Spatial Prediction of Undersampled Electromagnetic Fields

We present a method to accurately predict the electromagnetic fields in an indoor setting using a limited number of measurements of scattered electromagnetic fields. We consider a room with a single object inside and a transmitter at the center of the room, and generate synthetic measurements using a Boundary Integral (BI) solver. We use Huygens' principle to set up a linear relation between the measured fields and the tangential electric and magnetic fields on the scatterers' surfaces - the latter being the unknowns that we seek to estimate. Since we want to use as few measurements as possible, the above relation is ill-posed and we regularize the solution by means of seeking a minimum Ll-norm solution. The tangential fields are represented in various bases (Fourier series, Wavelet and Discrete Cosine Transform (DCT)) to determine the basis which results in the most sparse representation of the solution. Once the tangential fields are obtained, we use Huygens' principle once again to obtain the field everywhere in the domain and compare it with the true solution. We present a study of the variation in error with increasing observations, increasing number of basis functions and different types of basis functions. We also present a study of the sparsity of the solution in different bases.