\input zb-basic
\input zb-ioport
\iteman{io-port 06053445}
\itemau{Shi, Yan; Yang, Xiaoyuan}
\itemti{The lifting factorization of wavelet bi-frames with arbitrary generators.}
\itemso{Math. Comput. Simul. 82, No. 4, 570-589 (2011).}
\itemab
Motivated by the work of Sweldens, the authors study bi-frames with arbitrary generators. An Euclidean algorithm for arbitrary $n$ Laurent polynomials and a factorization algorithm of a polyphase matrix are proposed. It is proved that any wavelet bi-frame can be factorized into a finite number of alternating lifting and dual lifting steps. Based on this concept, a new idea is presented for constructing bi-frames by lifting. For the construction, by using generalized Bernstein basis functions, the authors realize a lifting scheme of wavelet bi-frames with arbitrary prediction and update filters and establish explicit formulas for wavelet bi-frame transforms. By combining the different designed filters for the prediction and update steps, they devise practically unlimited forms of wavelet bi-frames. The filters can be designed to satisfy some desirable properties such as symmetry and high vanishing moments by properly choosing the parameters. Several examples are constructed to illustrate the conclusion.
\itemrv{R\'emi Vaillancourt (Ottawa)}
\itemcc{}
\itemut{wavelet bi-frames; the lifting scheme; generalized Bernstein basis; symmetric framelets; vanishing moments}
\itemli{doi:10.1016/j.matcom.2011.10.001}
\end