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 radar 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:

Inverse Problems, Modeling and Simulation, May 21-25, Malta

International Conference on Sensing and Imaging, October, 2018

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Some Past Conferences and Workshops: 

Tomography Short Course (introduction to field) Atlanta AMS national meeting, January 3-4, 2005 Proceedings of short course available!

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)

100 years of the Radon Transform, Linz, Austria, March 27-31, 2017

International Conference on Sensing and Imaging, Chengdu, China, June 5-9, 2017

 

Scientific advisory Board: Mobius Imaging

 

Editor:

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)


Student Research:

My students do research on pure and applied mathematics.

Recent Undergraduate Research Students:

Jill Rennie (BA Summa cum Laude 2006) 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, 2006) 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,2008) 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.

 

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 [71].

Joshua Levy (BS 2014) 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.

 

Sarah Reitzes (BS 2016) 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.



Recent and current 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.

 

Anuj Abhishek (current) is working on the microlocal analysis of limited data problems in inverse problems.

 


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Last modified by Todd Quinto on
9/10/2017