Abstract. We consider the problem of locating point sources in the planar domain from overdetermined boundary measurements of solutions of Poisson's equation. In this paper, we propose a novel technique, termed "analytic sensing," which combines the application of Green's theorem to functions with vanishing Laplacian-known as the "reciprocity gap" principle-with the careful selection of analytic functions that "sense" the manifestation of the sources in order to determine their positions and intensities. Using this formalism we express the problem at hand as a generalized sampling problem, where the signal to be reconstructed is the source distribution. To determine the positions of the sources, which is a nonlinear problem, we extend the annihilating-filter method, which reduces the problem to solving a linear system of equations for a polynomial whose roots are the positions of the point sources. Once these positions are found, resolving the according intensities boils down to solving a linear system of equations. We demonstrate the performance of our technique in the presence of noise by comparing the achieved accuracy with the theoretical lower bound provided by Cramér-Rao theory. 1. Introduction. Source imaging from boundary Cauchy data satisfying the Laplace equation is a classical inverse problem that is of high interest to many fields in engineering. Unfortunately, the problem is ill-posed and additional assumptions about the source configuration are needed to make the solution unique. Typically, one can restrict the class of source distributions by imposing smoothness properties (e.g., Tikhonov regularization [19]) or by imposing a parametric source model.It is when the source configurations are expected to be "sparse" that parametric solutions may prove very useful. This situation is particularly attractive in the electromagnetic setting for applications such as electroencephalography (EEG); e.g., the localization of some type of epileptic foci can be reasonably modeled by pointwise (dipolar) sources [15]. Almost all known techniques rely on the forward model; i.e., computing the boundary data for a given source configuration and iteratively fitting its parameters by least-squares optimization [16]. The corresponding cost function has many local minima and makes the solution very sensitive to the initial guess. Therefore, successful recovery of the parametric sources is often limited to single-dipole models.Despite the practical difficulties in identifying parametric source models, the mathematical uniqueness of the solution has been proven [7], and stability results are available for the case of dipolar and point sources in 2D [5,16] and 3D [20]. Instead

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.

Source localization from EEG surface measurements is an important problem in neuro-imaging. We propose a new mathematical framework to estimate the parameters of a multidipole source model. To that aim, we perform 2-D analytic sensing in multiple planes. The estimation of the projection on each plane of the dipoles' positions, which is a non-linear problem, is reduced to polynomial root finding. The 3-D information is then recovered as a special case of tomographic reconstruction. The feasibility of the proposed approach is shown for both synthetic and experimental data.Index Terms-EEG, source localization, finite rate of innovation, annihilating filter, dipole models, analytic functions

Analytic sensing is a new mathematical framework to estimate the parameters of a multi-dipole source model from boundary measurements. The method deploys two working principles. First, the sensing principle relates the boundary measurements to the volumetric interactions of the sources with the so-called "analytic sensor," a test function that is concentrated around a singular point outside the domain of interest. Second, the annihilation principle allows retrieving the projection of the dipoles' positions in a single shot by polynomial root finding. Here, we propose to apply analytic sensing in a local way; i.e., the poles are not surrounding the complete domain. By combining two local projections of the (nearby) dipolar sources, we are able to reconstruct the full 3-D information. We demonstrate the feasibility of the proposed approach for both synthetic and experimental data, together with the theoretical lower bounds of the localization error.

Source imaging maps back boundary measurements to underlying generators within the domain; e.g., retrieving the parameters of the generating dipoles from electrical potential measurements on the scalp such as in electroencephalography EEG). Fitting such a parametric source model is non-linear in the positions of the sources and renewed interest in mathematical imaging has led to several promising approaches