Design of 9/7 biorthogonal wavelet filter banks with rational coefficients and high performance

A new technique to construct the 9/7 biorthogonal wavelets was presented. Given the 9/7 synthesis filter represented by a free parameter, the perfect reconstruction condition of the filter bank was formulated as a Diophantine equation, and was solved. Thus, the exact free parameter expressions for the 9/7 biorthogonal wavelet filter banks were derived. By assigning arbitrarily different real numbers to the free parameter, any linear phase 9/7 filter (a pair) with different features could be constructed. As a case study, two kinds of high performance filter banks were constructed, and one had optimal coding gain and rational coefficients whereas the other had coefficients that were approximated by dyadic fractions. Both theoretical analysis and extensive simulations show that the former filter bank has the same compression performance as the popular 9/7 filter bank by Cohen, Daubechies, and Feauveau for image transform coding, and yet requires lower computational complexity; the latter filter bank has the advantage that its discrete wavelet transform (DWT) can be realized by the shift-add operation, which benefits highly efficient implementation of DWT via very large scale integration.