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The lifting scheme for wavelet bi-frames: theory, structure, and algorithm.
Xiaoyuan Yang1, Yan Shi, Liuhe Chen
1Key Laboratory of Mathematics, Informatics, and Behavioral Semantics, Ministry of Education, China. xiaoyuanyang@vip.163.com
Summary
This study introduces a lifting scheme for wavelet bi-frames, enabling their decomposition into simple filtering steps. This method facilitates the custom design of symmetric bi-frames with enhanced properties like vanishing moments.
Area of Science:
- Applied Mathematics
- Signal Processing
- Harmonic Analysis
Background:
- Wavelet bi-frames offer advanced signal representation capabilities.
- Existing construction methods for wavelet bi-frames can be complex.
- Efficient and customizable design of wavelet bi-frames is an ongoing research area.
Purpose of the Study:
- To present a novel lifting scheme for wavelet bi-frames.
- To provide a theoretical analysis of wavelet bi-frame decomposition.
- To introduce a new method for constructing custom wavelet bi-frames, including symmetric ones.
Main Methods:
- Decomposition of wavelet bi-frames into finite filtering steps.
- Factorization of the polyphase matrix of wavelet bi-frames.
- Utilizing generalized Bernstein basis functions for symmetric filter design.
- Developing an iterative algorithm for increasing vanishing moments.
Main Results:
- Demonstrated that any wavelet bi-frame can be decomposed via a lifting scheme.
- Developed a new construction framework for wavelet bi-frames using prediction and update filters.
- Showcased the design of symmetric bi-frames using generalized Bernstein basis functions.
- Presented an algorithm to achieve arbitrary order of vanishing moments for bi-framelets.
Conclusions:
- The proposed lifting scheme offers a structured and efficient approach to wavelet bi-frame construction.
- The method allows for the custom design of wavelet bi-frames with specific properties, such as symmetry and vanishing moments.
- This work provides practical tools for creating a wide variety of wavelet bi-frames.
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