EEG cortical imaging: A vector field approach for laplacian denoising and missing data estimation

Abstract

The surface Laplacian is known to be a theoretical reliable approximation of the cortical activity. Unfortunately, because of its high pass character and the relative low density of the EEG caps, the estimation of the Laplacian itself tends to be very sensitive to noise.We introduce a method based on vector field regularization through diffusion for denoising the Laplacian data and thus obtain robust estimation. We use a forward-backward diffusion aiming for source energy minimization while preserving contrasts between active and non-active regions. This technique uses headcap geometry specific differential operators to counter the low sensor density. The comparison with classical denoising schemes clearly demonstrates the advantages of our method.We also propose an algorithm based on the GaussOstrogradsky theorem for estimation of the Laplacian on missing (bad) electrodes, which can be combined with the regularization technique in order to provide a joint validation framework.

Differential geometric approximation of the gradient and Hessian on a triangulated manifold

“…In the context of inverse problems, both the gradient and Laplacian have frequently been used to regularize the inverse problems of both modalities (e.g. Tikhonov regularization) by bounding the derivatives of solutions on the surface of interest [2, 3, 4, 5]. Identification of nodes on a surface mesh with high gradient magnitude, along with the gradient direction, can be used for edge detection or detection of spatio-temporal phenomena such as local bursts of activity on the cortex or cardiac wavefront arrival on the heart [9].…”

“…In particular, in electrocardiography (ECG) and electroencephalography (EEG), the body, heart, scalp, and cortical surfaces can all be usefully represented as manifolds when electric potentials are directly recorded by electrode arrays on those surfaces or estimated on them from remote measurements. First- and second-order spatial derivatives of these potentials are useful in a number of settings, including both the processing of recorded data and regularization of an inverse problem [1, 2, 3, 4, 5]. However, in practice, we do not have knowledge of complete manifolds, but rather a collection of measurement or estimation locations, taken to be points on a manifold, along with potential values at those locations, and we must approximate the derivatives of interest.…”

“…Typically the approach of these methods is to approximate the triangulated mesh locally by a surface (such as a plane [2], sphere [6, 5], or thin plate splines [7]), on which a local coordinate system is constructed. In the case of the approximated gradient, this requires an a posteriori set of rotations, one for each local coordinate system, to return to global coordinates.…”

“…In the case of the approximated gradient, this requires an a posteriori set of rotations, one for each local coordinate system, to return to global coordinates. The data on the surface is approximated by a function whose derivatives are known (such as polynomial approximation [6], first- and second-order Taylor polynomials [2, 5], or analytic splines [7]) on the local coordinate system.…”

Resistor mesh model of a spherical head: Part 2: A review of applications to cortical mapping

“…Unfortunately, because of its high pass character and the relatively low electrode density of the LEG caps, the estimation of the Laplacian tends to be very sensitive to noise. Different methods have been proposed for de-noising the Laplacian data and thus obtaining a robust estimation (ALECU et al, 2004). Surface Laplacians generated by spline functions appear to be robust against the unavoidable perturbations of measured potentials and errors of head geometry and resistivity.…”