Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for rⁿ

Although filter banks have been in use for more than a decade, only recently have some results emerged, setting up the theory of general, nonseparable multidimensional filter banks. At the same time, wavelet theory emerged as a useful tool 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 discrete 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 indicating orthogonality relations between the filters in the bank and their shifts on the sampling lattice. A linear phase condition 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: the form of a general orthogonal solution is given and possible linear phase solutions are presented, showing that orthogonality and symmetry are exclusive, independent of the number of dimensions (assuming real FIR filters). Attractive cascade structures 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 that is impossible using separable filters. We also discuss methods for obtaining multidimensional filters from their one-dimensional counterparts. Next, we make a connection to nonseparable wavelets through the construction of iterated filter banks. Assuming the L² convergence of the scaling function, we show that as in the one-dimensional case, the scaling function satisfies a two-scale equation, and the wavelets are orthogonal to each other and their scales and translates (as well as to the scaling function). Then, for the scaling function to exist, we show that it is necessary that the low-pass filter have a zero at aliasing frequencies. Following the discussion on the choice of the dilation matrix, an interesting “dragon” is constructed for the hexagonal case. For the two-channel case in multiple dimensions it is shown that the wavelets defined previously indeed constitute a basis for L²(Rⁿ) functions. Following the 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 filter is designed for which we conjecture that it would lead to a continuous scaling function, and thus, wavelet basis. A biorthogonal example is also given.