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.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Victor Roytburd
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Michigan State University
East Lansing
United States
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