Fourth-year graduate student Laura Petto recently returned from giving a talk at Bowdoin College, where she was drawn to the challenge of pure mathematics as an undergraduate. “I kept studying math because I enjoyed the material, the cooperative nature of the work, and the creativity needed to solve problems,” says Laura. After beginning her Ph.D. program at Dartmouth wanting to study combinatorics and algebra, she switched to applied math and now works with Professor Anne Gelb on image reconstruction problems related to the fields of medical and aerial imaging. Real-world applications of these problems include synthetic aperture radar (SAR), ultrasound, CT, and MRI.

There are “lots of really bad data in medical and aerial imaging,” she says. “The noise in the signal can be larger than the data.” Noise and error come from various sources, including transmission and detector error, environmental interference, and the problems associated with the discretization of a continuous image.

The Shepp-Logan phantom image used to evaluate different reconstruction algorithms, and an example of noisy reconstruction of a modified phantom.

Laura is collaborating with Professor Gelb and Dr. Theresa Scarnati, an applied mathematician at the Air Force Research Laboratory in Dayton, Ohio on improving computational algorithms for reconstructing images and recovering signals. The Air Force is interested in identifying and classifying targets from synthetic aperture radar, but there are a variety of applications for this research, including in medical imaging (MRI and ultrasound). Some of the challenges include very high levels of noise — so much that the signal can be overwhelmed by bad information. Another difficulty is that the collected data most often has to be processed before it can be visualized, and this process often causes information loss. Finally, the algorithms have to be fast, because the output is needed in “real time”. The new algorithm Laura is helping to design is unique in how it combines statistical inference and numerical optimization techniques. “Sometimes I work on application specific problems by exploiting features of the particular imaging device, such as the impact of individual transducer angles in ultrasounds. I love working on problems with interesting real-life constraints and thinking about whether our algorithms can be realistically implemented,” Laura adds.

When not working on image reconstruction, Laura finds plenty of time for outreach: she has been involved with the past two Mathematics REUs at Dartmouth; has led a workshop, been a panelist, or co-organized Sonia Kovalevsky Math Day each year since 2016; and currently volunteers at a local elementary school.

Laura finds time to be involved with Dartmouth Mathematics REU programs and Sonia Kovalevsky Math Day. Click for larger images.

As for what the future holds, “I hope to still be doing math research in 10 years in the areas of optimization and Bayesian statistics and to be using math to solve real world problems. I also hope to be somehow involved in teaching, sharing my love of math with others, and helping to make math more accessible.”