The MUSIC-algorithm and the factorization method in inverse scattering theory for inhomogeneous media

We consider the scattering of time-harmonic plane waves by an inhomogeneous medium. The far field patterns u^∞ of the scattered waves depend on the index of refraction 1 + q, the frequency, and directions and of observation and incidence, respectively. The inverse problem which is studied in this paper is to determine the support Ω of q from the knowledge of u^∞ (, ) for all , where the frequency is fixed (and known). Our new approach is based on the far field operator F which is the integral operator with kernel u^∞ (, ). It depends on the data only and is therefore known (at least approximately). The MUSIC algorithm in signal processing uses the discrete version of F, i.e. the matrix F = (u^∞ (_i, _j)) ∊ ^N×N, and determines the locations of the point scatterers. The key idea in both cases is to factorize F and F in the forms where the operator S and the matrix S are 'more explicit' than F and F, respectively, and T, T are suitable isomorphisms. In a first theoretical result we show that the ranges of S and F_# coincide, where F_# is some suitable combination of the real and imaginary parts of F. In the finite dimensional case a simple argument from matrix theory yields that the ranges of S and F coincide. Since F_# is known from the data we can decide for every function on the unit sphere whether it belongs to the range of S or not. We apply this test to the far field patterns of point sources and arrive at an explicit test whether a point z belongs to Ω or not. We will demonstrate that this method also leads to a fast visualization of the obstacle.