Single Frequency Inverse Scattering by Fourier Extrapolation

One of the main themes of inverse scattering theory for time-harmonic acoustic or electromagnetic waves is to determine information on unknown objects from measurements of scattered waves away from these objects. These inverse problems are non-linear and severely ill-posed.

Besides standard regularization methods, which are typically iterative, a completely different methodology - so-called qualitative reconstruction methods - has attracted a lot of interest recently. These algorithms recover specific qualitative properties of scattering objects in a reliable and fast way. They avoid the simulation of forward models and need no a priori information on physical or topological properties of the unknown objects to be reconstructed. One of the drawbacks of currently available qualitative reconstruction methods is the large amount of data required by most of these algorithms. It is usually assumed that measurement data of waves scattered by the unknown objects corresponding to infinitely many primary waves are given - at least theoretically.

We consider the inverse source problem for the Helmholtz equation as a means to provide a qualitative inversion algorithm for inverse scattering problems for acoustic or electromagnetic waves with a single excitation only. Probing an ensemble of obstacles by just one primary wave at a fixed frequency and measuring the far field of the corresponding scattered wave, the inverse scattering problem that we are interested in consists in reconstructing the support of the scatterers. To this end we rewrite the scattering problem as a source problem and apply two recently developed algorithms - the inverse Radon approximation and the convex scattering support - to recover information on the support of the corresponding source. The first method builds upon a windowed Fourier transform of the far field data followed by a filtered backprojection, and although this procedure yields a rather blurry reconstruction, it can be applied to identify the number and the positions of well separated source components. This information is then utilized to split the far field into individual far field patterns radiated by each of the well separated source components using a Galerkin scheme, and to compute the associated convex scattering supports as a reconstruction of the individual scatterers. We discuss this algorithm and present numerical results.