In this paper, an objective criterion on wavelet filters in proposed. Wavelet transforms are used in number of important signal and image industrial processing tasks including image coding and denoising. The choice of the wavelet filter bank is is very important and is directly linked to the efficiency of the application.
Some criteria have been proposed such as regularity, size of the support of the wavelet and number of vanishing moments. The size of the wavelet support increases with the number of vanishing moments. The wavelet regularity is important to reduce the artifacts. The choice of an optimal wavelet is thus the result of a trade-off between the number of vanishing moments and artifacts. But there is only a partial correlation between filter regularity and reconstructed image quality.
The proposed criterion is composed of two indexes. The first one is a frequency index computed from the aliasing of the filters. The second is a spatial index computed from the spread of the coefficients in spatial domain. From these indexes a filter set can be represented by a point in a criteria-plan. The abscissa is given by the frequency index and the ordinate by the spatial index. The quality of a wavelet filter bank is a trade-off between frequency and spatial quality. So the quality of a wavelet filter bank can be assessed from the position of the corresponding point in the criteria-plan.
The coding and denoising performances are estimated for various filters (including orthogonal splines and Daubechies). These performances are connected to the indexes of each filter bank. The results show that the two proposed indexes allow :
1/ a good estimation of the coding and denoising performances of the
wavelet filters, and 2/ an objective comparison of the filters.
Some clues on the connection between our indexes and the kernel size in the Heisenberg-Gabor formula are also given.