Parametrization construction of biorthogonal wavelet filter banks for image coding

We construct popular biorthogonal wavelet filter banks (BWFBs) having the linear phase and arbitrary multiplicity of vanishing moments (VMs). A novel parametrization construction technique, which is based on the theory of Diophantine equation, is presented and explicit one-parameter expressions of the BWFBs are derived. Using the expressions, any one-parameter family of BWFBs with different VMs can be constructed, and ten families, i.e., 5/7, 6/6, 9/7, 6/10, 5/11, 10/6, 13/7, 6/14, 17/11, and 10/18 families, are constructed here. The free parameter can be used to optimize the resulting BWFBs with respect to other criteria. In particular, in each family, three specific BWFBs with attractive features are obtained by adjusting the free parameter: the first has optimum coding gain and rational coefficients; the second which also has rational coefficients is very close to a quadrature mirror filter (QMF) bank; and the third which has binary coefficients can realize a multiplication-free discrete wavelet transform. In addition, four BWFBs are systematically verified to exhibit performance competitive to several state-of-the-art BWFBs for image compression, and yet require lower computational costs. © 2007 Springer-Verlag London Limited.