I. Cohen, S. Raz, and D. Malah. Orthonormal shift-invariant wavelet packet decomposition and representation. Signal Processing, 57(3):251--270, March 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....although computation is always performed using the fast DWT. The DWT is essentially a discretized version of the expansion in (1) The DWT provides a representation with respect to one wavelet basis whereas the NWT, WPT and NWPT provide representations with respect to a library of bases. Recently, Cohen et al. 1997) describe an O(N log N) best basis algorithm for the NWPT. Flexible decompositions of functions using a library of bases are known in the literature as atomic decompositions, see Mallat and Zhang (1993) and Chen et al. 1995) The non decimated wavelet transforms (NTs) have another useful ....

Cohen, I., Raz, S., & Malah, D. (1997). Orthonormal shift-invariant wavelet packet decomposition and representation. Sig. Proc., 57, 251--270.

....expansion in (2) The DWT provides a representation with respect to one wavelet basis whereas the NWT, WPT and NWPT provide representations with respect to a library of bases (wavelet packets) To represent functions in a library a basis needs to be selected. Coifman and Wickerhauser (1992) and Cohen et al. 1997) describe basis selection methods for wavelet packets and non decimated wavelet packets respectively. Decompositions of functions using a library of bases are known in the literature as atomic decompositions, see Mallat and Zhang (1993) and Chen et al. 1996) Property: translation equivariance ....

Cohen, I., Raz, S., & Malah, D. (1997). Orthonormal shift-invariant wavelet packet decomposition and representation. Sig. Proc., 57, 251--270.

.... statistical interest is the wavelet packet transform (see e.g. Wickerhauser (1994) which has many potential uses such as dimension reduction and classification (Saito (1994) Saito and Coifman (1996) Coifman and Saito (1994) Learned and Willsky (1995) the stationary wavelet packet transform (Cohen, Raz and Malah (1998), Nason, Sapatinas and Sawczenko (1997) and the multiple wavelet transform (this is where there is more than one mother and father wavelet, see Downie and Silverman (1996) 5 Estimators for data with correlated noise 5.1 Level dependent thresholding Johnstone and Silverman (1997) consider the ....

Cohen, I., Raz, S., & Malah, D. (1998). Orthonormal shift-invariant wavelet packet decomposition and representation. Sig. Proc., 57, (to appear).

.... limiting conditions on the scaling function [57, 3, 4] Recently, several authors proposed independently to extend the library of bases, in which the best representations are searched for, by introducing additional degrees of freedom that adjust the time localization of the basis functions [47, 11, 12, 23, 36]. It was proved that the proposed modifications of the wavelet transform and wavelet packet decomposition lead to orthonormal best basis representations which are shift invariant and characterized by lower information costs. The principal idea is to adapt the down sampling when expanding each ....

....parity properties of the seg mented signal at the end points. We showed that a locally adapted polarity yields a manageable computational complexity which is comparable to that of the LCD. A similar procedure is available for other types of bases, most notably wavelets and wavelet packet bases [47, 11, 12, 24, 36]. In these cases, the adaptive relative shift is equivalent to an adaptive down sampling when expanding each parent node. By choosing between the odd and even samples after the low pass and high pass filters, we obtain an orthonormal best basis representation which is shift invariant and of a ....

I. Cohen, S. Raz and D. Malah, "Orthonormal shift-invariant wavelet packet decomposition and representation", Technical Report, EE PUB No. 953, Technion - Israel Institute of Technology, Haifa, Israel, Jan. 1995.

....join neighboring basis functions, invariably degrades the energy concentration and may artificially split a given signal component into several frequency bands. In this paper, we propose an adaptive suppression of interference terms using the shift invariant wavelet packet decomposition (SIWPD) [6,9]. A prescribed signal is expanded on its best SIWPD basis, and subsequently transformed into the Wigner domain. The interference terms are controlled by adaptively thresholding the cross WD of interactive basis functions according to their distance and amplitudes in an idealized time frequency ....

....of the Fig. 1. Test signal g(t) consisting of a short pulse, a tone and a nonlinear chirp. input signal with respect to the basis functions. Furthermore, the time frequency tilings, produced by the best basis expansions, do not generally conform to standard time frequency energy distributions [9]. Hence we employ modified versions which induce shift invariance, lower information cost and improved time frequency resolution [6 8] Let us specifically consider the shift invariant wavelet packet decomposition (SIWPD) 6,9] The library of bases is extended by introducing an additional ....

[Article contains additional citation context not shown here]

I. Cohen, S. Raz, D. Malah, Orthonormal shift-invariant wavelet packet decomposition and representation, Signal Processing 57 (3) (March 1997) 251---270.

....C [22Pn2p,22P(n2pq l) 3) where g2p z g2 q 1 and n2p = naiv 2, implies ml mod 2 = m2. 4) Then E generates an orthonormal (ON) basis for V = B U i.e. U( eE . is an ON basis, and the set 0,0,0, of all E as specified above generates a SWP library. The proof is detailed in [10]. The SWP library thus created is larger than the WP library proposed in [1] by a square power. The SWP library can still be cast into a tree configuration, where each node is indexed by ( n, m) and represents the subspace U Condition (ii) above is equivalent to demanding that the relative shift ....

....to zero. Proposition 2 The best basis stemming from the previously described recursive algorithm is shift invariant. Proof: Let f, g V where f and g are identical to within a time shift , i.e. g(x) f(x q2 3) Let A and A denote the best bases for f and g, respectively. It can be shown [10] that B . C A (8) implies Bm, C As, h = m q) mod (2 ) 9) for all m,n Z and z . Hence, Ay and A s are identical to within a time shift . q. An alternative view of SIWPD is facilitated via filter bank terminology [11] Accordingly, each parent node is expanded by high pass and ....

I. Cohen, S. Raz and D. Malah, "Orthonormal shift- invariant wavelet packet decomposition and representation ", submitted.

....representations resulting from 2# di erent shift options, where # is the number of nodes in the best expansion tree, and choose that representation (shift option) which minimizes the entropy. This procedure is sub optimal compared with the shiftinvariant wavelet packet decomposition (SIWPD) [5,6], since the expansion tree is determined by the averaged entropy. Additionally, the shift options in step (iii) are examined one by one, whereas the SIWPD not only provides a recursive selection method for the optimal shift, but also o ers an inherent trade o between the computational complexity ....

....selection method for the optimal shift, but also o ers an inherent trade o between the computational complexity and the information cost. In this paper, we present a translation invariant denoising method, based on the SIWPD and the MDL criterion. An extended library of waveletpacket bases [6] is employed for generating a collection of competing models, and the MDL principle is applied for approximating the description length of the observed noisy data. We show that minimum description length is attainable by optimizing the expansion tree associated with the SIWPD. The optimal signal ....

[Article contains additional citation context not shown here]

I. Cohen, S. Raz, D. Malah, Orthonormal shift-invariant wavelet packet decomposition and representation, Signal Processing 57 (3) (March 1997) 251}270.

....representations resulting from 2 different shift options, where n denotes the number of nodes in the best expansion tree, and choose that representation (shiftoption) which minimizes the entropy. This procedure is sub optimal compared with the Shift Invariant Wavelet Packet Decomposition (SIWPD) [9], since the expansion tree is de termined by the averaged eniropy. Addiiionally, the shifi opiions in siep (iii) are examined one by one, whereas the SIWPD not only provides a recursire seleciion method for the optimal shill, but also offers an inhereni irade off bet veen the computational ....

....showy lhai lhis estimator, combined vith the recenlly iniroduced modified Wignet disiribuiion (MWD) 10] yields robusi time frequency represeniaiions lhai are characierized by high resoluiion, i,e, high conceniraiion and suppressed interference terms. An extendedlibrary of wavelet packet bases [9] is employed for generaling a collec tion of competing models, and the MDL principle is applied for approximating the descripiion lengih of the observed noisy daia. We showy lhai minimum descripiion lengih is aiiainable by optimizing the expansion iree associaied vith the SIWPD. The optimal ....

[Article contains additional citation context not shown here]

I. Cohen, S. Raz and D. Malah, "Orthonormal shift- invariant wavelet packet decomposition and representation ", Sig. Proc., Vol. 57, Mar. 1997, pp. 251 270.

....the shift invariant best basis decompositions. Finally, in Chapter 6 we conclude with a summary and discussion on future research directions. We would like to note that Chapters 2 and 3 and part of Chapter 4 are the detailed and expanded version of our published materials. Chapter 2 is based on [27, 28], Chapter 3 is based on [29, 30, 31, 32] and part of Chapter 4 is based on [31, 33] Additional manuscripts [34, 35, 36, 37, 38] based on Chapters 4 and 5, are about to be published. 1.4 Background A natural framework for the understanding of wavelet bases, and for the construction of new ....

I. Cohen, S. Raz and D. Malah, "Orthonormal shift-invariant wavelet packet decomposition and representation", Signal Processing, Vol. 57, No. 3, Mar.

....decomposition in an extended library of wavelet packets [12] The resultant best basis representation is preferable to the standard wavelet packet decomposition (WPD) 13] due to its desirable properties. Namely, shift invariance, lower information cost and improved timefrequency resolution [14]. The interference terms in the Wigner domain are controlled by adaptively thresholding the cross WD of interactive basis functions according to their distance in an idealized time frequency plane. The distance measure is related to a degree of adjacency by weighing the Euclidean timefrequency ....

....local cosine decomposition (LCD) 13] is the lack of shift invariance. Hence we employ modified versions which induce shift invariance, lower information cost and improved time frequency resolution [12, 16, 17] Let us specifically consider the shift invariant wavelet packet decomposition (SIWPD) [12, 14]. The library of bases is extended by introducing an additional degree of freedom that adjusts the time localization of the basis functions. This degree of freedom is practically incorporated into the search algorithm as an adaptive even odd downsampling. That is, following the low pass and ....

[Article contains additional citation context not shown here]

I. Cohen, S. Raz and D. Malah, "Orthonormal shiftinvariant wavelet packet decomposition and representation ", submitted to Signal Processing (also EE PUB No. 953, Technion - IIT, Haifa, Israel, Jan. 1995).

....to resolving multicomponent signals. Each component is determined as a partial sum of basis functions over an equivalence class in the time frequency plane. 2. MODIFIED WIGNER DISTRIBUTION Let g(t) P c (t) be the shift invariant wavelet packet decomposition (SIWPD) of the signal g [7]. Then, the MWD for g is defined by Tg = X 2 jc j 2 W 2 X f; 0 g2 Gamma Refcc 0 W ; 0 g (1) where W is the auto WD of and W ; 0 is the cross WD of and 0 : W ; 0 (t; Z (t =2) 0 (t Gamma =2)e Gammaj d ; 2) W (t; j W ; t; 3) ....

.... tile in the timefrequency plane which is positioned about t = 2 Gamma k 2 GammaL m (2 L Gamma Gamma 1)Ch (Ch Gamma Cg )R(n) 8) 2 GammaL [GC Gamma1 (n) 0:5] 9) where Ch and Cg are respectively the energy centers of the low pass and high pass quadrature filters [7], R(n) is an integer obtained by bit reversal of n in a L Gamma bits binary representation, and GC Gamma1 is the inverse Gray code permutation. The width and height of the tile are given by Deltat = 2 Gamma ; Delta = 2 GammaL : 10) The SIWPD [7] is preferable to the standard ....

[Article contains additional citation context not shown here]

I. Cohen, S. Raz and D. Malah, "Orthonormal shiftinvariant wavelet packet decomposition and representation ", Sig. Proc., Vol. 57, No. 3, Mar. 1997, pp. 251-- 270.

....of translations and dilations of wavelet packets [4, 5] The basis functions are localized in the time frequency plane, and organized in a binary tree structure where efficient search algorithms for the best basis can be implemented. The shift invariant wavelet packet decomposition (SIWPD) [6] is an adaptive representation in an extended library of wavelet packet bases. The extended library includes an additional degree of freedom that adjusts the time localization of the basis functions. This degree of freedom is incorporated into the best basis search algorithm by adaptively ....

.... resolution level, and B ;n;m j n 2 =2 n h 2 (t Gamma 2 GammaL m) Gamma k i : 0 k 2 o : 5) This library is larger than the standard wavelet packet library by a square power, but can be still structured into a tree configuration which supports fast search algorithms [6]. The additional parameter m provides the crucial degrees of freedom required for the time adjustment of the basis functions. When an analyzed signal is translated in time by = q Delta 2 GammaL (q 2 ZZ) a new best basis is selected whose elements are also translated by compared to the ....

[Article contains additional citation context not shown here]

I. Cohen, S. Raz and D. Malah, "Orthonormal shift-invariant wavelet packet decomposition and representation", Signal Processing, Vol. 57, No. 3, Mar. 1997, pp. 251--270.

....representations resulting from 2 different shift options, where denotes the number of nodes in the best expansion tree, and choose that representation (shiftoption) which minimizes the entropy. This procedure is suboptimal compared with the Shift Invariant Wavelet Packet Decomposition (SIWPD) [9], since the expansion tree is determined by the averaged entropy. Additionally, the shiftoptions in step (iii) are examined one by one, whereas the SIWPD not only provides a recursive selection method for the optimal shift, but also offers an inherent trade off between the computational complexity ....

....show that this estimator, combined with the recently introduced modified Wigner distribution (MWD) 10] yields robust time frequency representations that are characterized by high resolution, i.e. high concentration and suppressed interference terms. An extended library of wavelet packet bases [9] is employed for generating a collection of competing models, and the MDL principle is applied for approximating the description length of the observed noisy data. We show that minimum description length is attainable by optimizing the expansion tree associated with the SIWPD. The optimal signal ....

[Article contains additional citation context not shown here]

I. Cohen, S. Raz and D. Malah, "Orthonormal shiftinvariant wavelet packet decomposition and representation ", Sig. Proc., Vol. 57, Mar. 1997, pp. 251--270.

....representations resulting from 2 different shiftoptions, where is the number of nodes in the best expansion tree, and choose that representation (shift option) which minimizes the entropy. This procedure is sub optimal compared with the ShiftInvariant Wavelet Packet Decomposition (SIWPD) [5, 6], since the expansion tree is determined by the averaged entropy. Additionally, the shift options in step (iii) are examined one by one, whereas the SIWPD not only provides a recursive selection method for the optimal shift, but also offers an inherent trade off between the computational ....

....selection method for the optimal shift, but also offers an inherent trade off between the computational complexity and the information cost. In this paper, we present a translation invariant denoising method, based on the SIWPD and the MDL criterion. An extended library of wavelet packet bases [6] is employed for generating a collection of competing models, and the MDL principle is applied for approximating the description length of the observed noisy data. We show that minimum description length is attainable by optimizing the expansion tree associated with the SIWPD. The optimal signal ....

[Article contains additional citation context not shown here]

I. Cohen, S. Raz and D. Malah, "Orthonormal shift-invariant wavelet packet decomposition and representation", Signal Processing, Vol. 57, No. 3, Mar. 1997, pp. 251--270.

....join neighboring basis functions, invariably degrades the energy concentration and may artificially split a given signal component into several frequency bands. In this paper, we propose an adaptive suppression of interference terms using the Shift Invariant Wavelet Packet Decomposition (SIWPD) [6, 9]. A prescribed signal is expanded on its best SIWPD basis, and subsequently transformed into the Wigner domain. The interference terms are controlled by adaptively thresholding the cross WD of interactive basis functions according to their distance and amplitudes in an idealized time frequency ....

....for a particular application, may be significantly influenced by the alignment of the input signal with respect to the basis functions. Furthermore, the time frequency tilings, produced by the best basis expansions, do not generally conform to standard time frequency energy distributions [9]. Hence we employ modified versions which induce shift invariance, lower information cost and improved time frequency resolution [6, 8, 7] Let us specifically consider the shift invariant wavelet packet decomposition (SIWPD) 6, 9] The library of bases is extended by introducing an additional ....

[Article contains additional citation context not shown here]

I. Cohen, S. Raz and D. Malah, "Orthonormal shift-invariant wavelet packet decomposition and representation", Signal Processing, Vol. 57, No. 3, Mar. 1997, pp. 251--270.

.... 1) 3) where 2p = 2 1 and n2p = n2div 2, implies 1 m1 mod 2 Gamma 2 = m2 : 4) Then E generates an orthonormal (ON) basis for V j j U j 0;0;0 , i.e. S ( n;m)2E B j ;n;m is an ON basis, and the set of all E as specified above generates a SWP library. The proof is detailed in [10]. The SWP library thus created is larger than the WP library proposed in [1] by a square power. The SWP library can still be cast into a tree configuration, where each node is indexed by ( n; m) and represents the subspace U j ;n;m . Condition (ii) above is equivalent to demanding that the ....

....2 The best basis stemming from the previously described recursive algorithm is shift invariant. Proof: Let f; g 2 V j where f and g are identical to within a time shift , i.e. g(x) f(x Gamma q2 Gammaj ) Let Af and Ag denote the best bases for f and g, respectively. It can be shown [10] that B ;n;m ae Af (8) implies B ;n; m ae Ag ; m = m q) mod (2 Gamma ) 9) for all m;n 2 ZZ and 2 ZZ Gamma . Hence, Af and Ag are identical to within a time shift . 2: An alternative view of SIWPD is facilitated via filter bank terminology [11] Accordingly, each parent node ....

I. Cohen, S. Raz and D. Malah, "Orthonormal shiftinvariant wavelet packet decomposition and representation ", submitted.

.... limiting conditions on the scaling function [57, 3, 4] Recently, several authors proposed independently to extend the library of bases, in which the best representations are searched for, by introducing additional degrees of freedom that adjust the time localization of the basis functions [47, 11, 12, 23, 36]. It was proved that the proposed modifications of the wavelet transform and wavelet packet decomposition lead to orthonormal bestbasis representations which are shift invariant and characterized by lower information costs. The principal idea is to adapt the down sampling when expanding each ....

....the parity properties of the segmented signal at the end points. We showed that a locally adapted polarity yields a manageable computational complexity which is comparable to that of the LCD. A similar procedure is available for other types of bases, most notably wavelets and waveletpacket bases [47, 11, 12, 24, 36]. In these cases, the adaptive relative shift is equivalent to an adaptive down sampling when expanding each parent node. By choosing between the odd and even samples after the low pass and high pass filters, we obtain an orthonormal best basis representation which is shift invariant and of a ....

I. Cohen, S. Raz and D. Malah, "Orthonormal shift-invariant wavelet packet decomposition and representation", Technical Report, EE PUB No. 953, Technion - Israel Institute of Technology, Haifa, Israel, Jan. 1995.

No context found.

I. Cohen, S. Raz, and D. Malah. Orthonormal shift-invariant wavelet packet decomposition and representation. Signal Processing, 57(3):251--270, March 1997.

No context found.

I. Cohen, S. Raz, and D. Malah, "Orthonormal Shift-Invariant WaveletPacketDecompositionandRepresentation,"Signal Processing, vol. 57, no. 3, pp. 251-270, Mar. 1997.

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