Research Interests:
My research involves both pure and applied mathematics: integral geometry and tomography. Integral geometry is the study of transforms that integrate (average) functions over sets in the plane, space, and more complicated sets. Tomography involves finding densities of objects from data such as X-rays from a CT scanner, and I develop algorithms for industrial, scientific, and medical tomography. I am now working on algorithms for electron microscopy and emission tomography.
Publications: For selected publications and other professional information with links to dvi and pdf p/reprints, click here. For all the information, please see my resume (pdf).
Upcoming Conferences and Workshops:
Stay tuned for some sort of electron microscopy conference and
perhaps a conference in Germany.
Recent Conferences:
Tomography Short Course (introduction to field) Atlanta AMS national meeting, January 3-4, 2005
**Proceedings of short course available!**
New Mathematics and Algorithms for 3-D Image Analysis
IMA Imaging year, January 9-12, 2006
Oberwolfach tomography meeting, July 31-August 4, 2006, joint with Alfred Louis and Frank Natterer
PIMS conference on Applied Inverse Problems, minisymposium on New Problems in Tomography
ICIAM minisymposium on Trends in Tomography (with Adel Faridani and Andreas Rieder)
AMS special session on Radon Transforms, Tomography, and related Geometric Analysis, LSU, Baton Rouge, March 26-28, 2008
SIAM Imaging Sciences meeting and SIAM national meeting (I anticipate organizing a tomography minisymposia)
Member Advisory Board of “Integral Geometry and Tomography,” a conference in honor of Jan Boman’s 75th birthday.
Editor: Journal of Fourier Analysis and Applications (Editor-in-Chief: Hans Feichtinger, Publ.: Birkhauser)
Documenta Mathematica (Editor in Chief: Ulf Rehmann)
New: SIAM Journal of Imaging Science (Editor in Chief: Guillermo Sapiro)
Student Research:
My students do research on pure and applied mathematics.
Recent Undergraduates:
Jill Rennie (BA Summa cum Laude '06) did research on stationary sets for the wave equation and showed how stationary sets for the square behave [Properties of stationary sets for the wave equation, Contemporary Mathematics 405(2006)149-155]. Stationary sets are sets on (in this case) a square drum that never move. She created many pictures showing the range of stationary sets. This link shows stationary sets generated by sound waves on a drum that look much like hers. Her work was supported by an NSF REU
Sohhyun (Holly) Chung (BS Summa cum Laude, senior honors thesis, '06) did research on slant-hole SPECT, a new type of emission tomography in which the scanner takes data over lines a fixed angle from the vertical. She developed and tested local algorithms of mine and showed strengths and limitations. She proposed better data acquisition methods [56].
Tania Bakhos (BS Summa cum Laude, senior honors thesis, '08) has been continuing this exciting research on slant-hole SPECT. She has been developing the algorithm so that the reconstructions are exclelent, even with 10% or more noise. Any such backprojection algorithm addes singularities (see [56]). She developed a geometric description of the added singularities, and learned how this came about from microlocal analysis. Her work was supported by an NSF REU
Dan Cuzzocreo (BS '09) is working with me on electron microscopy as a Tufts Summer Scholar, and he has already shown the feasibility of a novel method to get better reconstructions
Recent Graduate Students:
Aleksei Beltukov (Ph.D. '04) developed beautiful and clever inversion methods for the sonar transform on hyperbolic spaces, and he is now at University of the Pacific.
Natalie Velasco (MS '08) did research on better data acquisition geometries and algorithms for novel interoperative cone beam CT scanners. She developed and tested a Lambda CT algorithm for cone-beam CT and discovered better data acquisition geometries.
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Last modified by Todd Quinto on 9/14/08