Syllabus of Math 664, Seminar in Applied Mathematics
Mathematical Methods of Computerized Tomography, Fall 2008

Instructor Peter Kuchment

Office Rm. Blocker 614A

Telephone (979)862-3257

E-mail: kuchment@math.tamu.edu

Home Page: http://www.math.tamu.edu/~kuchment

About the course

Tomography (or computerized tomography, or computer assisted tomography) is a technology (in fact, a complex of different technologies) that enables one to see inside of a non-transparent body. Many of you have probably heard about CAT scanners currently available in most hospitals. CAT here stands for Computer Assisted Tomography. One can easily imagine that if such a technique is available, it is extremely useful in all kinds of applications, e.g. in medical diagnostics (search for tumors, lung deceases, etc.), non-destructive evaluation in industry (checking for interior cracks in materials), oil and water prospection, deep Earth geophysics imaging, and border inspection. The crucial thing about tomography is that there is no ``film" there like in the case of X-ray pictures, so the final high quality images (we present some of them below) are the results of an intricate MATHEMATICAL procedure that belongs to the general area of inverse problems. The mathematics of tomography is extremely beautiful and diverse. It involves manifold techniques that are of general importance for mathematicians (either pure or applied), engineers (especially electrical engineers, biomedical engineers), physicists, and other scientists. Among these one can especially distinguish the so called Harmonic (or Fourier) Analysis, which is one of the most important ideas of the whole mathematics. As the name Harmonic Analysis suggests, it has some relations with music and sound propagation, but in fact it is of much more general significance for most of mathematics and for engineering topics like digital filtration, information transmission, heat conduction, and many others. Differential equations also play a significant role in most of the tomographic fields. Algebraic and computer programming aspects come into play as well.

New tomographic methods that require new engineering and mathematics solutions are being constantly developed (in particular at the mathematics, biomedical engineering, and nuclear engineering departments at TAMU). The class will touch upon various well established techniques (X-ray CAT scan, emission tomography, MRI, ultrasound imaging, etc.), as well as of those that are being developed (or improved) now (optical imaging, diffraction tomography, electrical impedance imaging, electron tomography, neutron tomography, hybrid methods such as thermo/photo-acoustic tomography). Some Matlab codes will be also written.

Recommended texts:

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Last revised August 20th, 2008