One Shot Inverse Scattering via Rational Approximation
We consider the two-dimensional inverse obstacle problem for the Helmholtz equation and aim for localizing several scatterers from the far field of the scattered wave for one fixed incident field and fixed frequency. Our method is independent of the physical properties of the scatterers and is based on a careful investigation of the decay of the tail of the Fourier coefficients of the given far field. Using Prony's method or, equivalently, certain rational Pade approximants, we determine a discrete set of point sources that produces a far field with approximately the same tail of Fourier coefficients. We further show how a repetition of this procedure for different virtual points of origin can be turned into a means for imaging the scatterers. Although this method suffers from a certain lack of stability in the presence of noise, it may provide a useful alternative imaging technique when the scatterers are small inhomogeneities and the number of measurements is as limited as described above.
inverse scattering helmholtz equation prony's method music