Home page for Math 584, Mathematics of Medical Imaging, Fall 2006

The course formerly known as Math 582

Instructor: Charles L. Epstein

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The Course

In the last 25 years there has been a revolution in image reconstruction techniques in fields from astrophysics to electron microscopy and most notably in medical imaging. In each of these fields one would like to have a precise picture of a 2 or 3 dimensional object that cannot be obtained directly. The data that is accesible is typically some collection of averages. The problem of image reconstruction is to build an object out of the averaged data and then estimate how close the reconstruction is to the actual object. In this course we introduce the mathematical techniques used to model measurements and reconstruct images. As a simple representative case we study transmission X-ray tomography (CT). In this context we cover some of the basic principles of mathematical analysis, the Fourier transform, interpolation and approximation of functions, sampling theory, digital filtering and noise analysis. The prerequisites are calculus through math 241, some linear algebra as well as basic physics. We will have several field trips to see various types of imaging equipment, in action, at the hospital.

Coursework: Problem sets will be assigned every other week and each student will be asked to prepare a presentation for the class on a subject related to medical imaging.

The textbook is Introduction to the Mathematics of Medical Imaging by C.L. Epstein. Additional material may be taken from the following sources:

• *Principles of Computerized Tomographic Imaging by Avinash C. Kak and Malcolm Slaney This book is now available online at Kak and Slaney
• *Computerized tomography: The new medical X-ray technology by L.A. Shepp and J.B. Kruskal in Amer. Math. Monthly 1978, pages 421-439
• Image reconstruction from Projections by Gabor T. Herman
• Foundations of Medical Imaging by Cho, Jones, Singh
• Intermediate Physics for Medicine and Biology by Russell K. Hobbie
• Radiological Imaging by Barrett and Swindell
• The Mathematics of Computerized Tomography by Frank Natterer
• *Numerical Linear Algebra by Trefethen and Bau
• Foundations of Image Science by Barrett and Meyers. This is an encyclopedic treatment of mathematical methods used in imaging.
  

An excellent general reference for background material in physics is Physics with Illustrative Examples from Medicine and Biology by George Benedek and Felix Villars.

• The class meets on TTh from 10:30-12:00 in DRL, room 3C6.   
• My office hour is 3:30 to 5:00 on Mondays. Contact me by e-mail if you have any questions.
• My office in the Math Department is 4E7 DRL, tel. 8-8476.
• email: cle@math.upenn.edu.  Send e-mail if you have a question or need to contact me.

Announcements

• My office hours for October 2, 2006 are canceled due to the Jewish holiday.
  

Problem Sets

• Problem set 1, due September 26, 2006
• MATLAB worksheet 1, due September 26, 2006.
  
• Problem set 2, due October 10, 2006
• Problem set 3, due October 24, 2006
• MATLAB worksheet 2, due October 24, 2006.
• Problem set 4, due November 7, 2006
• Problem set 5, due November 21, 2006
• Problem set 6, due December 7, 2006
• MATLAB worksheet 3, due December 7, 2006
  

Projects

• The project reports will be given on Wednesday, December 6 at 6:00PM in room A7 of DRL.
• We will visit HUP on Monday, Dec. 4 at 11AM. Please assemble by the entrance to the Gates Pavillion (across from Houston Hall) a little before 11.
  

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