Simplified Framework for Designing Biorthogonal and Orthogonal Wavelets

We initially discuss a new and simple method of parameterization of compactly supported biorthogonal wavelet systems with more than one vanishing moment. To this end we express both primal and dual scaling function filters (low pass) as products of two Laurent polynomials. The first factor ensures required vanishing moments and the second factor is parameterized and adjusted to provide required length and other low pass filter requirements. We then impose double shift orthogonality conditions on the resulting two sets of filter coefficients that make them ‘Perfect Reconstruction’ filters. This modification avoids the use of Diophantine equations and associated spectral factorization method[1,2,3,4] for its derivation. The method is then modified for the parametric and non-parametric orthogonal cases, which includes the derivation for Daubechies filters.