Inverse Boundary Problems

逆边界问题

• 批准号: 1265958
• 负责人: Gunther Uhlmann
• 金额: $ 88万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-06-01 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1265958&HistoricalAwards=false
• 关键词: Inverse; Boundary; Problems

This project will address the mathematical theory of several fundamental inverse problems arising in many areas of science and technology, including medical imaging, geophysics, and nondestructive testing. Four major topics of research are proposed. The first is travel-time tomography in anisotropic media. In mathematical terms this involves the determination of a Riemannian metric (anisotropic sound speed) in the interior of a domain from the lengths of geodesics joining points of the boundary (travel times) and from other kinematic information. The second topic is electric impedance tomography (EIT), also called Calderon's problem. In this inverse method one attempts to determine the conductivity of a medium by making voltage and current measurements at its boundary. The third topic focuses on coupled-physics or hybrid inverse problems. In this type of inverse problem one attempts to determine the internal properties of a medium by combining two types of waves through a physical principle; namely, one wave that has high resolution (e.g., ultrasound) and another that provides high contrast. Examples are photoacoustic tomography (PAT), thermoacoustic tomography (TAT), and transient elastography (TE). The fourth topic is on invisibility and cloaking: how to make objects invisible to different types of waves.In inverse boundary problems one attempts to determine the internal properties of a medium by making measurements at the boundary of the medium. In other words, can one "see" what is inside by making measurements on the outside? An example is a CT scan, a commonly used medical imaging technique. One measures the response of the body to X-rays and makes an image of what is inside from this information. The project will investigate new proposed medical imaging techniques, such as photoacoustic tomography. This combines the high resolution of one imaging method with the high contrast capabilities of another. One important medical imaging application is breast cancer detection. Ultrasound provides a high (sub-millimeter) resolution, but suffers from low contrast. On the other hand, many tumors absorb much more energy from electromagnetic waves than do healthy cells. Photoacoustic tomography consists of sending relatively harmless optical radiation into tissues. This causes heating (with increases of the temperature in the millikelvin range), which results in the generation of propagating ultrasound waves (the photo-acoustic effect). Another area that the project will explore is to image the Earth's interior by measuring the time that it takes for seismic waves to traverse it. In this way one attempts a journey through the center of the Earth with the help of information provided by earthquakes. A final major topic of research in the project is the study of cloaking and invisibility. Can one make objects invisible to light, sound, and other types of waves?

该项目将解决许多科学技术领域中出现的几个基本反问题的数学理论，包括医学成像、地球物理学和无损检测。提出了四个主要研究课题。第一个是各向异性介质中的走时断层扫描。用数学术语来说，这涉及根据边界的测地线连接点的长度（行程时间）和其他运动学信息来确定域内部的黎曼度量（各向异性声速）。第二个主题是电阻抗断层扫描（EIT），也称为卡尔德隆问题。在这种逆方法中，人们试图通过在介质边界处进行电压和电流测量来确定介质的电导率。 第三个主题侧重于耦合物理或混合反问题。在这类反演问题中，人们试图通过物理原理结合两种类型的波来确定介质的内部属性。即，一种波具有高分辨率（例如超声波），另一种波提供高对比度。例如光声断层扫描 (PAT)、热声断层扫描 (TAT) 和瞬态弹性成像 (TE)。第四个主题是隐形和隐身：如何使物体对不同类型的波不可见。在逆边界问题中，人们试图通过在介质边界进行测量来确定介质的内部属性。换句话说，我们能否通过测量外部来“看到”内部的情况？ CT 扫描就是一个例子，这是一种常用的医学成像技术。人们测量身体对 X 射线的反应，并根据这些信息绘制出身体内部的图像。该项目将研究新提出的医学成像技术，例如光声断层扫描。这将一种成像方法的高分辨率与另一种成像方法的高对比度功能结合起来。一项重要的医学成像应用是乳腺癌检测。超声波提供高（亚毫米）分辨率，但对比度低。 另一方面，许多肿瘤比健康细胞吸收更多的电磁波能量。光声断层扫描包括向组织发送相对无害的光辐射。这会导致加热（随着毫开尔文范围内的温度升高），从而产生传播超声波（光声效应）。该项目将探索的另一个领域是通过测量地震波穿过地球所需的时间来对地球内部进行成像。通过这种方式，人们可以借助地震提供的信息尝试穿越地心。该项目的最后一个主要研究课题是隐形和隐形的研究。能否使物体对光、声音和其他类型的波不可见？