Analytic Sensing: Noniterative Retrieval of Point Sources from Boundary Measurements

Abstract

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

Sparse Sampling of Signal Innovations

“…EEG measurements during neuronal events like epileptic seizures can be modeled reasonably well by an FRI excitation to a Poisson equation, and it turns out that these measurements satisfy an annihilation property [16]. Obviously, accurate localization of the activation loci is important for the surgical treatment of such impairment.…”

Estimating Localized Sources of Diffusion Fields Using Spatiotemporal Sensor Measurements

“…This is the basis of the reciprocity gap method [42] used in non-destructive testing of solids [42], [43]; it has also been exploited for the identification of heat sources from boundary measurements [44] and for estimating the sources of static fields governed by Poisson's equation in [45]. In this contribution, we propose an extension of the reciprocity gap method to the identification of instantaneous and non-instantaneous sources of diffusion in both space and time, whilst exploiting the use of more stable sensing functions.…”

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

“…Algorithmical and numerical aspects are described, most of them requiring (best constrained quadratic) optimization techniques. Our approach relies on harmonic analysis and function theory (the link with holomorphy comes from harmonicity), as does the work [15]. Compared to other methods (dipole fitting, MUSIC algorithms, [18]), it has the desired feature of providing an estimate of the number of sources (sources that may be correlated, in time).…”