My primary research interests are the analysis of partial differential equations (PDE) and mathematical problems in computer vision and image processing. I have, in particular, worked on viscosity solutions and numerical schemes for Hamilton-Jacobi equations, and problems that have an intersection with applied probability. I am also interested in exploring how mathematical analysis of problems can lead to efficient algorithms. Please see my detailed research statement. Below is a short version.
In my doctoral thesis, supervised by Professor Selim Esedoglu and Professor Alfred Hero at the University of Michigan, I studied the asymptotics of non-dominated sorting, which is an important combinatorial problem in multiobjective optimization. Multiobjective optimization is ubiquitous in many engineering and scientific contexts, ranging from the analysis of DNA sequences in bioinformatics to the sorting of hits in web searches. As it turns out, non-dominated sorting is also equivalent to the longest chain problem in probability, which has a long history beginning with Ulam’s famous problem of finding the length of a longest increasing subsequence. The longest chain problem is also intimately related to problems in combinatorics and graph theory, and materials science.
Non-dominated sorting can be viewed as arranging points in Euclidean space into layers, or fronts, by repeated removal of the set of minimal elements with respect to a partial order. I proved that in the large sample size limit, the fronts converge almost surely to the level sets of a function that satisfies a Hamilton-Jacobi PDE in the viscosity sense. I also proved convergence of a fast numerical scheme for this Hamilton-Jacobi PDE, and used this scheme to develop a fast approximate non-dominated sorting algorithm. This work has the potential to be useful in the context of big data streaming problems, which involve constant re-sorting of massive datasets upon the arrival of new samples.
During my doctoral studies, I also spent time researching integral invariants of curves, which are used to describe shapes for computer vision tasks such as shape recognition, matching, and classification. One of the more popular integral invariants is the circular area signature. It is computed by measuring the area of intersection of a ball of radius r > 0 centered on each boundary point with the interior of the shape; the area of intersection is reported as a function of arclength. Despite its widespread use in computer vision, it is currently unknown whether the circular area signature uniquely identifies curves. Recent work has shown uniqueness only in neighborhoods of circles.
I studied the circular area signature for graphs of periodic functions on ℝ, which is akin to considering shapes with only one “side”. The area of intersection is reported as a function of the x-axis in this case. I proved a global uniqueness result and a stability estimate on the inverse mapping. This result helps to rule out false positives for this signature that is so prominent in computer vision. Click here for the full version of my research statement.
In my Master's at Queen's University, I studied Sobolev gradient flows for image diffusion and sharpening applications. Click here for an overview.