Coupled Physics Inverse Problems in Medical and Geophysical Imaging

无损成像技术为物体内部成像提供了多种方法。对医学临床诊断和石油勘探具有重要意义。例如，对于医学成像方法来说，成功主要取决于两个因素：一方面，这些方法应该显示健康和不健康组织之间的巨大对比度，另一方面，它们应该具有足够高的分辨率，通常是亚毫米级的分辨率。计算机断层扫描（CT）、磁共振成像（MRI）或超声成像（UI）是具有这种分辨率的模式的典型例子。然而，这些模式不能表现出足够的对比。其他模式，例如那些基于组织的光学、弹性或电学特性的模式，确实显示出高对比度，但分辨率较低。通过结合两种现有的模式：一种具有大对比度，另一种具有高分辨率，在克服这些限制的道路上取得了突破。这样的组合被称为混合模态。在过去的几十年里，各种混合模式已经被发明和测试。它们的有效性在实验中得到了证明，但其背后的数学理论仍然不完整。该项目旨在补充现有的几种混合模式的数学知识。更好地理解所涉及的数学可以促进模式的进一步改进，为下一代医学和地球物理成像方法提供潜在的候选方法。在其他混合模式中，光声断层扫描（PAT）和热声断层扫描（TAT）是医学成像的典型方法；地震电成像（SE）和电震成像（ES）是地球物理成像的例子。PAT和TAT利用了超声波的高分辨率和光学/电磁波的高对比度，而SE和ES成像将地震波和电磁波耦合起来。出现在不同的领域，它们的数学模型共有两个阶段：双曲系统的逆源问题和具有内部数据的逆问题。研究者将在每个模式中研究这两个阶段以及不同模式之间的关系。研究主要从唯一性、稳定性、局部测量案例、重构算法等方面对目标物的光学、电磁、地震参数进行恢复。研究反双曲型问题的主要工具是微局部分析，微局部分析是跟踪奇异传播的一种有效方法。考虑具有内部数据的逆问题将基于具体方程的性质。该研究还有望产生将在数值上实现和测试的重建算法。

Non-destructive imaging technology provides a variety of methods to image the interior of an object. It is especially important for clinical diagnosis in medicine and for oil prospecting. For medical imaging methods, for example, the success rests primarily on two ingredients: on one hand, the methods should display a large contrast between healthy and unhealthy tissues, on the other hand, they should have a sufficiently high, typically sub-millimeter, resolution. Computerized Tomography (CT), Magnetic Resonance Imaging (MRI), or Ultrasound Imaging (UI) are typical examples of modalities with such a resolution. These modalities however fail to exhibit a sufficient contrast. Other modalities, for example those based on the optical, elastic, or electrical properties of tissues, do display a high contrast but suffer from low resolution capabilities. A breakthrough has been made on the way to overcoming such limitations by coupling two existing modalities: one with large contrast while the other with high resolution. A combination like this is referred to as a hybrid modality. Various hybrid modalities have been invented and tested in the past few decades. Their effectiveness is justified in experiments, yet the underlying mathematical theory is still incomplete. This project aims to complement the extant mathematical knowledge of several hybrid modalities. Better understanding of the involved mathematics could facilitate further improvement of the modalities, providing potential candidates for the next generation of medical and geophysical imaging methods.Among other hybrid modalities, Photo-Acoustic Tomography (PAT) and Thermo-Acoustic Tomography (TAT) are typical methods in medical imaging; and Seismo-Electric (SE) and Electro-Seismic (ES) imagings are examples in geophysical imaging. PAT and TAT take advantage of the high resolution of ultrasound waves and high contrast of optical/electromagnetic waves, while SE and ES imagings couple seismic waves and electromagnetic waves. Arising in different areas, their mathematical models consist in common of two stages: an inverse source problem for a hyperbolic system and an inverse problem with internal data. The investigator will study these two stages in each modality as well as the relations between different models. The research concerns primarily recovery of optical, electromagnetic and seismic parameters of the targeted object from the aspects of uniqueness, stability, partial measurement case, and reconstructive algorithms. The main tool to study the inverse hyperbolic problems is microlocal analysis which is a powerful method for tracking propagation of singularities. Consideration of the inverse problems with internal data will be based on properties of the specific equations. The research is also expected to result in reconstructive algorithms which will be numerically implemented and tested.