Localization of point sources for systems governed by the wave equation

Analytic sensing has recently been proposed for source localization from boundary measurements using a generalization of the finite-rate-of-innovation framework. The method is tailored to the quasi-static electromagnetic approximation, which is commonly used in electroencephalography. In this work, we extend analytic sensing for physical systems that are governed by the wave equation; i.e., the sources emit signals that travel as waves through the volume and that are measured at the boundary over time. This source localization problem is highly ill-posed (i.e., the unicity of the source distribution is not guaranteed) and additional assumptions about the sources are needed. We assume that the sources can be described with finite number of parameters, particularly, we consider point sources that are characterized by their position and strength. This assumption makes the solution unique and turns the problem into parametric estimation. Following the framework of analytic sensing, we propose a two-step method. In the first step, we extend the reciprocity gap functional concept to wave-equation based test functions; i.e., well-chosen test functions can relate the boundary measurements to generalized measure that contain volumetric information about the sources within the domain. In the second step-again due to the choice of the test functions - we can apply the finite-rate-of-innovation principle; i.e., the generalized samples can be annihilated by a known filter, thus turning the non-linear source localization problem into an equivalent root-finding one. We demonstrate the feasibility of our technique for a 3-D spherical geometry. The performance of the reconstruction algorithm is evaluated in the presence of noise and compared with the theoretical limit given by Cramer-Rao lower bounds.