Coupled Physics Inverse Problems in Medical and Geophysical Imaging

无损成像技术提供了多种方法来对物体内部进行成像。它在医学上的临床诊断和石油勘探中尤其重要。例如，对于医学成像方法，成功主要取决于两个因素：一方面，这些方法应该在健康和不健康组织之间显示出强烈的对比度，另一方面，它们应该具有足够高的分辨率，通常是亚毫米级。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，这些模式未能显示出充分的对比。其他模式，例如那些基于组织的光学、弹性或电学属性的模式，确实显示出高对比度，但受到低分辨率能力的影响。在克服这些限制的道路上取得了突破，将现有的两种模式结合在一起：一种具有大对比度，另一种具有高分辨率。像这样的组合被称为混合情态。在过去的几十年里，各种混合模式被发明和测试。它们的有效性在实验中是合理的，但基本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。在众多的混合成像方法中，光声层析成像(PAT)和热声层析成像(TAT)是医学成像的典型方法，地震电成像(SE)和电震成像(ES)是地球物理成像的典型方法。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在一起。他们的数学模型产生于不同的领域，包括两个共同的阶段：双曲组的源反问题和内数据的反问题。研究者将研究每种模式下的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、局部测量情况、重建算法等方面恢复目标物体的光学参数、电磁参数和地震参数。研究双曲型反问题的主要工具是微局部分析，它是跟踪奇点传播的有力方法。考虑具有内部数据的反问题将基于特定方程的性质。这项研究还有望产生重建算法，这些算法将得到数值实现和测试。

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.