Wavelet Frames and Image Restorations

One of the major driven forces in the area of applied and computational harmonic analysis over the last decade or longer is the development of redundant systems that have sparse approximations of various classes of functions. Such redundant systems include framelet (tight wavelet frame), ridgelet, curvelet, shearlet and so on. This paper mainly focuses on a special class of such redundant systems: tight wavelet frames, especially, those tight wavelet frames generated via a multiresolution analysis. In particular, we will survey the development of the unitary extension principle and its generalizations. A few examples of tight wavelet frame systems generated by the unitary extension principle are given. The unitary extension principle makes constructions of tight wavelet frame systems straightforward and painless which, in turn, makes a wide usage of the tight wavelet frames possible. Applications of wavelet frame, especially frame based image restorations, are also discussed in details. Mathematics Subject Classification (2000). Primary 42C15; 42C40; 94A08 Secondary 42C30; 65T60; 90C90.