We summarize recent approaches to estimate electromagnetic fields in an arbitrary two-dimensional scattering domain. Only an estimate of the objects' spatial support in the scattering environment is required in our approach. Using the Huygens' principle and compressive sensing, the tangential electromagnetic fields on the scatterers' surface are estimated by making few field measurements in the scattering domain. Based on the field measurements, two inverse formulations are discussed in the paper: (a) scattered field inverse (SFI) and (b) total field inverse (TFI). For an indoor scattering domain with three scattering objects, the total electric field is recovered with an accuracy of 92%, and the incident field is recovered with an accuracy of 96% when the number of measurements is 0.3 times the number of unknowns in which the problem is formulated. © 2021 EurAAP.