On the Digital Filter Associated with Bivariate Box Spline Wavelets

Battle-Lemari e's wavelet has a nice generalization in the bivariate setting. This generalization is called bivariate box spline wavelets. The magnitude of the lters associated with the bivariate box spline wavelets is shown to converge to an ideal high-pass lter when the degree of the bivariate box spline functions increases to 1. The passing and stopping bands of the ideal lter are dependent on the structure of the box spline function. Several possible ideal lters are shown. While these lters work for rectangularly sampled images, hexagonal box spline wavelets and lters are constructed in order to process hexagonally sampled image. The magnitude of the hexagonal lters converges to an ideal lter. Both convergence is shown to be exponentially fast. Finally, the computation and approximation of these lters are discussed. In recent papers ((Saito and Beyklin'93], Aldroubi and Unser'94], and Lai'95]), the asym-potic properties of the lters associated with Daubechies' and Battle-Lemar e's wavelets have been studied. It was shown that the magnitude of the lters associated with Daubechies' wavelet and Battle-Lemari e's wavelet converges to an ideal lter. The Battle-Lemari e wavelet has a nice generalization in the bivariate setting, which is called bivariate box spline wavelets (cf. Riemenschneider and Shen'91]). It is interesting to see the asympotic properties of the lter associated with these bivariate wavelets. Since a bivariate box spline wavelet is not a tensor product of Battle-Lemari e's wavelets, the study of the asympotic properties of bivariate box spline wavelet is not a simple generalization of the study carried out in Aldroubi and Unser'94]. To be more precise about what we are going to study in this paper, we have to introduce some necessary notation and deenitions. Let e 1 = (1; 0) and e 2 = (0; 1) be the standard