Algorithms in Limited Data Tomography

Directed by Todd Quinto

Project Description

I am excited to work with VERSEIM students this summer on tomography. Tomography is the mathematics, science, and engineering that uses indirect data to image the internal structure of objects. It is the math behind X-ray CT scanners, radar, and other cool technologies. We will focus on limited data tomography–when some data are missing. For example, in X-ray CT of large objects, the X-ray scanner can often only X-ray a small part of the object, and this produces limited data. Because of this, algorithms have a hard time imaging the underlying objects and artifacts and missing features can occur. Here are some examples from my VERSEIM research students from last summer.

Figure 1. LEFT: Reconstruction of disk of radius 1/4 at ( 1/2, 0) from limited Compton data (R. Dennis, VERSEIM, 2021). RIGHT: Reconstruction of a disk from truncated Sonar data. Note the missing parts of the disks and the extra artifacts (K. Tsingas, VERSEIM, 2021). The goal of our research is to understand and decrease such artifacts to get better images.

Depending on your interest and expertise, we will work on limited data problems for new types of tomography, Compton tomography or Sonar. During our research project, I will first teach you about the field and the underlying math. Then, you will program an algorithm that we will develop together. You will test it and evaluate the performance of the algorithm–how well it images objects, i.e., reconstructs the inside structure of objects. The images of the object are called reconstructions. Typically, reconstructions from limited data show some parts of the object better than other parts, and they can have streak artifacts. You will conjecture which limitations in the reconstruction are caused by the algorithm and which are intrinsic to the problem, and you will use this information to refine the algorithm. You will learn the mathematics that explains these strengths and limitations. At the end, you will write a report, and some years, students write a journal article.

Desired Background

An understanding of calculus plus some knowledge of Matlab or a high-level computer language are needed for these projects.

About the Professor

I am a professor at Tufts University, and I do research 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) [1], to Compton Tomography [4], to photoacoustic tomography [2].

The article[3] is a basic introduction to my work analyzing visible singularities and artifacts in X-ray tomography is

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 32 years.

REferences
  1. 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.
  2. 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.
  3. E. T. Quinto. Artifacts and Visible Singularities in Limited Data Tomography. Sensing and Imaging, 18, 2017. http://rdcu.be/oRYJ.
  4. 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, july 2020.