The lifting factorization and construction of wavelet bi-frames with arbitrary generators and scaling

Area of Science:

• Signal Processing
• Applied Mathematics
• Harmonic Analysis

Background:

• Wavelet frames are essential in signal processing for analyzing signals at different scales and resolutions.
• Traditional wavelet frame constructions can be complex and lack flexibility.
• Redundant filter banks offer more flexibility than classical two-channel filter banks.

Purpose of the Study:

• To present a novel lifting factorization method for constructing wavelet bi-frames.
• To develop a general construction of bi-frames based on this factorization.
• To provide an explicit formula for constructing specific symmetric bi-frames and an algorithm for increasing vanishing moments.

Main Methods:

• Factorization of wavelet bi-frame polyphase matrices into lifting steps.
• Algebraic manipulation of Laurent polynomials relevant to redundant filter banks.
• Iterative lifting algorithm for enhancing vanishing moments.

Main Results:

• Demonstration that arbitrary wavelet bi-frame polyphase matrices can be factorized via lifting.
• A general construction for wavelet bi-frames is proposed.
• An explicit formula for bi-frames with two scaling, two generators, symmetry, and one vanishing moment is derived.
• An algorithm to arbitrarily increase the order of vanishing moments is presented.

Conclusions:

• The lifting factorization provides an efficient and flexible approach to constructing wavelet bi-frames.
• The proposed methods simplify the construction and enhance the properties of wavelet bi-frames.
• The developed algorithm offers a straightforward way to achieve higher-order vanishing moments.