Digital filters associated with bivariate box spline wavelets

Wenjie He
1997 Journal of Electronic Imaging (JEI)  
Battle-Lemarié 's wavelet has a nice generalization in a bivariate setting. This generalization is called bivariate box spline wavelets. The magnitude of the filters associated with the bivariate box spline wavelets is shown to converge to an ideal high-pass filter when the degree of the bivariate box spline functions increases to ϱ. The passing and stopping bands of the ideal filter are dependent on the structure of the box spline function. Several possible ideal filters are shown. While these
more » ... shown. While these filters work for rectangularly sampled images, hexagonal box spline wavelets and filters are constructed to process hexagonally sampled images. The magnitude of the hexagonal filters converges to an ideal filter. Both convergences are shown to be exponentially fast. Finally, the computation and approximation of these filters are discussed. © 1997 SPIE and IS&T. [S1017-9909 (97) 00604-1] ͮ n . ͑1͒ This expression resembles the Fourier transform of the well-known B-spline function. ͑For this and the other properties of box spline functions, see, e.g., Refs. 5 and 6. For computation with box spline functions, see Refs. 7 and 8.͒ Furthermore, let M l,m,n (x,y)ϭB l,m,n ͓(x,y)ϩc l,m,n ͔, with c l,m,n ϭ͓͑lϩn͒/2,͑mϩn ͒/2͔, where M l,m,n stands for the centered box spline function. The Fourier transform of M l,m,n is M l,m,n ͑ 1 , 2 ͒ϭ͓sinc ͑ 1 /2͔͒ l ͓sinc ͑ 2 /2͔͒ m ϫ͓sinc ͑ 1 ϩ 2 ͒/2͒] n , Paper IST-06
doi:10.1117/12.276850 fatcat:w3yox4alefaenijo5unjrmjvdu