D. P. Hardin and D. W. Roach, "Multiwavelets prefilters I: Orthogonal prefilters preserving approximation order p 2," preprint, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the multiwavelet case. When the multiscaling functions differ significantly, a direct mapping of samples to coefficients introduces high frequency artifacts. The problem of approximating multiscaling function coefficients from a given set of samples has been the object of a number of recent papers [32, 24, 13, 31, 29, 7, 1]. Lifting provides a way to construct families of multiwavelets for which such preprocessing reduces to a simple polyphase split. Consider the scaling function coefficients for a smooth function f(x) Expanding f(x) into a Taylor series, we obtain Z f(x) x Gamma k)dx = Z [f(k) xf 0 ....

D. P. Hardin and D. W. Roach, Multiwavelet prefilters I: Orthogonal prefilters preserving approximation order p 2, preprint (1997).

....of research partly because they made possible the construction of wavelet systems that are simultaneously orthogonal, symmetric, and FIR. However, it became clear that the implementation of the discrete multiwavelet transform required the design of specialized prefilters, see for example [6] [11], 15] 23] 25] 27] 28] In [14] 18] it was noted that if one wishes to avoid the prefiltering procedure, the multiwavelet basis should have extra approximation properties that they be K balanced . For example, even though the GHM scaling functions OE i;0 are acceptable, the GHM ....

D. P. Hardin and D. W. Roach. Multiwavelet prefilters I: Orthogonal prefilters preserving approximation order p != 2. IEEE Trans. on Circuits and Systems II, 45(8):1106--1112, August 1998.

....4.1 Length The length of a prefilter is the span of the non zero coefficients in the filter. The longer a prefilter s length the worse is the time localisation of the resulting wavelet decomposition. 4. 2 Orthogonality A matrix prefilter is orthogonal if P n Q n Q n Gammak = ffi k I [5]. Hence each postfilter coefficient of an orthogonal prefilter is the transpose of the prefilter coefficient. No repeated signal prefilter may be orthogonal as Gamma doubles the dimensionality of the signal. The only exceptions being trivial, e.g. fl n = ffi n (1; 0) T : 4.3 Degree If a single ....

....lowest degree of polynomial which gives non zero wavelet coefficients, after preprocessing by Q (or Gamma) and transformation by the DMWT. A minimal prefilter for a given multiwavelet has length one and degree greater than or equal to the degree of any prefilter of length one. Hardin and Roach [5] describe a prefilter with same degree as the multiwavelet has vanishing moments, v; as preserving the approximation order v. 4.4 Frequency Response The frequency response of a prefilter is derived below. Assume that the starting coefficients C 0;k are defined on the integers. Thus when a ....

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D.P. Hardin and D.W. Roach. Multiwavelet prefilters I: Orthogonal prefilters preserving approximation order p 2. IEEE Transactions on Circuits and Systems, 1998. To appear.

....will need several initial (boundary) values of v (0) k; Gamma1 ; v (0) k; Gamma2 ; In the case of finite length signals, these numbers can be obtained from the conditions of periodization or symmetric extension (Section 4. 3) Other multiwavelet preprocessing techniques are discussed in [7, 19, 27, 41, 43, 44, 45, 46]. 4.3 Symmetric extension of finite length signals In practice all signals have finite length, so we must devise techniques for filtering such signals at their boundaries. There are two common methods for filtering at the boundary that preserve critical sampling. The first is circular ....

D. Hardin and D. Roach, "Multiwavelet prefilters I: orthogonal prefilters preserving approximation order p 2," preprint, 1997.

....of research partly because they made possible the construction of wavelet systems that are simultaneously orthogonal, symmetric, and FIR. However, it became clear that the implementation of the discrete multiwavelet transform required the design of specialized prefilters, see for example [6] [11], 17] 25] 27] 28] In [15] 19] it was noted that if one wishes to avoid the prefiltering procedure, it is necessary that the multiwavelet basis possesses extra approximation properties that they are K balanced . But in this paper, examples of (symmetric) balanced orthogonal ....

D. P. Hardin and D. W. Roach. Multiwavelet prefilters I: Orthogonal prefilters preserving approximation order p != 2. IEEE Trans. on Circuits and Systems II, 45(8):1106--1112, August 1998.

....: to be preserved by the operator L i.e. L[ 1; 1; 1; 1; 1; 1; 1; 1; 13) However, most of the multiwavelets constructed so far don t even verify this simple requirement as illustrated in Fig. 5. A solution proposed in [12] and generalized in [15] [5], is to add some pre post filtering of the input output signal to adapt it to the spectral imbalance of the filter bank. A. Critical sampling A natural way of prefiltering is to partition the input signal into size 2 vectors chunks and apply on the sequence of vectors the refinement mask A(z) ....

....1; p 2; 1; p 2; 14) 4 The results obtained (Fig. 4) using this trick are of the same order as the ones obtained using a plain Daubechies filter bank with 4 taps. However, the prefilters constructed so far are either destroying the orthogonality [15] or the linear phase [5] of the system thus reducing the interest of multiwavelet based filter banks against usual biorthogonal wavelets based filter banks. B. Non critical sampling Another way of doing pre post filtering is to allow non critical sampling and to construct some projection of the input signal on V 0 . For ....

D.P.Hardin and D.W.Roach, "Multiwavelets prefilters I: Orthogonal prefilters preserving approximation order p 2", Preprint, 1997.

....1] we will need several initial (boundary) values of v (0) k, 1 , v (0) k, 2 , In the case of finite length signals, these numbers can be obtained from the conditions of periodization or symmetric extension (Section 4. 3) Other multiwavelet preprocessing techniques are discussed in [7, 19, 27, 41, 43, 44, 45, 46]. 4.3 Symmetric extension of finite length signals In practice all signals have finite length, so we must devise techniques for filtering such signals at their boundaries. There are two common methods for filtering at the boundary that preserve critical sampling. The first is circular ....

D. Hardin and D. Roach, "Multiwavelet prefilters I: orthogonal prefilters preserving approximation order p # 2," preprint, 1997.

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D. P. Hardin and D. W. Roach, "Multiwavelet prefilters I: Orthogonal prefilters preserving approximation order p 2,"IEEE Trans. on Circuits and System-II, vol. 45, pp. 11061112, August 1998.

....to be an essential and necessary step in applications using unbalanced multiwavelets. In this paper, we develop 2 methods to obtain optimal 2 nd order approximation preserving prefilters for a given orthogonal multiwavelet basis. These procedures use the prefilter construction introduced in [4]. The first prefilter optimization scheme exploits the Taylor series expansion of the prefilter combined with the multiwavelet. The second one is achieved by minimizing the energy compaction ratio of the wavelet coefficients for an experimentally determined average input spectrum. We use both ....

....this corresponds to the property of polynomial reproduction. In applications, one must associate a given discrete signal with a function in the scaling space V 0 (see [2] 10] 12] 14] 13] Such a process is referred to as prefiltering. In a companion to this paper, Hardin and Roach [4] develop a theory for constructing prefilters which preserve both orthogonality and approximation order up to order 2 (that is up to linear polynomial reproduction) In this paper, we use the results in [4] to parameterize all length 3 approximation order preserving prefilters. We then develop 2 ....

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D. P. Hardin and D. W. Roach,"Multiwavelet Prefilters I: Orthogonal prefilters preserving approximation order p 2,"IEEE Trans. on Circuits and System-II, vol 45, no 8, pp 11061112, August 1998.

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D. W. Roach, Multiwavelet prefilters: orthogonal prefilters preserving approximation order p != 3, PhD thesis, Vanderbilt University, Nashville, Tennesee, May 1997.

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D. P. Hardin, D. W. Roach, "Multiwavelet Prefilters I: Orthogonal prefilters preserving approximation order p 2," submitted IEEE Systems and Circuits 1997

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D. P. Hardin and D. W. Roach, "Multiwavelets prefilters I: Orthogonal prefilters preserving approximation order p 2," preprint, 1997.

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D. P. Hardin and D. W. Roach, "Multiwavelets prefilters I: Orthogonal prefilters preserving approximation order p 2," IEEE Trans. Circuits Syst. II, vol. 45, pp. 1106--1112, 1997.

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D. P. Hardin and D. W. Roach, "Multiwavelets prefilters I: Orthogonal prefilters preserving approximation order p != 2 ", preprint, 1997.

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