Localization of Diffusive Sources Using Spatiotemporal Measurements

Consider a diffusion field induced by a finite number of localized and instantaneous sources. In this paper, we study the problem of estimating these sources (including their intensities, spatial locations, and activation time) from the spatiotemporal samples taken by a network of spatially distributed sensors. We start by considering the case of a single source but with unknown activation time. We show that the diffusion field at any spatial location is a scaled and shifted version of a common prototype function, and that this function is the unique solution to a particular differential equation. This observation leads to an efficient algorithm that can estimate the unknown parameters of the source by solving a system of linear equations. We then extend this result to the case of multiple sources with different activation time. For the algorithm proposed in this work, the minimum number of sensors required is d + 1, where d is the spatial dimension of the field. This requirement is independent of the number of active sources.