Shape reconstruction of an impenetrable scattering body via the Rayleigh hypothesis

Abstract

This work deals with the determination of the shape of a generally non-circular impenetrable cylinder from the way it scatters incident sound. A complete family (of generally non-orthogonal functions) representation of the scattered field is employed to match the total measured field. The data equation and state equation, derived from the Rayleigh hypothesis, are grouped into a single nonlinear cost functional which is minimized by means of the modified Levenberg-Marquardt algorithm to obtain the parametric equation of the boundary of the body in the cross section plane. Numerical examples of the results of the inversion scheme are given for cylinders with both convex and non-convex boundaries illuminated by a plane wave with frequency or angle-of-incidence diversity. Potential applications include robotics (artificial vision), non-destructive evaluation and medical imagery.

Recovery of the location, size, orientation and shape of a rigid cylindrical body from simulated and experimental scattered acoustic field data

“…A1 The great majority of studies concerning inverse scattering problems have relied on synthetic data (e.g., [1,[6][7][8][9]). Often, reconstructions of the boundary of the target have turned out to be surprisingly accurate, considering that it is known that the inverse problem is fundamentally ill-posed ( [6,14]), and that, in some cases, accurate reconstructions were obtained with relatively small amounts of data.…”

“…This may have been one of the reasons why researchers in this field have been admonished by the authors of [6] not to commit the inverse crime, i.e., not to employ the same theoretical model for generating data as for estimating the latter. To avoid this ''crime'', researchers generally add random noise to their synthetic data (e.g., [7,9]), with the additional implication that, in so doing, they simulate experimental error. R12 shows that this may not be a valid assumption.…”

“…One notes therein a pronounced similarity of the main features of the scattered fields, but significant quantitative differences, especially in the phases. Most of these differences appear to have a systematic (i.e., nonstochastic) nature, which suggests that adding random noise to synthetic data is not the proper way (often employed in inverse scattering studies such as [7,9]) of simulating the errors in real data. …”

“…An oft-used method for solving such problems [1,6] is to generate a cost functional expressing the discrepancy between the actual (simulated or measured) data and what the estimator predicts the scattered field to be for a set of trial location and boundary parameters of the body; the sought-for parameters are then chosen to be those trial parameters which give rise to the minimum of the cost functional. A major difficulty with this approach, rarely alluded-to in previous studies (see [7][8][9] for exceptions), is that the cost functional often exhibits not one, but many minima (in [8] the global minimum is chosen to pinpoint the solution, but as shown hereafter, this method for resolving the nonuniqueness is not foolproof), so that the procedure cannot provide a unique solution. We exhibit this fact empirically for simulated data, and observe that the distribution of minima follows a regular pattern, which in fact suggests that there exists a simple law relating the position of the minima to some or all of the soughtfor parameters.…”

Identification of objects in an acoustic wave guide inversion II: Robin–Dirichlet conditions

“…In one of its forms, the procedure enables the reconstruction of the local radius of the body, for a given polar angle, by solving a single non-linear 15 equation. In this regard see the seminal works by Scotti and Wirgin [2][3][4][5][6]. Another variant consists in ÿnding this local radius by minimizing the L 2 cost functional of the aforementioned 17 discrepancy.…”