We consider localized instantaneous sources that reside in a 2D dif- fusive environment. Our goal is to reconstruct the induced field from the measurements obtained by distributed sensors. Although the field is non-bandlimited, we capitalize on the fact that it is com- pletely determined by a finite number of parameters to develop a method that allows perfect reconstruction. We demonstrate how these results can be applied in practice in the particular case of heat diffusion. Simulation results confirm the effectiveness of the method.