Results and Conclusions

The most flat and regular nonseparable wavelets ever designed have been computed. We give for each flatness order the minimal size for achieving it, the number of remaining free degrees, a parametrization of the family of the minimal size filters for this flatness order, the optimization of the remaining free degrees and the analysis of the resulting filter banks in terms of regularity, frequency characteristics and performance in image compression.

The results of our study concerns four scientific communities :

• From a computer algebra point of view, it is one of the major applications of computer algebra in digital signal processing. This area looks like yielding promising applications, especially in the field of filter banks. In addition, the problem has been solved using the most recent techniques for the Gröbner bases computation.
• From the point of view of the wavelet field, it is the first example of bidimensional orthogonal linear-phase wavelets with such regularity.
• From the point of view of filter bank design, the design examples are convincing: the resulting filters have good frequency characteristics, and it has been shown that the remaining free degrees can be chosen so as to optimize various criteria.
• From the image compression point of view, it is interesting to have new filter banks exhibiting simultaneously orthogonality, centrosymmetry and regularity, and in addition with good frequency characteristics. They give as good results as the best filters known up till know in terms of compression efficiency, without any global optimization of the compression scheme.

Further work might include the study of other cascade forms for filter banks, or a proof that arbitrarily high flatness and regularity can be achieved in this filter bank family.