Perfect Reconstruction QMF Banks for Two-Dimensional Applications

The application of subband coding techniques for images has recently received considerable attention [l]-[4]. Vetterli [l] has extended the idea of quadrature mirror filtering [5] for the two-dimensional case: The usefulness of such techniques has been well demonstrated in recent contributions by Woods and O’Neil PI. The purpose of this correspondence is to introduce certain methods whereby two-dimensional quadrature mirror filter (QMF) banks can be designed with complete freedom from all linear distortions (namely, aliasing, amplitude, and phase distortions [S]). Our results here are extensions of the one-dimensional results in [7]; the presentation here can, however, be understood in a self-contained manner, as the developments do not depend on those in [7]. A two-dimensional QMF band with MN channels [l]-[4] is basically a parallel interconnection of MN branches of the form shown in Fig. 1. The branch labeled (m, n) takes the input signal x( n,, n,),-forms a subband signal by~passing through the two-dimensional filter H,,,, (z, , z2) (analysis filters), and decimates the filtered signal by the factor (M, N), where M and N are the decimation ratios in the horizontal and vertical directions, respectively. At the synthesis end, the signal is interpolated, filtered by the synthesis filters F,,(zi, z2) (to remove the images [6]), and then recombined. The operations of the decimators are described as r,,(n,, nz) = v,,( Mn,, Nn,). The interpolators are described by tmn(nl,n2) = r,,(n,/M, n,/N) if n, and n2 are multiples of ‘M and N, respectively, and fmn(nl, ns) = 0 otherwise. In the transform domain, the decimator and interpolator are described, respectively, by the input-output relations R,,(zl,z2) = (~/MN)~~~~~~‘V,,(Z~‘~~~,Z:‘~W~) (where W, = e-“‘jl”, etc.), and Tm,(zl, z2) = Rmn(ziM, z,"). Accordingly, the reconstructed output signal n( n,, n2) (which is the sum of the MN signals p,,,,( n,, n2) in Fig. 1) is related to the input