Multidimensional causal, stable, perfect reconstruction filter banks

Abstract

Following our earlier one-dimensional (1-D) work, we show that it is possible to have a large family of biorthogonal perfect reconstruction multidimensional (n-D) subband coding filter banks, which are causal and IIR both at the analysis and at the synthesis ends. While the need for perfect reconstruction IIR filter banks are more apparent for multidimensional signals than for 1-D signals, lack of design techniques have made it impossible to use such schemes. We argue that such filter banks, including their multichannel counterparts, should not only exist in a large variety, but we also indicate a number of tools potentially useful in their design. Specifically, a complete parameterization of such filters leading to design methods in 2-D are given. Parameterizations in frequency domain terms as well as in terms of state space descriptions of filters are provided. These latter techniques have potential for better numerical implementation. The development is valid for two-band as well as for multiband subband coding schemes. Examples of bi-orthogonal version of continuous multidimensional wavelets generated by these iterated filter banks, that have not been previously constructed, are worked out. Our development is valid for dilation matrices other than the conventional quincunx subsampling matrix. Thus, for the maximally decimated case considered here, sampling density factor other than two are implicit in the development. Consequently, we are led to consider multiband subband coding schemes for arbitrary sampling decimation and interpolation matrices.