Abstract
This paper deals with an inverse scattering problem under a linearized scattering model for a multi-static/multi-frequency configuration. The focus is on the determination of a sampling strategy that allows the reduction of the number of measurement points and frequencies and at the same time keeping the same achievable performance in the reconstructions as for full data acquisition. For the sake of simplicity, a 2D scalar geometry is addressed, and the scattered far-field data are collected. The relevant scattering operator exhibits a singular value spectrum that abruptly decays (i.e., a step-like behavior) beyond a certain index, which identifies the so-called number of degrees of freedom (NDF) of the problem. Accordingly, the sampling strategy is derived by looking for a discrete finite set of data points for which the arising semi-discrete scattering operator approximation can reproduce the most significant part of the singular spectrum, i.e., the singular values preceding the abrupt decay. To this end, the observation variables are suitably transformed so that Fourier-based arguments can be used. The arising sampling grid returns several data that is close to the NDF. Unfortunately, the resulting data points (in the angle-frequency domain) leading to a complicated measurement configuration which requires collecting the data at different spatial positions for each different frequency. To simplify the measurement configuration, a suboptimal sampling strategy is then proposed which, by an iterative procedure, enforces the sampling points to belong to a rectangular grid in the angle-frequency domain. As a result of this procedure, the overall data points (i.e., the couples angle-frequency) actually increase but the number of different angles and frequencies reduce and lead to a measurement configuration that is more practical to implement. A few numerical examples are included to check the proposed sampling scheme.
Highlights
A measurement collection problem has been addressed in the framework of inverse scattering
An optimal sampling strategy for the case of the field collected with a multi-static and multi-frequency configuration in far zone has been proposed. The latter allows minimizing the number of both frequency and spatial measurements by returning several data close to number of degrees of freedom (NDF)
Such a strategy could lead to a complicated measurement configuration which requires collecting the data at different spatial positions for each frequency
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Our problem can be recast as the determination of a sampling representation which is able to approximate the “first” NDF left singular functions To this end, the sampling approach developed in [34] can be exploited along with the classical. An optimal sampling strategy which minimizes the number of data in the anglefrequency domain is proposed for far-field data This is done by suitably transforming the observation variables so that sampling approach mentioned above can be still exploited. To simplify the measurement configuration, a suboptimal sampling strategy is introduced, which through an iterative procedure, enforces the sampling points to belong to a rectangular (not necessarily uniform) grid in the angle-frequency domain In this way, the total number of scattered field data is increased (because they are not the optimum ones). The resulting measurement configuration is easier to implement and what is more the number of angle points at which to collect the data reduces
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