Dates: June 28 - July 16, 2010 Looking for this summer's school? Click here. Overview This summer school gave advanced undergraduates and beginning graduate students the opportunity to study with the University of Washington's integrated Inverse Problems/PDE group. Students attended lectures in the morning, recitation sections in the afternoon, groups with mentors in the afternoon, and introductory tutorials on MATLAB, Fourier Analysis, and Distribution Theory. The RTG grant paid for travel and lodging for US National students, together with a stipend to cover local expenses. Students were housed in campus dormitories for the 3 week period. Access to computer facilities and the student computer lab were also provided.

Minicourses

The main activity of the summer school consisted of two concurrent mini-courses, with a total of 10 hours of lecture each week. A course on the Radon and X-Ray Transforms was taught by Gunther Uhlmann and Peter Kuchment; this course met daily, with one hour of lecture each morning and two lectures on Wednesday. A course on Orthogonal Bases and Multiscale Analysis was taught by Hart Smith , and met for one hour each day except Wednesday.

Outline

The following is an outline of the topics covered in the primary lectures, ordered by week. RT indicates the course of Uhlmann and Kuchment, and OB the course of Smith.

• Week 1:
• Introduction to X-ray tomography (RT)
• Fourier transform and Fourier series (RT)
• Inversion formula for the X-ray transform in two dimensions (RT)
• The Discrete Fourier Transform (OB)
• The Fast Fourier Transform algorithm (OB)
• The Discrete Cosine Transform and applications (OB)
• Week 2:
  • Characterization of the range and support theorem (RT)
  • The Filtered Backprojection algorithm (RT)
  • Convolution and filtering (OB)
  • The Haar basis and signal compression (OB)
• Week 3:
• Introduction to Microlocal Analysis and local tomography (RT)
• Funk and geodesic X-ray transforms on hyperbolic space (RT)
• Filter Banks and Discrete Wavelet transforms (OB)
• 2-dimensional Wavelet transforms and image compression (OB)

Accompanying the morning lectures, recitation sessions were held in the afternoon by RTG supported graduate students. Mark Hubenthal held a series of introductory tutorials on the MATLAB computer package. James Vargo and Gregory Drugan gave a series of lectures to introduce students to the Fourier analysis and distribution theory underlying the lectures of Uhlmann and Kuchment. Additionally, two afternoon workshop series provided students with experience in implementation of the lecture material in MATLAB. Hart Smith held a series of problem sessions covering the discrete Fourier transform, the JPEG algorithm, and the use of wavelets for image and signal analysis. Guest lecturer Leonid Kunyansky, from the University of Arizona, Tucson, gave a two-week series of hands-on workshops that guided students through the implementation of the discrete Radon transform in MATLAB. Students had access to a computer lab with MATLAB equipped workstations throughout the summer school, as well as remote access to MATLAB on University of Washington servers through their laptops.

References

The following were used as textbooks to accompany the mini-courses:

• Introduction to the Mathematics of Medical Imaging by C. Epstein.
• Discrete Fourier Analysis and Wavelets , by S.A. Broughton and K. Bryan.

Please direct questions or comments about the IPDE Summer School to ipdemail@math.washington.edu.