Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for ⁿ

Abstract-Although filler banks have been in use for more than a decade, only recently have some results emerged, selling up the theory of general, nonseparable multidimensional tiller banks. At the same time, wavelet theory emerged as a useful lool in many different fields of pure and applied mathematics as well as in signal analysis. Recently, it has been shown that the two theories are closely related. Not only does the filter bank perform a discrele wavelet transform, but also under certain conditions it can be used to construct continuous bases of compactly supported wavelets. For multidimensional filter banks, using arbitrary sampling lattices, conditions for perfect reconstruction are given. The orthogonal case is analyzed indi-cating orthogonality relations between the filters in the bank and their shifts on the sampling lattice. A linear phase conditio! follows, as a tool for testing or building banks containing linear phase (symmetric) filters. It is shown how, in some cases, nonseparable filters can be implemented in a separable fashion. The two-channel case in multiple dimensions is studied in detail: (he form of a general orthogonal solution is given and possible linear phase solutions are presented, showing thai orthogonality and symmetry are exclusive, independent of (he number of dimensions (assuming real FIR filters). Attractive cascade struc-tures with specific properties (orthogonality and linear phase) are proposed. For the four-channel two-dimensional case, filters being orthogonal and symmetric are obtained, a solution lhal is impossible using separable filters. We also discuss methods for obtaining multidimensional filters from their one-dimensional counterparts. Next, we make a connection lo nonseparable wavelets through the construction of iterated filter banks. As-suming the L² convergence of the scaling funclion, we show that as in Ihe one-dimensional case, the scaling funclion satisfies a Iwo-scale equation, and the wavelels are orthogonal 10 each other and their scales and translates (as well as to the scaling function). Then, for the scaling function lo exist, we show thai it is necessary that the low-pass filter have a zero at aliasing frequencies. Following Ihe discussion on the choice of the dila-tion matrix, an interesting "dragon" is constructed for (he hexagonal case. For the two-channel case in multiple dimensions it is shown that the wavelets defined previously indeed constitute a basis for L²( ⁿ) functions. Following Ihe result on necessity of a zero, we conjecture that the low-pass filter can be made regular by putting a zero of sufficiently high order at aliasing frequencies. Based on this, a small orthonormal low-pass filler is designed for which we conjecture that it would lead to a contin-uous scaling function, and thus, wavelet basis. A biorthogonal example is also given.