Math 899: Advances of Frames and Wavelets with Applications
Department of Mathematics
A regular-class format
Tu 2:00 -4:00
Advances of frames and wavelets will be discussed in this class. Frames and wavelets are closely related. For instance, the most useful biorthogonal wavelets (bi-wavelets) are in fact part of extensions of frames. They are known for their superior smoothness and regularity under the same (filter) length. The construction of such useful biorthogonal wavelets has gone beyond conventional basis approach. This includes the dual multiresolution analysis method and the lifting scheme. It turns out that these techniques can be unified in a more general theory of pseudoframes for subspaces (PFFS). PFFS with its great flexibility is believed to have profound impact to applications in the signal processing and communication society. In this class, we will present the theory of PFFS and its applications to the construction of new biorthogonal wavelets of greater smoothness and regularity (while keeping the same filter length). Applications of PFFS in noise removal, deconvolution will also be discussed. Open problems on these applications will be addressed. Some of these studies are supported by the National Science Foundation of America. As part of the purpose of the class, I would be extremely delighted if it can stimulate some interests in collaborative studies of many open problems in this area. The topics of the class will include (but may not be limited to) the following:
¡P Fundamentals of frames and applications.
¡P Theory of pseudoframes for subspaces (PFFS).
¡P Constructions of bi-wavelets using PFFS and applications.
¡P Relationship of bi-wavelets constructed via PFFS and those derived via lifting schemes.
¡P Open problems on applications of PFFS in signal processing and image representation.