Mathematical Problems in Imaging in Random Media

William Marsh Rice University, Houston TX

Investigators

Abstract

We consider inverse problems for the acoustic wave equation, where the goal is to image strong reflectors in a medium from measurements of the scattered echoes at a remote array of transducers. In known and smooth environments, imaging is done well with coherent migration type imaging techniques. We are concerned with imaging in cluttered media, with rapid and unknown fluctuations of the sound speed, which we model with random processes. The research is focused on theoretical and numerical studies of statistically stable imaging methods in such media, in regimes of significant interaction of the waves with the inhomogeneities in the clutter. Explicitly, we consider a coherent interferometric imaging approach that uses statistical smoothing techniques for suppressing the unwanted clutter effects and emphasizing the coherent part of the data, which can then be processed to get robust images that are independent of the realization of the clutter. The research has the following main parts: (1) Theoretical studies of both statistical stability and resolution of coherent interferometric imaging in random media, for particular wave scattering regimes. (2) Development of algorithms that are capable of estimating adaptively the needed amount of smoothing, without any a priori knowledge of the statistics of the clutter. (3) Theoretical and numerical studies of optimal illumination strategies for achieving the best possible resolution of coherent interferometric images. (4) Coupling of the coherent interferometric imaging of strong reflectors with the estimation of the background (average) sound speed in the medium. This will be done in the context of a mine detection problem. (5) Theoretical and numerical studies of statistically stable imaging techniques for noisy acoustic waveguides. <br><br> We study robust array imaging techniques in cluttered media that arise naturally in applications such as ground or foliage penetrating radar, nondestructive evaluation of aging concrete structures, medical ultrasound, imaging in noisy ocean waveguides, etc. These media consist of a smooth part, which is known or can be estimated, and a fluctuating part, which is due to the presence of small inhomogeneities that are not known and that cannot be estimated. When the interaction of the waves with the clutter is weak, there is a lot of coherence in the scattered echoes, and classic migration (radar) imaging techniques work well. However, these techniques fail in richly scattering environments, in the sense that they give speckled images that are difficult to interpret and that change unpredictably from one clutter to another. Our goal is to develop a robust imaging framework for such scattering environments and to quantify the effect of the inhomogeneities in the clutter on the resolution of the images. The study brings together a combination of ideas from statistics, asymptotic stochastic analysis, numerical simulations, and signal processing and considers specific problems in the following applications: (1) Ultrasound, nondestructive evaluation of aging concrete structures. (2) Land mine detection. (3) Imaging through foliage. (4) Imaging in noisy ocean waveguides. We are presently collaborating with experimentalists on all these applications and an important part of the study will be the testing of our imaging techniques on experimental data.