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, X-ray CT, and emission tomography as well as the pure mathematics that helps one understand and refine the algorithms.
Publications: For 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:
Banff International Research Station Conference on Mathematical Methods in Emerging Modalities of Medical Imaging, October 25-30, 2009
Oberwolfach Conference on Mathematical Problems in Tomography, April 12-16, 2010 (Coorganizer with Martin Burger and Alfred Louis)
SIAM 2010 Imaging Sciences Conference, Chicago, April 11-14, 2010
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, January 9-12, 2006.
Oberwolfach tomography meeting, July 31-August 4, 2006, joint with Alfred Louis and Frank Natterer .
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, July 7-11, 2008.
Integral Geometry and Tomography, a conference in honor of Jan Boman’s 75th birthday.
Mathematical Research Communities Conference on Inverse Problems, June 20-26, 2009.
AIP 2009 Conference, Vienna, July 20-24, 2009 (minisymposium on Tomographic Inverse Problems)
Editor:
Journal of Fourier Analysis and Applications (Editor in Chief: Hans Feichtinger, Publ.: Birkhauser)
Documenta Mathematica (Editors in Chief: Alfred Louis, Ulf Rehmann, and Peter Schneider)
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, Highest Thesis Honors for her 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 and proposed better data acquisition methods. This work appeared in [56].
Tania Bakhos (BS Summa cum Laude, Highest Thesis Honors for her senior honors thesis,'08) has been continuing this exciting research on slant-hole SPECT. She developed the algorithm so that the reconstructions are excellent, even with 10% or more noise. Any such backprojection algorithm adds 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) worked with me on electron microscopy as a Tufts Summer Scholar in 2008. He programmed a novel method of mine to get better reconstructions for single object data. He developed efficient numerical methods and showed that refined methods work effectively even with limited data.
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 over arbitrary curves and demonstrated a better data acquisition curve than the standard ones for local CT.
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Last modified by Todd Quinto on 9/22/09