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 radar as well as the pure mathematics that helps one understand and refine the algorithms.
Upcoming Conferences and Workshops:
International Conference on Sensing and Imaging, October, 2018
Some Past Conferences and Workshops:
Integral Geometry and Tomography, a conference in honor of Jan Boman’s 75the birthday.
Mathematical Research Communities Conference on Inverse Problems, June 20-26, 2009. (Coorganizer with Gunther Uhlmann, Chair, Guillaume Bal, and Allan Greenleaf)
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)
Summer School on Image Reconstruction, Mathematics & Applications, Munich Germany, July 23-27, 2012
7th international conference, “Inverse Problems: Modeling and Simulation,” May 26-31, 2014, Antalya, Turkey, minisymposium (coorganized with Peter Maass) to honor Alfred Louis.
Oberwolfach Conference on Mathematical Problems in Tomography, August 11-15, 2014 (Coorganizer with Martin Burger and Alfred Louis)
Applied Inverse Problems (Two Minisymposia on Current Applications of Tomography), May 25-29,2015.
Computational and Analytical Aspects of Image Reconstruction, ICERM workshop, July 13-17, 2015.
Eighth international conference, “Inverse Problems: Modeling and Simulation,” May 23-28, 2016, Antalya, Turkey, (birthday minisymposium)
Scientific advisory Board: Mobius Imaging
Inverse Problems (Chief Editor: Simon Arridge)
Journal of Fourier Analysis and Applications (Editor in Chief: Hans Feichtinger, Publ.: Birkhauser)
SIAM Journal of Imaging Science (Editor-in-Chief: Michael Elad)
Howard Levinson (BS Summa cum Laude and Highest Thesis Honors 2011) 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 wrote .
worked with me to rewrite my electron microscopy algorithm so it will read the standard ET data format and he learned the microlocal and Fourier analysis behind the problem. He improved the algorithm and its implementation.
worked with me to develop algorithms and analyze limited data problems in thermoacoustic tomography. She studied graduate microlocal analysis and distribution theory and applied this to rigorously justify her conjectures about limitations of the problem.
Adrian Devitt-Lee (BS/MA 2016) became intrigued by Compton tomography and is showing that the forward operator in several Compton problems is a Fourier Integral Operator. He is using this to analyze singularities.
Ivan Tsenov (BS 2019) developed clever seminorms to quantify local singularities and relate those of functions to those of their Radon transforms. He was supported as a Tufts Summer Scholar.
Michael Thramann (BS 2020) did important algorithm development for problems in Compton CT supported by my NSF grant.
Anuj Abhishek (current) is working on the microlocal analysis of limited data problems in inverse problems.
Last modified by Todd Quinto on 9/10/2017