Our group's general research area is in mathematical modeling
and computational techniques for image processing, with some
connections to computer vision and computer graphics applications.
Traditional methods in image processing are based on transform
(either Fourier or wavelets) or stochastic/statistical models.
These methodologies have been highly successful. Recently, there
has been increased interest in a new and complementary approach,
using partial differential equations (PDEs) and differential-geometric
models and techniques. This approach offers a more systematic
treatment of geometric features of images, such as shapes, contours
and curvatures, etc., as well as allowing the wealth of techniques
developed for PDEs and Computational Fluid Dynamics (CFD) to be
brought to bear on image processing tasks.
Notable examples of such
techniques are total variation regularized image restoration methods
and variational level set methods for image segmentation and active
contours. Ideas from shock capturing can also be borrowed to handle
compression of image data with sharp discontinuities (e.g. edges).
These new variational PDE models also call for new computational
techniques to deal with their inherent nonlinearities, singularities
and ill-conditioning. These pose new challenges for computational
mathematics. Another challenge is to combine effectively these PDE
models with the statistical and transform based models, in order to
reap the advantages of each approach.
The latest of our research is portraited in this homepage.
We welcome your feedback and suggestions.
Our research has been supported by the Office of Naval Research
and the National Science Foundation.