Mathematics for Imaging with Waves

Inverse problems arise in all fields of science and technology when one seeks a cause for an observed effect or wants to produce a desired effect. The increase in computing power and the development of powerful algorithms have made it possible to apply the techniques of inverse problems to real-world problems of growing complexity. This research will focus on inverse problems with applications that include a number of medical techniques as well as other problems in imaging, such as locating oil and mineral deposits in the Earth's substructure, creating of astrophysical images from telescope data, finding cracks and interfaces within materials, shape optimization, and many others.

The familiar medical imaging technologies of computed tomography (CT) scans, magnetic resonance imaging (MRI) and ultrasound are examples where inverse problems have played a fundamental role and have helped to save lives.

This research will develop the mathematical theory of several fundamental inverse problems. The first project deals with a relatively new medical imaging technique called electrical impedance tomography. In EIT, one attempts to determine an object's electrical properties by making voltage and current measurements at the boundary of the object.

One potential application is in medical imaging, particularly in detecting pulmonary edema or a cancerous tumor since such anomalies have very different electrical properties in normal situations. The second project is travel time tomography. In this imaging technique, the Principal Investigator (PI) will probe the object with different types of waves, like electromagnetic waves or sound waves.

By measuring the travel times of the waves going through the medium, the PI will attempt to determine the properties of the medium. The third project is an inverse problem arising in nonlinear acoustics. It has applications in ultrasound, particularly in a medical imaging technique called tissue harmonic imaging (THI).

THI is a routinely used component of diagnostic ultrasonography (US), and higher-frequency harmonic waves produced by nonlinear fundamental US wave propagation in the method generate images containing fewer artifacts.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.