Analytic sensing: direct recovery of point sources from planar Cauchy boundary measurements

Abstract

Inverse problems play an important role in engineering. A problem that often occurs in electromagnetics (e.g. EEG) is the estimation of the locations and strengths of point sources from boundary data.We propose a new technique, for which we coin the term "analytic sensing". First, generalized measures are obtained by applying Green's theorem to selected functions that are analytic in a given domain and at the same time localized to "sense" the sources. Second, we use the finite-rate-of-innovation framework to determine the locations of the sources. Hence, we construct a polynomial whose roots are the sources' locations. Finally, the strengths of the sources are found by solving a linear system of equations. Preliminary results, using synthetic data, demonstrate the feasibility of the proposed method.

Compressive Sampling of EEG Signals with Finite Rate of Innovation

“…In order to address the nonbandlimitedness of the EEG signals, our compression method will be based on the theory of sampling signals with finite rate of innovation (FRI) [12]. This theory has recently been investigated for a compression technique for electrocardiogram (ECG) signals [13] and neonatal EEG seizure signals [14] as well as for EEG seizure source localisation [15].…”

Compression of neonatal EEG seizure signalswith finite rate of innovation

“…Recently a new theory on sampling and perfectly reconstructing signals with finite rate of innovation (FRI) has been developed [4], [5], and a compression technique has been formulated for electrocardiogram (ECG) signals [6]. This theory has also been applied to EEG seizure source localisation recently [7]. In this paper, a new lossy compression approach which closely models the morphology of the EEG signal is presented based on the theory presented in [4].…”

EEG source localization by multi-planar analytic sensing

“…Parameter relative error (10dB) relative error (5dB) x 1 0.2% 4.6% p 1 3.6% 12% x 2 0.1% 4% p 2 3.2% 10% Table 1. Relative errors with respect to the sphere's radius of the estimated dipoles' positions and moments at different noise levels.…”

“…Finally, the dipolar moments are retrieved by solving a linear system of equations. We extend the 2-D approach, which has been presented before [2], to 3-D by applying it to multiple planes. Each 2-D localization provides us with the projection of the dipoles' positions and moments onto the respective plane.…”