SOURCES IN DIFFUSION FIELDS FROM SPATIOTEMPORAL SAMPLES

Consider a diffusion field induced by a finite number of localized and instantaneous sources. In this paper, we study the p roblem of estimating these sources (including their intensiti e , spatial locations, and activation time) from the spatiotempor al samples taken by a network of spatially distributed sensors. We propose two estimation algorithms, depending on whether the ac tivation time of the sources is known. For the case of known acti vation time, we present an annihilating filter based method t o estimate the Euclidean distances between the sources and sens or , which can be subsequently used to localize the sources. For t he case of a single source but with unknown activation time, we show that the diffusion field at any spatial location is a scal ed and shifted version of a common prototype function, and that this function is the unique solution to a particular differe ntial equation. This observation leads to an efficient algorithm t at can estimate the unknown parameters of the source by solving a system of linear equations. For both algorithms proposed i n this work, the minimum number of sensors required is + 1, whered is the spatial dimension of the field. This requirement is independent of the number of active sources. Keywords— diffusion field, source localization, finite rate of innovation, spatiotemporal sampling