In the fall of 2012, I started working with Leonid Kunyansky on inverse problems associated with medical imaging. Our first project developed fast reconstruction techniques to process data gathered from a new imaging apparatus for thermo-acoustic tomography. Most conventional reconstruction algorithms for photo-acoustic (and thermo-acoustic) tomography assume that acoustic waves propagate in a free-space and leave a region of interest after a finite amount of time. This assumption is not accurate when acoustic waves bounce off of detectors or walls. We considered a situation in which a rectangular cavity was surrounded by acoustically hard material. We developed a fast reconstruction algorithm that converges in the L-two norm. A link to our results is provided below.
Our most recent work concerns TAT in reflecting cavities that may not be rectangular and may not have a constant speed of sound. We developed a modified form of time reversal that produces reconstructions which converge weakly in certain situations. These results are summarized in a poster that I presented at the conference ``Inverse Problems and Spectral Theory''. My attendance at this conference was made possible by two funding sources, a travel grant from the Don Wilson Travel Fund, and support from the conference organizers. A link to my poster is provided.
Details of these reults are in the following paper (a link to the pre print is below).
In the summer of 2013, I received funding to be a research assistant to Moysey Brio and Pavel Polynkin. We worked in conjunction with Luke McGuire to develop a computational model of laser ablation. My contribution to the project was to write C code to simulate hydrodynamic processes involving multiple fluids that are modelled by the Euler equations. This work is still in progress and has not yet been published.