Target reconstruction from scattered far field data

Shape reconstruction of an impenetrable scattering body via the Rayleigh hypothesis

“…At this point we underline the main features of the other Rayleigh hypothesis-based inversion schemes by treating the case of a single realization (only in [17] is the case of more than one realization taken into account) for measurements of the field over the totality of a circle at infinity as in [15][16][17][18].…”

“…wherein σ is the (positive) penalty (regularization) factor. Unfortunately, Angell et al [15][16][17] do not offer a recipe for choosing this factor, although they usually take σ 1. Let us now turn to the JM variant [18] of the Rayleigh hypothesis method.…”

“…In the following we shall be concerned with a new variant of what is termed, for historical reasons (in the context of forward-scattering problems), the Rayleigh hypothesis method for treating the inverse problem. This variant leads to a better state equation than in previous studies [15][16][17][18] employing the Rayleigh hypothesis and alleviates the arbitrariness associated with the choice of regularization parameters at the inversion stage. To answer forthwith the question of why it is of interest to employ the Rayleigh hypothesis in inverse scattering problems, and reveal some of the results obtained herein, it should be underlined that: (i) the Rayleigh hypothesis gives reconstructions with accuracy comparable to that obtained with the extinction theorem and equivalent source methods for a somewhat diminished computational effort; (ii) the Rayleigh hypothesis yields satisfactory reconstructions more simply and at less cost as compared to the integral equation and dual space methods; (iii) the Rayleigh hypothesis gives rise to better reconstructions than by means of all the presently-known methods which rely on explicit (e.g., Kirchoff, intersecting canonical body) approximations of the forward-scattered field.…”

“…It will be noticed that the inversion technique proposed herein is a one-step process (except for the iterations in the black-box subroutine DUNSLF) which does not appeal to any penalty (regularization) parameter. The resolution of the associated forward-scattering problem is implicit in the above scheme and makes itself felt by the presence of the inverse of the matrix E in the function G. In the following section we show how our method differs, at the theoretical level, from the two heretofore-proposed Rayleigh hypothesis methods: the Angell, Kleinman, Kok and Roach (AKKR) technique [15][16][17] and the Jones and Mao (JM) method [18].…”

“…An altogether different, although likewise rigorous, technique makes use of analytic continuation of partial wave representations of the scattered field [14]. The use of a single complete family of functions (partial waves as in the analytic continuation of partial wave representations technique) representation of the scattered field outside the scattering body forms the basis of the Rayleigh hypothesis method for solving both the forward-scattering and inverse-scattering problems [15][16][17][18]. The so-called equivalent source or multiple multipole methods rely on another single complete family of functions representation of 0266-5611/96/061027+29$19.50 c 1996 IOP Publishing Ltd the scattered field involving response functions to fictitious sources within the body for the exterior-boundary-value problem [19][20][21][22].…”