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 and the pure microlocal analysis to understand and refine algorithms for X-ray CT, radar, sonar, seismic imaging, electron microscopy, PAT, and Compton CT.
Upcoming Conferences, Semesters, and Workshops
- Special Semester on Tomography Across the Scales, Johannes Kepler Universität Linz, fall 2021. There will be 5 week-long workshops in different types of tomography.
- Inverse Problems Modeling and Simulation May 22-29, 2022, Mellieha, Malta
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
- Inverse Problems, Modeling and Simulation, May 21-25, Malta
- International Conference on Sensing and Imaging, October, 2018
- Modern Challenges in Imaging In the Footsteps of Allan Cormack, August 5-9, 2019, Tufts University Check out an article about the conference in Tufts Now.
- Geometric Analysis on Euclidean and Homogeneous Spaces, NSF supported workshop in conjunction with AMS special session honoring Sigurdur Helgason, January 8-9, 2012.
- Inverse Problems (Chief Editor: Otmar Scherzer)
- Journal of Fourier Analysis and Applications (Editor in Chief: Hans Feichtinger, Publ.: Birkhauser)
- Journal of Inverse and Ill-posed Problems (Editor-in-Chief: Sergey Kabanikhin)
- Sensing and Imaging advisory board (Chief Editors: Ming Jiang and Nathan Ida)
My students do research on pure and applied mathematics.
- 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 moves. 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 . 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 back-projection algorithm adds singularities (see ). 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 2009) 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 and Highest Thesis Honors 2011) worked with TQin 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 the highest thesis honors. This work is in cooperation with Venky Krishnan, and the three of us wrote .
- Stephen Bidwell (BS Summa cum Laude and Highest Thesis Honors 2012) worked with TQ to understand numerical and microlocal reasons for artifacts in limited angle tomography and he developed algorithms to decrease them. He wrote a senior honors thesis that received the highest thesis honors. It included microlocal analysis of the Radon line transform as well as his experiments to decrease the artifacts and his explanations of the mathematics.
- Joshua Levy (2014) worked with TQ 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 (2015) worked with TQ 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 showed that the forward operator in several Compton problems is a Fourier Integral Operator. He used this to analyze singularities.
- Ivan Tsenov (2018) 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 (2019) did important algorithm development for problems in Compton CT including dramatically improving an Compton CT algorithm supported by TQ’s NSF grant.
- Ally Lee (2019) developed and tested an algorithm of mine for bistatic radar. She analyzed different receiver paths and showed that a circular path is better than a linear path. Her work was supported by my NSF grant and leads to some of TQ’scurrent research.
- Madeleine Duke (2019) developed and tested Limited data filtered back-projection for fan beam CT. She studied novel limited data that don’t come up for parallel beam CT. Maddy worked with the VERSE program supported by TQ’s NSF grant and leads to some of my current research.
- Gloria Kitchens (2019) developed and tested an algorithm for bistatic synthetic aperture radar with a circular receiver path. She highlighted a limitation in the algorithm and developed a fix so the algorithm now images the ground more accurately. Gloria worked with the VERSE program at Tufts supported by my NSF grant and leads to some of my current research. Check out this article that includes info about her.
- Elise (Zetty) Cho (2020) developed and tested a limited data tomography algorithm for sonar and photoacoustic tomography. She found new artifacts in the reconstruction that were not predicted by our previous theory and demonstrated how analytic artifacts can be viewed as limits of numerical artifacts. This points to new microlocal research for TQ.
- Brian Parkes (2020) studied, developed and refined a bistatic synthetic aperture algorithm following from the work of Ally Lee and Gloria Kitchens. He researched artifacts for limited data radar problems and conjectured when the artifacts occurred. This points to new microlocal research for TQ. Brian was a part of the VERSE program at Tufts supported by my NSF grant. Brian is now writing a senior honors thesis at Penn State on his work.
Recent and current Graduate Students:
- Aleksei Beltukov (Ph.D. 2004) 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 2008) 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 (Ph.D. 2018) proved deep support and uniqueness theorems for geodesic transforms on tensors. He was a postdoc at Drexel working with Shari Moskow and is now at University of North Carolina, Charlotte.
- Jon Warneke, (M.S., 2021) worked on the microlocal analysis of bistatic synthetic aperture radar when the transmitter is fixed. He is in the process of showing that the forward operator is a Fourier integral operator, and he showed that the operator satisfies the semi-global Bolker Assumption, so the normal operator (with a cutoff) is a pseudodifferential operator. We are working together on analyzing the microlocal properties of limited data problems.
- Alejandro Coyoli-Valencia (Ph.D., current) is working with Fulton Gonzalez and TQ on spherical Radon transforms using microlocal and harmonic techniques. He already has proven uniqueness theorems and a null space result.