Wavelet Frames and Applications

Zuowei Shen, National University of Singapore

One of the major driving forces in the area of applied and computational harmonic analysis during
the last two decades is the development and the analysis of redundant systems that produce sparse
approximations for classes of functions of interest.  Such redundant systems include wavelet
frames, ridgelets, curvelets and shearlets, to name a few.  This talk focuses on tight wavelet
frames that are derived from multiresolution analysis and their applications in imaging.
The pillar of this theory is the unitary extension principle and its various generalizations,
hence we will first give a brief survey on the development of extension principles.
The extension principles allow for systematic constructions of wavelet frames that can be
tailored to, and effectively used in, various problems in imaging science. We will discuss some of
these applications of wavelet frames.  The discussion will include frame-based image analysis and
restorations, image inpainting, image denosing, image deblurring and blind deblurring, image
decomposition, and image segmentation.