Efficient recursive computation of 1D and 2D-quincunx IIR wavelets

In this paper, we consider the problem of computation of wavelet functions generated from perfect reconstruction filter banks with rational filters. We derive the dilation equations in recursive form and show how to exploit the recursive equations to compute the limit functions. We also derive a recursive equation for the cascade algorithm introduced by Daubechies. The recursive equations link the limit functions directly to the coefficients of the difference equation instead of impulse responses. In both cases, the proposed recursive method has significant computational saving. Furthermore, all the low sensitivity and efficient structures (such as lattice structure) developed in signal processing can be directly applied for wavelet computation. As an application, we show that the class of IIR orthogonal cardinal scaling functions can be computed exactly at all dyadic rationals by using the recursive equations. Recursive formulas for the 2D quincunx wavelets are also derived.