We present a novel method based on Huygens' principle and compressive sensing to predict the electromagnetic (EM) fields in arbitrary scattering environments by making a few measurements of the field. In doing so, we assume a homogeneous medium between the scatterers, though we do not assume prior knowledge of the permittivities or the exact geometry of the scatterers. The major contribution of this work is a compressive sensing-based subspace optimization method (CS-SOM). Using this, we show that the EM fields in an indoor situation with up to four scattering objects can be reconstructed with approximately 12% error, when the number of measurements is only 55% of the number of variables used to formulate the problem. Our technique departs significantly from traditional ray tracing approaches. We use a surface integral formulation which captures wave-matter interactions exactly, leverage compressive sensing techniques so that field measurements at a few random locations suffice, and apply Huygens' principle to predict the fields at any location in space.