The lifting factorization of wavelet bi-frames with arbitrary generators (Q433614)

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.