Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

24-52 Passive, Active, and Digital FiltersInputInputWaveletsWaveletsLevel 1Level 1Level 2Level 2Level 3Level 3Level 4Level 4Scaling functionScaling functionLevel 4Level 4(a)DT-CWT(b)Real DWTFIGURE 24.32 Wavelet and scaling-function components at levels 1–4 of 16 shifted step responses of the DT-CWT(a) with 18-tap Q-shift filters and the real DWT (b) with 13,19-tap filters.of Ref. [27], part of which is reproduced here in Figure 24.32. Good transforms have responses that arevisually almost identical (Figure 24.32a), whereas poor transforms (e.g., the DWT) have dramaticallyfluctuating responses (Figure 24.32b).In this relatively short discussion of dual-tree=Hilbert pair ideas, we have not discussed the importantextension of these ideas to two- and three-dimensional datasets, in which a second and very importantadvantage of the complex nature of the wavelet coefficients lies in their ability to produce stronglydirectionally selective wavelets while still retaining the computational advantages of separable filtering[22]. In addition, extension of the dual-tree ideas to M-band filter banks has been achieved by Chauxet al. [28], making even greater directional selectivity possible in two and three dimensions.References1. I. Daubechies. Ten Lectures on Wavelets. Society for Industrial and Applied MathematicsPhiladelphia, PA, 1992.2. M. Vetterli and J. Kovacevic. Wavelets and Subband Coding. Prentice-Hall, Englewood Cliffs, NJ,1995.3. G. Strang and T. Nguyen. Wavelets and Filter Banks. Wellesley-Cambridge Press, Wellesley, MA,1996.4. H. Caglar and A. N. Akansu. A generalized parametric PR-QMF design technique based on Bernsteinpolynomial approximation. IEEE Trans. Signal Proc., 41(7):2314–2321, July 1993.5. D. B. H. Tay, N. G. Kingsbury, and M. Palaniswami. Orthonormal Hilbert-pair of wavelets with(almost) maximum vanishing moments. IEEE Signal Proc. Lett. 13(9):533–536, September 2006.6. M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies. Image coding using wavelet transform.IEEE Trans. Image Proc., 1(2):205–220, April 1992.