Projection method for the inverse source problem of the Poisson equation

This paper proposes an algebraic method for reconstructing the position, strength and number of point sources in a three-dimensional (3-D) Poisson field from its boundary data. Our algorithm uses projection of the 3-D sources onto the xy-plane or the Riemann sphere. It is shown that the xy-plane projection using the lower harmonic components of the potential is appropriate for the case where the sources are scattered in the region, whereas the Riemann sphere projection, which uses algebraic relations involving infinitely higher harmonic components, is appropriate for the case where the sources are concentrated around the surface of the region.