Design of regular 2D nonseparable wavelets/filter banks using transformations of variables

In this paper we are concerned with the design of 2D biorthogonal, 2-channel filter banks where the sampling is on the quincunx lattice. Such systems can be used to implement the nonseparable Discrete Wavelet Transform and also to construct the nonseparable scaling and wavelet functions. One important consideration of such systems (brought into attention by wavelet theory) is the regularity or smoothness of the scaling and wavelet functions. The regularity is related to the zero-property - the number of zeros of the filter transfer function at the aliasing frequency ((Ω1, Ω2) = (π,π) for the quincunx lattice). In general the greater the number of zeros, the greater the regularity. It has been shown previously by the authors that the transformation of variables is an effective and flexible way of designing multidimensional filter banks. However the wavelet aspects of the filter banks (i.e., regularity) were not considered. In this paper we shall show how the zero-property can be easily imposed through the transformation of variables technique. A large number of zeros can be imposed with ease. Arbitrarily smooth scaling and wavelet functions can be constructed. Several design examples will be given to illustrate this.