| Wavelets are used in such areas of modern science as signal processing, image recognitions, data compressing, etc. With the increasing popularity of wavelets, new methods of creating wavelet bases with predefined properties from elementary wavelets such as the Lazy wavelet, defined in the text, are being developed. This work represents a substantial expansion of one of the popular such methods, called the Lifting Scheme. It is known that any biorthogonal wavelet basis whose matrix functions associated with the low-pass and high-pass filters have Laurent polynomial entries and determinant equal to 1, can be built from the Lazy wavelet using the alternating lifting and dual lifting steps. We extend the method to the case when the entries of the matrix functions associated with the wavelet filters are quotients of Laurent polynomials and relax the determinant condition to the requirement that determinant is not identically equal to zero, but a certain function of the frequency variable. Hence, we show that the extended Lifting Scheme can be used for building of any biorthogonal wavelet basis whose low-pass and high-pass filters are quotients of Laurent polynomials. |
