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:
Radon Transforms and Geometric Analysis in Honor of Sigurdur Helgason's 85th Birthday, Special Session, AMS National Meeting, Boston, January 6-7, 2012
Geometric Analysis on Euclidean and Homogeneous Spaces, workshop at Tufts University, January 8-9, 2012
(coorganizers Jens Christensen and Fulton Gonzalez)
Recent Conferences:
Tomography Short Course (introduction to field) Atlanta AMS national meeting, January 3-4, 2005 Proceedings of short course available!
Oberwolfach tomography meeting, July 31-August 4, 2006, (Coorganizer with Alfred Louis and Frank Natterer)
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. (Coorganizer with Gunther Uhlmann, Chair, Guillaume Bal, and Allan Greenleaf)
Oberwolfach Conference on Mathematical Problems in Tomography, April 12-16, 2010 (Coorganizer with Martin Burger and Alfred Louis)
Integral Geometry: Analysis and Applications, special session January 6, 2011 at the AMS national meeting at New Orleans (joint with Gaik Ambartsoumian, Gestur Olafsson and Boris Rubin)
Workshop in Analysis and Geometry, at LSU, January 4-5, 2011, (Advisory Committee, organizers: Gestur Olafsson and Boris Rubin).
ICIAM minisymposium, Tomography: the road ahead, July 18-22, 2011 (with Adel Faridani and Andreas Rieder)
Editor:
Inverse Problems (Chief Editor: Alfred Louis)
Journal of Fourier Analysis and Applications (Editor in Chief: Hans Feichtinger, Publ.: Birkhauser)
SIAM Journal of Imaging Science (Editor in Chief: Guillermo Sapiro)
Student Research:
My students do research on pure and applied mathematics.
Recent Undergraduate Research Students:
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 that Jill created. One can generate similar standing waves by putting sand on a drum and inducing vibrations. 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]. Her work was supported by a Tufts Summer Scholarship.
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 Summa cum Laude '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. His work was supported by a Tufts Summer Scholarship.
Howard Levinson (BS Summa cum Laude '11) worked with me in summer 2010 on an REU to develop novel local reconstruction methods for the common offset problem in bistatic radar. He found an optimal differential operator and cutoff function for the algorithm and explained mathematically why they were optimal. He developed the basic microlocal analysis to understand singularities and how the algorithm adds singularities. His senior honors thesis received Highest Thesis Honors. This work is in cooperation with Venky Krishnan, and the three of us are finishing an article now.
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 a professor at the University of the Pacific where he continues his research into sonar transforms.
Natalie Velasco (MS '08) did research on math for novel interoperative cone beam CT scanners. She found data acquisition geometries that are more effective than the standard ones, and she analyzed the effectiveness using microlocal analysis. . She developed and tested a local tomography algorithm for cone-beam CT over arbitrary curves and demonstrated it's efficacy on a new X-ray source curve that she developed.
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Last modified by Todd Quinto on 8/7/11