Mathematical Problems in Imaging in Random Media

随机介质成像的数学问题

• 批准号: 0604008
• 负责人: Liliana Borcea
• 金额: $ 27.8万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-08-01 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604008&HistoricalAwards=false
• 关键词: Mathematical; Problems; Imaging; Random; Media

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.

我们考虑声波方程的逆问题，其目标是通过测量远程换能器阵列的散射回波来成像介质中的强反射器。在已知和光滑的环境中，相干偏移成像技术可以很好地完成成像。我们关心的是在混乱的介质中成像，声速的快速和未知的波动，我们用随机过程来建模。研究的重点是在这种介质中统计稳定成像方法的理论和数值研究，在波与杂波中的不均匀性的显著相互作用制度下。明确地，我们考虑了相干干涉成像方法，该方法使用统计平滑技术来抑制不必要的杂波影响并强调数据的相干部分，然后可以对其进行处理以获得独立于杂波实现的鲁棒图像。本研究主要包括以下几个方面：(1)随机介质中特定波散射条件下相干干涉成像的统计稳定性和分辨率的理论研究。(2)开发能够自适应估计所需平滑量的算法，而不需要任何先验的杂波统计知识。(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)强反射镜相干干涉成像与介质中背景（平均）声速估计的耦合。这将在地雷探测问题的范围内进行。(5)噪声波导统计稳定成像技术的理论与数值研究。brbr我们研究了在杂乱介质中自然出现的强大阵列成像技术，如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、嘈杂海洋波导成像等应用。这些介质由已知或可估计的光滑部分和波动部分组成，这是由于存在未知且无法估计的小的不均匀性。当波与杂波的相互作用较弱时，散射回波具有较强的相干性，传统的偏移（雷达）成像技术效果良好。然而，这些技术在高度散射的环境中失败了，因为它们给出的斑点图像很难解释，并且从一个杂波到另一个杂波的变化是不可预测的。我们的目标是为这种散射环境开发一个强大的成像框架，并量化杂波中的不均匀性对图像分辨率的影响。本研究结合了统计学、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)超声、老化混凝土结构的无损评估。(2)地雷探测。(3)树叶成像。(4)噪声海洋波导成像。目前，我们正在与实验人员就所有这些应用进行合作，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。