Optimized orthogonal and biorthogonal wavelets using linear parameterization of halfband filters

A new method for the construction of compactly supported orthogonal and biorthogonal wavelets that can be optimized for specific applications is proposed. Unlike existing methods for the parameterization of orthogonal wavelets, the new method is based on a linear parameterization of all halfband filters. The linear dependence of the halfband filters on a “free” parameter vector considerably facilitates the subsequent numerical optimization through which optimized wavelets are constructed. Biorthogonal wavelets are obtained by factorization of a polynomial representing the parameterized halfband filters while orthogonal wavelets are constructed by imposing nonnegativity on the parameterized halfband filters followed by spectral factorization. The well-known Daubechies orthogonal wavelets and the Cohen-Daubechies-Feauveau biorthogonal wavelets can be viewed as subsets in the parameterized orthogonal and biorthogonal wavelet classes, respectively. Applications of this parameterization-based optimization of wavelets to signal coding are discussed