Prof. Guowei Wei , Michigan State University 

PDE Transform --- A Unified Paradigm for Image Analysis and Multiscale Modeling

The past two decades have witnessed increasing interest in geometric partial differential equations (PDEs).  However, much attention is paid to the use of second-order geometric PDEs as low-pass filters in signal, image and data analysis. My talk focuses on some non-conventional aspects of geometric PDEs.  First, I discuss the construction of arbitrarily high-order geometric PDEs and their utility for image and surface analysis.  Additionally, the design of nonlinear high-pass filters from a coupled PDE system is illustrated. Appropriate combination of geometric PDEs gives rise to the PDE transform.  Like the wavelet transform, the PDE transform is able to decompose signal, image and data into functional modes with controllable time-frequency localizations.  The inverse PDE transform leads to a perfect reconstruction.  Finally, I analyze the geometric feature of the PDE transform that offers a powerful means for the multiscale modeling of biomolecular systems.  The resulting differential geometry based multiscale models encompass discrete atomistic descriptions of macromolecules and continuum macroscopic descriptions of solvent. Applications are discussed to biomedical images, molecular solvation, virus surface formation, protein-protein interactions, multiscale molecular dynamics, and ion channel transport.