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 )) ∊ ℂ 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.
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Andreas Kirsch (2002) studied this question.