Subband coding based on filter banks is a very efficient technique, but the design of the filter bank is still an open issue, since two main properties, orthogonality and linear phase, cannot be simultaneously achieved in the commonly used dyadic separable schemes, except for the Haar wavelet. Therefore designing efficient filter banks yielding simultaneously orthogonality and phase-linearity is a relevant issue in subband coding. It is possible to achieve these properties with the standard separable sampling scheme with nonseparable filters. The design of such filters is however a difficult task. Three main approaches have been considered in the design of orthogonal nonseparable wavelets :

- Straightforward formulation of the design as an optimization of the filters' coefficients under the quadratic constraints for orthogonality (implying perfect-reconstruction) [9].
- Optimization of cascade forms ensuring structurally orthogonality [4], [6], [5]. It is to be noticed that no complete cascade exists in the multidimensional case, due to the lack of a factorization theorem. The main difficulty here appears when trying to optimize the parameters of the cascade.
- State-space matrix representations [7] look a very promising approach to the design of orthogonal multidimensional filter banks [8].

We consider the design of *M*-band bidimensional FIR filter banks, i.e. *M*
polynomials achieving some desirable properties. A
filter bank implements signal decomposition onto a basis, which we want to
be orthogonal. It is also desirable in image processing to use linear-phase
filters, which means (at least) centrosymmetric or centro-antisymmetric. It
is well known that orthogonality and phase-linearity cannot be achieved for
2-band systems, expect in the case of Haar filters. Thus these properties
cannot hold simultaneously for separable filter banks with sampling matrix
2*I*, which are the most common in image processing. This means that the
simplest system allowing simultaneously orthogonality and phase-linearity is
made of nonseparable filters and sampling matrix
2*I*. Examples under cascade form are proposed in [4]. We consider a
particular family of nonseparable filter banks for sampling matrix 2*I*,
holding structurally orthogonality and centrosymmetry. This family [4] is
defined by polynomial matrix products. These matrices include some angles
that can be chosen arbitrarily.

It is well known that such filter banks may generate wavelet bases [1], [3],
and a necessary condition for that is that the filters vanish at some
aliasing frequencies. In addition, the resulting wavelets can be *N* times
continuously differentiable only if the polynomials
vanish as well as their derivatives up to order *N* at
their aliasing frequencies. It is known for 1-D dyadic systems that in
practice imposing these vanishing moments is an efficient way to obtain some
regularity [2], so that we expect that designing maximally flat nonseparable
filter banks will provide regular wavelet bases.