On the optimal choice of wavelet bases and signal compression using the wavelet lifting scheme

A simple method is proposed for the design of biorthogonal wavelet bases. Starting from any simple orthogonal wavelets and using the wavelet lifting scheme, we show that new, completely arbitrary, biorthogonal wavelets can be constructed to suit any specific design specification. Two specific applications are considered. The first is to design biorthogonal filter banks with a perfect reconstruction (PR) property. The second application lies in choosing the best wavelet bases for signal compression. In the first application, it is shown that, unlike traditional orthogonal QMF banks, where the analysis and synthesis filters are closely related, biorthogonal filter banks have the feature that the analysis and synthesis filters can be freely chosen to meet any design specifications, while maintaining the perfect reconstruction property. The flexibility in designing these filter banks is achieved by optimizing the parameters of the lifting scheme used in their construction. In the denoising application, the strategy is to use the lifting scheme to get the optimum wavelet bases that lead to the minimum energy norm L₂ up to scale L. Illustrative examples are given to demonstrate the simplicity and efficiency of the proposed design approach.