Due to recent progress in digital signal processing, radars, sonars, or ultrasound displays can now handle a large amount of antennas at the same time. Consequently, the quality of the detection and imaging has been greatly improved in simple media. Nevertheless, imaging in complex media is still very challenging. Indeed, in such media, echoes of targets can be strongly and randomly distorted. In this special issue, we focus on a specific problem: detecting or imaging an object which is embedded in a random collection of scatterers. The multiple scattering induces two concomitant effects that blur images. First, ballistic echoes from objects are strongly attenuated. Second, complex interferences between scatterers produce strong clutters. To efficiently image a target in such media, new approaches are proposed. The most efficient ones use multiple antennas, playing both the role of transmitters and receivers and giving access to the so-called multi-static inter-antenna matrix. In the case of a wide-band reference signal, even more information on the medium can be collected.
To take benefit of all this information, original array processing techniques have been developed. All of them are based on a coherent analysis of the inter-element array matrix. One of the oldest is beamforming, where the received signals are simply time-shifted before being summed coherently. At the position of the object, the time-shifted signals coherently interfere. In more complex media, where the propagation does not only induce time shifting, adaptive beamforming has to be used. Then the Green's functions of the medium, which are assumed to be known, at least partially are used as steering vectors. Apart from these two basic approaches, much more efficient imaging techniques are being developed in terms of resolution, signal to noise ratio, etc. Many of them are based on the singular value decomposition (SVD) of the inter-antenna matrix. This generalization of the diagonalization of non-squared matrices is related to a physical invariant. The singular vectors worked out from SVD are the time-reversal invariants of the system, even in the presence of strong multiple scattering. One of the well-known applications of SVD is MUSIC (MUltiple SIgnal Classification). MUSIC takes benefit of the noise to localize a scatterer. The resolution of this nonlinear algorithm overcomes the diffraction limit. However, this technique is very sensitive to mismatch, non-Gaussian noise, etc.
The five articles of this special issue provide a rather complete overview of the recent developments on this topic. Four of them are theoretical and numerical papers and the last one is an experimental one. The work of Park and Lesselier shows a non-iterative algorithm based on SVD and a rigorous asymptotic expansion formula of the scattering amplitude in the presence of inclusions. Thanks to a wide-band SVD, they successfully detect a dielectric or magnetic curved scatterer embedded in a strongly scattering media. The paper by Ishimaru et al. introduces a wide-band generalization of the MUSIC algorithm. With a steering vector corresponding to the coherent Green's function, they successfully localize a scatterer within a medium that is five times thicker than the mean free path. The article of Ammari et al. focuses on detection tests based on the singular values of the response matrix, on the one hand, and on weighted-subspace migration functionals, on the other hand, by using recent tools of random matrix theory and extreme value theory. They show that reflector localization should be performed with reverse-time migration rather than any other form of weighted-subspace migration.
The last two papers have a quite different approach. The goal is to detect or localize a moving target embedded in a multiple scattering media. The basic idea is to acquire two inter-antenna matrices for two different positions of the target. The difference between the two matrices only contains the paths that are scattered at least once by the target. In this way, all the other contributions are washed out. In other words, the clutter is canceled out. Nevertheless the localization is still limited by the multiple scattering. The paper by Fouda and Teixeira suggests using some reference scatterers to estimate the steering vector in order to efficiently localize the target. Finally, Boneau et al. propose an experimental demonstration at ultrasonic scale. Using SVD and differential acquisition, they localize a flow embedded in a thick multiple scattering medium.