My main interest lies in statistical analysis of manifold-valued processes. Manifold-valued responses in curved spaces frequently arises in many disciplines including computer vision, medical imaging, computational biology, among many others. Many problems in these areas are naturally posed as problems of optimization or statistical inference on nonlinear manifolds. This is because there are some intrinsic constraints on the pertinent features that force the corresponding representations to these manifolds.

More specifically, I have been working on estimation, registration, comparison, analysis, modeling and evaluation of manifold-valued temporal trajectories. These require multiple tools including geometry, modern functional analysis, computer science and statistics. There are many difficulties when analyzing temporal trajectories on nonlinear manifolds. First, the observed data are always noisy and discrete at unsynchronized times. Second, trajectories are observed under arbitrary temporal evolutions. We first addressed the problem of estimating full smooth trajectories on nonlinear manifolds using only a set of time-indexed points, for use in interpolation, smoothing, and prediction of dynamic systems. Furthermore, we introduced a quantity that provides both a cost function for temporal registration and a proper distance for comparison of trajectories. This distance, in turn, is used to define statistical summaries, such as the sample means and covariances, of given trajectories and Gaussian-type models to capture their variability.

My other previous work include shape detection and estimation from 2D and 3D point clouds, shape analysis of parametrized curves.