Inverse Boundary Problems

本计画将探讨在许多科技领域，包括医学影像学、生物物理学及非破坏性检测等，所产生的几个基本反问题的数学理论。提出了四个主要的研究课题。第一种是各向异性介质中的走时层析成像。在数学术语中，这涉及到从连接边界点的测地线的长度（旅行时间）和其他运动学信息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成像（EIT），也称为卡尔德龙问题。在这种逆方法中，人们试图通过在介质的边界处测量电压和电流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。在这种类型的逆问题中，人们试图通过物理原理组合两种类型的波来确定介质的内部性质;即，一种波具有高分辨率（例如，超声波）和另一种提供高对比度。实例是光声断层成像（PAT）、热声断层成像（达特）和瞬态弹性成像（TE）。第四个主题是关于隐形和伪装：如何使物体对不同类型的波不可见。在反边界问题中，人们试图通过在介质边界处进行测量来确定介质的内部属性。换句话说，一个人能通过测量外部来“看到”内部吗？一个例子是CT扫描，一种常用的医学成像技术。一种是测量身体对X射线的反应，并根据这些信息制作出身体内部的图像。该项目将研究新提出的医学成像技术，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能力相结合。一个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，但具有低对比度。 另一方面，许多肿瘤从电磁波中吸收的能量比健康细胞多得多。光声断层扫描包括将相对无害的光辐射发送到组织中。这导致加热（随着温度在毫开尔文范围内的增加），这导致产生传播的超声波（光声效应）。另一个研究领域是通过测量地震波穿越地球的时间来拍摄地球内部的图像。这样，人们就可以借助地震提供的信息来尝试穿越地球中心的旅行。该项目的最后一个主要研究课题是隐形和隐形研究。人们能使物体在光、声和其他类型的波中不可见吗？

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?