Algorithms in Limited Data Tomography
Directed by Todd Quinto
The goal of tomography is to reconstruct or image the internal structure of objects from indirect data, such as from X-ray CT machines. Good reconstruction algorithms are known if all data are available. However, often one has limited data—some data cannot be acquired, for example, when the sensors are on only one side of the object.
You will research a practical problem in limited data in Compton tomography, ultrasound, or Sonar. We will first discuss the background of your problem and the goals of the project. Then I will describe an algorithm for limited data that you will program. Your algorithm will produce a picture or reconstruction of the object. Typically, reconstructions from limited data show some parts of the object better than others, and often they add artifacts–streaks in the reconstructions that are not part of the object. Check out the artifacts below, discovered by my 2022 VERSEIM students, Brandon Mukadziwashe and Claire Callon.
You will produce reconstructions of various objects from a range of limited data sets we will describe. You will evaluate your reconstructions visually to determine what parts of the object are reconstructed well and whether there are artifacts. You will conjecture how this relates to data that are missing. We will use this to improve the algorithm and, together, to better understand the problem and suppress the artifacts. You will learn some of the deep mathematics behind this.
Calculus plus some knowledge of MATLAB or a high-level computer language.
ABOUT THE PROFESSOR
I teach and do research at Tufts on tomography, the math behind X-ray CT scanners, radar, sonar, seismic imaging, and many other imaging techniques. I use the pure mathematical theory of singularities, microlocal analysis, to rigorously understand the strengths and weaknesses in limited tomographic data in a range of problems from X-ray CT (Click the number to read the article) , to photoacoustic tomography  to Compton Tomography . The article  is a basic introduction to my work analyzing visible singularities and artifacts in X-ray tomography.
Doing research with undergrads is one of my favorite activities, and I am looking forward to working with VERSEIM students this summer. Another favorite activity is volunteering at Boston Children’s Hospital, which I have been doing for 33 years.
- L. Borg, J. Frikel, J. S. Jørgensen, and E. T. Quinto. Analyzing reconstruction artifacts from arbitrary incomplete X-ray CT data. SIAM J. Imaging Sci., 11(4):2786–2814, 2018.
- J. Frikel and E. T. Quinto. Artifacts in incomplete data tomography with applications to photoacoustic tomography and sonar. SIAM J. Appl. Math., 75(2):703–725, 2015.
- E. T. Quinto. Limited-Data Tomography, chapter in Computed Tomography: Algorithms, Insight, and just enough Theory, Society for Industrial and Applied Mathematics, 2021.
- J. W. Webber, E. T. Quinto, and E. L. Miller. A joint reconstruction and lambda tomography regularization technique for energy-resolved x-ray imaging. Inverse Problems, 36(7):074002, 2020.