Mathematical Problems in Imaging in Random Media

我们考虑声波方程的反问题，其中的目标是通过测量远程换能器阵列上的散射回波来成像介质中的强反射体。在已知和平滑的环境中，相干偏移类型成像技术可以很好地进行成像。我们关注的是杂乱介质中的成像，以及声速的快速和未知波动，我们用随机过程建模。这项研究主要集中在这类介质中统计稳定成像方法的理论和数值研究，在这种介质中，波与杂波中的非均匀性存在显著的相互作用。明确地说，我们考虑一种相干干涉成像方法，该方法使用统计平滑技术来抑制不想要的杂波效应，并强调数据的相干部分，然后可以对其进行处理以获得与杂波的实现无关的稳健图像。主要研究内容如下：(1)随机介质中特定波散射区域相干干涉成像的统计稳定性和分辨率的理论研究。(2)开发能够自适应地估计所需的平滑量的算法，而不需要任何关于杂波统计的先验知识。(3)实现相干干涉图像最佳分辨率的最佳照明策略的理论和数值研究。(4)强反射体的相干干涉成像与介质中背景(平均)声速的估计相结合。这将在地雷探测问题的背景下进行。(5)噪声声波导统计稳定成像技术的理论和数值研究。Brbr我们研究杂乱介质中的稳健阵列成像技术，这些杂乱介质自然会出现在地面或树叶穿透雷达、老化混凝土结构的无损评估、医学超声波、在嘈杂的海洋波导中成像等应用中。这些介质由已知或可以估计的平滑部分和波动部分组成，波动部分是由于存在未知和无法估计的微小不均匀。当波与杂波的相互作用很弱时，散射回波中存在大量的相干性，经典的偏移(雷达)成像技术可以很好地工作。然而，这些技术在高度分散的环境中失败了，因为它们产生的斑点图像很难解释，并且从一个杂乱到另一个杂乱发生不可预测的变化。我们的目标是为这种散射环境开发一个稳健的成像框架，并量化杂波中的不均匀对图像分辨率的影响。该研究综合了统计学、渐近随机分析、数值模拟和信号处理的思想，并考虑了以下应用中的具体问题：(1)老化混凝土结构的超声无损评估。(2)地雷探测。(3)透过树叶成像。(4)噪声海洋波导中的成像。我们目前正在与实验者就所有这些应用进行合作，这项研究的一个重要部分将是在实验数据上测试我们的成像技术。

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. brbrWe 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.