Source localization using rational approximation on plane sections

Abstract

In functional neuroimaging, a crucial problem is to localize active sources within the brain non-invasively, from knowledge of electromagnetic measurements outside the head. Identification of point sources from boundary measurements is an ill-posed inverse problem. In the case of electroencephalography (EEG), measurements are only available at electrode positions, the number of sources is not known in advance and the medium within the head is inhomogeneous. This paper presents a new method for EEG source localization, based on rational approximation techniques in the complex plane. The method is used in the context of a nested sphere head model, in combination with a cortical mapping procedure. Results on simulated data prove the applicability of the method in the context of realistic measurement configurations.In memory of Line Garnero, and of her communicative dedication to unveiling the mysteries of the brain.

Identifiability Properties for Inverse Problems in EEG Data Processing Medical Engineering with Observability and Optimization Issues

“…Observe further that similar deconvolution issues also appear in automatic control (on the boundary of domains of dimension 2, however), concerning harmonic identification in frequency domain [5]. For dipolar point sources, we review some identifiability results related to the EEG inverse problem [9], that we also formulate as observability properties. Algorithmical and numerical aspects are described, most of them requiring (best constrained quadratic) optimization techniques.…”

“…Spherical head models are classically considered and supposed to be made of 3 spherical homogeneous layers [9]. Put then Ω = B for the unit ball and ∂Ω = S for the unit sphere.…”

Dipole recovery from sparse measurements of its field on a cylindrical geometry

“…In view of recovering the location X d of the dipole, and since the field is known on circles, we adapt and apply the ideas of [1] to our context (see also the references there and [4] for other localization procedures). The key observation is that the denominator of Equation (1) is not polluted by the moment of the dipole, together with the fact that it can be linked to the pole of some rational function whose definition only relies on the data available on each of the 9 circles.…”

“…We conducted experiments using Matlab, with several locations X d and moments M d . For each configuration, we built our data using Equation (1). The parameters values are realistic: R = 111 mm and the sections heights are h 1 = 0 mm, h 2 = 15 mm, h 3 = 30 mm.…”

Weighted extremal domains and best rational approximation

“…To prove the first equality, we appeal to another type of approximation, namely, meromorphic approximation in L 2 -norm on T , for which asymptotics of the error and the poles are obtained below. This type of approximation turns out to be useful in certain inverse source problems [9,34,16]. Observe that |T | 1/p−1/2 h 2,T ≤ h p,T ≤ |T | 1/p h T for any p ∈ (2, ∞) and any bounded function h on T by Hölder inequality, where · p,T is the usual p-norm on T with respect to ds and |T | is the arclength of T .…”