Inverse Source Problems for the Helmholtz Equation and the Windowed Fourier Transform

In this work we extend the qualitative reconstruction method for inverse source problems for time-harmonic acoustic and electromagnetic waves in free space, recently developed in [R. Griesmaier, M. Hanke, and T. Raasch, SIAM J. Sci. Comput., 34 (2012), pp. A1544–A1562], to a relevant three-dimensional setting. The reconstruction algorithm relies on the fact that a windowed Fourier transform of the far field pattern of the wave radiated by a compactly supported source approximates an exponential ray transform with a purely imaginary exponent of a mollified version of the source. A filtered backprojection scheme for the standard ray transform applied to the absolute values of the windowed Fourier transform of the far field pattern is used to recover information on the support of the source. We provide the theoretical foundation of the method, discuss a numerical implementation of the fully three-dimensional algorithm, and present a series of numerical examples, including an inverse scattering problem, to support our theoretical results.