Department of Mathematics Mini Course

Multiwave Imaging


Multi-wave imaging methods, also called hybrid methods, attempt to combine the high resolution of one imaging method with the high contrast capabilities of another through a physical principle. 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 of electromagnetic waves (in some specific energy bands) than healthy cells. Photoacoustic tomography (PAT) consists of sending relatively harmless optical radiation into tissues that causes heating which results in the generation of propagating ultrasound waves (the photo-acoustic effect). Such ultrasonic waves are readily measurable. The inverse problem then consists of reconstructing the optical properties of the tissue from these measurements. In Thermoacoustic tomography (TAT) low frequency microwaves, with wavelengths on the order of 1m, are sent into the medium. The rationale for using the latter frequencies is that they are less absorbed than optical frequencies. Transient Elastography (TE) images the propagation of shear waves using ultrasound. Multi-wave imaging methods lead to a rich supply of new mathematical questions that involve elliptic and hyperbolic partial differential equations. The speaker will discuss some of the inverse problems arising in these imaging techniques with emphasis on PAT.

The fourth and fifth lectures will be completely independent of the previous three.

We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arise in several applications in geophysics and medical imaging among others.

The problem can be recast as a geometric problem: Can one determine a Riemannian metric of a Riemannian metric with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will also consider the problem of determining the metric from the scattering relation, the so-called lens rigidity problem. The linearization of these problems involve the integration of a tensor along geodesics, similar to the X-ray transform.


About the speaker

Prof. Gunther Uhlmann is the world’s leading mathematician on inverse problem which focuses on determining the identity of an object by measuring how the object scatters incoming light, sound waves or other types of waves. Such problems are of immense interest to mathematicians, scientists and engineers and have important applications in daily life, including shedding light on the elusive phenomenon of invisibility. In 2003, Prof Uhlmann has proved that by measuring the scattered waves at a boundary, a person may not be able to tell what that object is as he can construct two different objects that give exactly the same scattering. His theorem actually implies that Harry Potter’s cloak can become real. In 2006, physicists proved (independently) that the mathematical transform introduced by Prof. Uhlmann can be used to design Harry Potter’s invisibility cloak.

Prof. Uhlmann obtained his PhD from the Massachusetts Institute of Technology in 1976. After postdoctoral positions at Harvard University, the Courant Institute and MIT, he was on the MIT faculty from 1980 to 1984. He joined the University of Washington in 1984, where he became Walker Family Endowed Professor in Mathematics in 2006.

Prof. Uhlmann was elected to be the Simons Fellow, Fellow of the American Mathematical Society, Finnish Distinguished Professor 2013-17, Rothschild Distinguished Visiting Fellow at Cambridge University and Isaac Newton Institute of Mathematical Sciences 2011 and Chair of Excellence 2012-13 of the Fondation Sciences Mathématiques de Paris. He is also the member of American Academy of Arts and Sciences and the foreign member of the Finnish Academy of Sciences. In 2011, he was awarded the prestigious Bôcher Memorial Prize (awarded once every three or five years) by the American Mathematical Society and the Kleinman Prize (awarded to one person every other year) by the Society of Industrial and Applied Mathematics.

He has a long and distinguished record in serving on the editorial boards of many mathematical journals, including “Inverse and Ill-Posed Problems”, “SIAM Journal on Mathematical Analysis” and “Analysis and PDE”. He was named a Highly Cited Researcher by Institute for Scientific Information (ISI) in 2004.

Lecture notes (HKUST login required)

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