adults. Our results show that for a dipolar source model, the effect of the head shape variations on the EEG/MEG inverse problem due to the random head model is slightly larger than the effect of the electronic noise present in the sensors. The variations in the EEG inverse problem solutions are due to the variations in the shape of the volume conductor, while the variations in the MEG inverse problem solutions, larger than the EEG ones, are caused mainly by the variations of the absolute position of the sources in a coordinate system based on anatomical landmarks, in which the magnetometers have a fixed position.

MEG forward problem formulation using equivalent surface current densities

We present a formulation for the magnetoencephalography (MEG) forward problem with a layered head model. Traditionally the magnetic field is computed based on the electric potential on the interfaces between the layers. We propose to express the effect of the volumetric currents in terms of an equivalent surface current density on each interface, and obtain the magnetic field based on them. The boundary elements method is used to compute the equivalent current density and the magnetic field for a realistic head geometry. We present numerical results showing that the MEG forward problem is solved correctly with this formulation, and compare it with the performance of the traditional formulation. We conclude that the traditional formulation generally performs better, but still the new formulation is useful in certain situations.

General bounds for electrode mislocation on the EEG inverse problem

We analyze the effect of electrode mislocation on the electroencephalography (EEG) inverse problem using the Cramér-Rao bound (CRB) for single dipolar source parameters. We adopt a realistic head shape model, and solve the forward problem using the Boundary Element Method; the use of the CRB allows us to obtain general results which do not depend on the algorithm used for solving the inverse problem. We consider two possible causes for the electrode mislocation, errors in the measurement of the electrode positions and an imperfect registration between the electrodes and the scalp surfaces. For 120 electrodes placed in the scalp according to the 10-20 standard, and errors on the electrode location with a standard deviation of 5mm, the lower bound on the standard deviation in the source depth estimation is approximately 1mm in the worst case. Therefore, we conclude that errors in the electrode location may be tolerated since their effect on the EEG inverse problem are negligible from a practical point of view.

Radar detection algorithm for GARCH clutter model

We propose a GARCH model to represent the clutter in radar applications. We fit this model to real sea clutter data and we show that it represents adequately the statistics of the data. Then, we develop a detection test based on this model. Using synthetic and real radar data, we evaluate its performance and we show that the proposed detector offers higher probability of detection for a specified value of probability of false alarm than tests based on Gaussian and Weibull models, especially for low signal to clutter ratios.

Waveform Selection for Electrical Impedance Tomography

Electrical Impedance Tomography (EIT) is a noninvasive method that can be used to estimate the electrical conductivity of the head tissues. It is based on the measurement of electric potential on the scalp generated by the injection of a small electric current. If the generated electric potential distribution is measured with an Electroencephalography (EEG) equipment, the neural activity of the brain will produce signals that may affect the EIT measurements. In the present work we propose a method to reduce the effect of these signals and show a procedure to obtain the minimum number of samples that is needed to neglect the effect of the brain activity. The method requires the obtention of the optimum waveform for the applied current to minimize the variance of the electric potential estimation. As an example, the method is applied to two sets of EEG measurements of two patients, and we determine the optimum waveform and minimum number of samples for each measurement set. We also show that the replacement of the optimum waveform by a sinusoid with arbitrary phase does not significantly affect the estimations, but a previous spectral analysis of the brain activity must be performed in order to determine convenient frequencies.

Sparse modeling for polarimetric radar

In this paper, we develop a sparse model to represent the data recorded by a polarimetric, coherent radar. We produce this model defining an over-complete library of possible target responses in the range-polarization space. Then, we employ compressive sensing methods to infer the position and the scattering matrix of the target. Using real radar data, we show that this new approach offers better interference rejection over other methods.

Spatial filtering in electrical impedance tomography

We propose a new method to localize electrical conductivity changes in the human head using Electrical Impedance Tomography (EIT). In EIT, a localized conductivity change produces a change in the electric potential distribution which is equivalent to a potential generated by a dipole at the same location. We propose to estimate the location of conductivity changes with the same techniques used to solve the Electroencephalography source localization problem. In particular, we show an adaptation of the Linear Constrain Minimum Variance Beamforming technique to perform this estimation. We simulate a localized 10% conductivity change in a realistic model of the human head to apply the method. Results show that depending on the noise level, the method can accurately localize the conductivity change.

Linearly constrained minimum variance spatial filtering for localization of conductivity changes in electrical impedance tomography.

We localize dynamic electrical conductivity changes and reconstruct their time evolution introducing the spatial filtering technique to electrical impedance tomography (EIT). More precisely, we use the unit-noise-gain constrained variation of the distortionless-response linearly constrained minimum variance spatial filter. We address the effects of interference and the use of zero gain constraints. The approach is successfully tested in simulated and real tank phantoms. We compute the position error and resolution to compare the localization performance of the proposed method with the one-step Gauss-Newton reconstruction with Laplacian prior. We also study the effects of sensor position errors. Our results show that EIT spatial filtering is useful for localizing conductivity changes of relatively small size and for estimating their time-courses. Some potential dynamic EIT applications such as acute ischemic stroke detection and neuronal activity localization may benefit from the higher resolution of spatial filters as compared to conventional tomographic reconstruction algorithms.