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Description of the problem

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 :

The design technique we propose derives from the cascade form approaches. It consists in looking at the optimization issue as solving a set of polynomial equations, and in solving these equations using the computer algebra techniques known as Gröbner bases.

We consider the design of M-band bidimensional FIR filter banks, i.e. M polynomials $H_0,\ldots H_{M-1}$ 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 2I, 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 2I. Examples under cascade form are proposed in [4]. We consider a particular family of nonseparable filter banks for sampling matrix 2I, 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 $H_0,\ldots H_{M-1}$ 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.


next up previous contents
Next: The polynomial system of Up: The filter bank design Previous: The filter bank design