Algorithms in Limited Data Tomography

Directed by Todd Quinto

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. Here are some examples from my research students from last summer.

Figure 1. LEFT: Elliptical Radar reconstruction of Square and Circle with full data. RIGHT: Sonar reconstruction of box with partial data. Note how the vertical boundaries are missing.

Depending on your interest and expertise, we will work on limited data problems for novel problems in Radar or new types of tomography, Compton tomography or Photoacoustic tomography. 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 data X-ray CT [1, 4], Radar [3], Compton Tomography [6], and photoacoustic tomography [2].

The best introductions to my work are (Click here for article) [5] for the basic ideas and (Click here for article) [1] for X-ray CT.

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 over 30 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]    C. Grathwohl, P. Kunstmann, E. T. Quinto, and A. Rieder. Microlocal analysis of imaging operators for effective common offset seismic reconstruction. Inverse Problems, 34(11):114001 (24 pages), 2018.

[4]    E. T. Quinto. Singularities of the X-ray transform and limited data tomography in ℝ2 and ℝ3. SIAM J. Math. Anal., 24:1215–1225, 1993.

[5]    E. T. Quinto. Artifacts and Visible Singularities in Limited Data Tomography. Sensing and Imaging, 18, 2017. http://rdcu.be/oRYJ.

[6]    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, jul 2020.