Source localization by gradient estimation based on Poisson integral

We consider the problem of localizing the source of a diffusion process. The source is supposed to be isotropic, and several sensors, equipped on a vehicle moving without position information, provide pointwise measures of the quantity being emitted. The solution we propose is based on computing the gradient–and higher-order derivatives such as the Hessian–from Poisson integrals: in opposition to other solutions previously proposed, this computation does neither require specific knowledge of the solution of the diffusion process, nor the use of probing signals, but only exploits properties of the PDE describing the diffusion process. The theoretical results are illustrated by simulations.