I. Cohen, S. Raz and D. Malah "Orthonormal shift- invariant adaptive local trigonometric decomposition", to appear in Signal Processing, Vol. 57, No. 1.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....they map the local trigonometric functions onto an orthonormal basis of trigonometric functions. Thanks to this property results from classical harmonical analysis can be applied. Furthermore, we want to mention the application of best basis algorithms for the adaptive choice of the window size in [20, 23, 45, 48] and the construction of bivariate bases in [11, 16, 50, 51] Results on the approximation with local trigonometric bases can be found in [14, 15, 29, 33] For the design of optimized window functions we refer to [12, 14, 15, 36, 41] To reduce blocking e ects local trigonometric bases have been ....

I. Cohen, Shalom Raz, and David Malah. Orthonormal shift-invariant adaptive local trigonometric decomposition. Signal Processing, 57:43{ 64, 1997.

....they map the local trigonometric functions onto an orthonormal basis of trigonometric functions. Thanks to this property results from classical harmonical analysis can be applied. Furthermore, we want to mention the application of best basis algorithms for the adaptive choice of the window size in [20, 23, 45, 48] and the construction of bivariate bases in [11, 16, 50, 51] Results on the approximation with local trigonometric bases can be found in [14, 15, 29, 33] For the design of optimized window functions we refer to [12, 14, 15, 36, 41] To reduce blocking e#ects local trigonometric bases have been ....

I. Cohen, Shalom Raz, and David Malah. Orthonormal shift-invariant adaptive local trigonometric decomposition. Signal Processing, 57:43-- 64, 1997.

.... 80 A comparative study is presented, focused on two linear transform types, the wavelet packet transform (WPT) and the cosine packet transform (CPT) Shiftinvariance is also implemented for the wavelet packet transform (SIWPT) whereas, although an SI algorithm exists for CPT, the SICPT algorithm [1] will only be mentioned for the reader s information but not applied, for some reasons explained later in the paper. Shiftinvariance is often introduced as a criterion for classification robustness and one objective of this work is to compare the SI transforms with their non SI relatives. WPT was ....

....the least costly one (for a given information cost) would be shift invariant, but the memory and computation time for such a transform would be far too high. A fast and lowmemory SI CPT algorithm was recently introduced, as part of a wider family of SI adaptive local trigonometric decompositions [1]. The algorithm complexity is O(N(J 2 logN J 1 )log 2 N) which is generally higher than SIWPT but still allows for real time decomposition. The SICPT algorithm results in selecting the timeshift, which produces the cheapest discrete packet transform, according to a given information cost ....

I.Cohen, S. Raz and D. Malah, "Orthonormal shift-invariant adaptive local trigonometric decomposition, " Signal Processing,vol. 57, pp. 43-64, 1997

....basis, but f(t Gamma 1 3 ) cannot have a representation with an absolutely summable coefficient series. Note that we do not achieve perfect translation invariance of the adapted transform, only a bound on the variation of compressibility. For true shift invariance, we refer to the methods of [CRM], which for a discrete signal of length N are of computational complexity between O(N log 2 N) and O(N 2 log N ) depending on the choice of minimal time step. 2) With a similar technique one proves easily that for all 2 R, e i t f(t) belongs to K 1 when f(t) does. In other words, K 1 is ....

I. Cohen, S. Raz, and D. Malah, Orthonormal shift-invariant adaptive local trigonometric decomposition, Signal Processing, 57, no.1, 43--64, (1997).

....basis can be implemented. A serious drawback of the wavelet packet decomposition (WPD) and local cosine decomposition (LCD) 13] is the lack of shift invariance. Hence ve elnploy lnodified versions vhich induce shift invariance, lover inforlnation cost and ilnproved tilne frequency resolution [12, 16, 17]. Let us specifically consider the shift mvariat wavelet packet decompositio (SIWPD) 12, 14] The library of bases is extended by introducing an additional degree of freedoln that adjusts the tilne localization of the basis functions. This degree of freedoln is practically incorporated into the ....

I. Cohen, S. Raz and D. Malah "Orthonormal shift- invariant adaptive local trigonometric decomposition", to appear in Signal Processing, Vol. 57, No. 1.

.... 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 degree of freedom that adjusts the time localization of the basis functions. This degree of freedom is practically incorporated into the ....

....the interference terms. Fig. 10 illustrates the best basis expansion and MWD for g(t) obtained using an extended library of local trigonometric bases and a corresponding best basis search algorithm, namely the shift invariant adapted polarity local trigonometric decomposition (SIAP LTD) [7,8]. Here, the basis functions fail to represent the signal e#ciently. We may compare the entropy ( 2.81) with that obtained with the SIWPD (1.88 with C , 2.09 with D , and 2.32 with S ) The reduced performance of the SIAPLTD for this particular signal stems from the fact that short pulses, ....

[Article contains additional citation context not shown here]

I. Cohen, S. Raz, D. Malah, Orthonormal shift-invariant adaptive local trigonometric decomposition, Signal Processing 57 (1) (February 1997) 43---64.

....the expansion tree associated with the SIWPD. The optimal signal estimate is subsequently obtained by thresholding the resulting coe cients. The proposed method is extendable to other adaptive transforms, e.g. the shift invariant adaptive polarity local trigonometric decomposition (SIAP LTD) [8]. A corresponding procedure to optimize either the library of bases or the lter banks used at each node of the expansion tree is described as well. The signal estimator is independent of the alignment of the observed signal with respect to the basis functions. Furthermore, the intrinsic ....

....MWD, as shown in Figs. 9(c) and (d) are potentially valuable when analyzing the measurements and studying the non stationary phenomena, such as mode build up and competition and pulse shortening [1] which are common in such high power microwave tubes. In this example, we employed the SIAP LTD [8], since it yielded a shorter description length than the SIWPD (probably because the energy of the pulse is concentrated in the cavity modes of the magnetron, and local trigonometric bases are more appropriate for describing oscillations [7] The residual between the noisy measurement and the ....

I. Cohen, S. Raz, D. Malah, Orthonormal shift-invariant adaptive local trigonometric decomposition, Signal Processing 57 (1) (February 1997) 43}64.

....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 examples, is provided by the multiresolution ....

....However, as f(x) V j is decomposed into orthonormal wavelet packets using the best basis algorithm of Coifman and Wickerhauser [45] the often crucial property of shift invariance is no longer valid. One way to achieve shift invariance is to adjust the time localization of the basis functions [111, 27, 32, 90]. That is, when an analyzed signal is translated in time by # , a new best basis is selected whose elements are also translated by # compared to the former best basis. Consequently, the expansion coe#cients, that are now associated with translated basis functions, stay unchanged and the ....

I. Cohen, S. Raz and D. Malah, "Orthonormal shift-invariant adaptive local trigonometric decomposition", Signal Processing, Vol. 57, No. 1, Feb. 1997, pp. 43--64.

.... limiting conditions on the scaling function [49, 1, 2] 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 [40, 8, 12, 24, 33, 14]. 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 ....

....E ) However, as f(x) E is decomposed into orthonormal wavelet packets using the best basis algorithm of Coifinan and Wickerhauser [18] the often crucial property of shift invariance is no longer valid. One way to achieve shift invariance is to adjust the time localization of the basis functions [40, 8, 12, 32]. That is, when an analyzed signal is translated in time by , a new best basis is selected whose elements are also translated by compared to the former best basis. Consequently, the expansion coefficients, that are now associated with translated basis functions, stay unchanged and the ....

[Article contains additional citation context not shown here]

I. Cohen, S. Raz and D. Malah "Orthonormal shift-invariant adaptive local trigonometric decomposition", to appear in Signal Processing, Vol. 57, No. 1.

....the best basis can be implemented. A serious drawback of the wavelet packet decomposition (WPD) and 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 ....

I. Cohen, S. Raz and D. Malah "Orthonormal shiftinvariant adaptive local trigonometric decomposition", to appear in Signal Processing, Vol. 57, No. 1.

....the expansion tree associated with the SIWPD. The optimal signal estimate is subsequently obtained by thresholding the resulting coefficients. The proposed method is extendable to other adaptive transforms, e.g. the Shift Invariant Adaptive Polarity Local Trigonometric Decomposition (SIAP LTD) [8]. A corresponding procedure to optimize either the library of bases or the filter banks used at each node of the expansion tree is described as well. The signal estimator is independent of the alignment of the observed signal with respect to the basis functions. Furthermore, the intrinsic ....

....MWD, as shown in Figs. 9(c) and (d) are potentially valuable when analyzing the measurements and studying the non stationary phenomena, such as mode build up and competition and pulse shortening [1] which are common in such high power microwave tubes. In this example, we employed the SIAP LTD [8], since it yielded a shorter description length than the SIWPD (probably because the energy of the pulse is concentrated in the cavity modes of the magnetron, and local trigonometric bases are more appropriate for describing oscillations) The residual between the noisy measurement and the signal ....

I. Cohen, S. Raz and D. Malah, "Orthonormal shift-invariant adaptive local trigonometric decomposition", Signal Processing, Vol. 57, No. 1, Feb. 1997, pp. 43--64.

....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 degree of freedom that adjusts the time localization of the basis functions. This degree of freedom is practically incorporated into the ....

....the interference terms. Fig. 10 illustrates the best basis expansion and MWD for g(t) obtained using an extended library of local trigonometric bases and a corresponding best basis search algorithm, namely the shift invariant adapted polarity local trigonometric decomposition (SIAP LTD) [7, 8]. Here, the basis functions fail to represent the signal efficiently. We may compare the entropy ( 2:81) with that obtained with the SIWPD (1:88 with C 12 , 2:09 with D 6 , and 2:32 with S 9 ) The reduced performance of the SIAP LTD for this particular signal stems from the fact that short ....

[Article contains additional citation context not shown here]

I. Cohen, S. Raz and D. Malah, "Orthonormal shift-invariant adaptive local trigonometric decomposition", Signal Processing, Vol. 57, No. 1, Feb. 1997, pp. 43--64.

.... limiting conditions on the scaling function [49, 1, 2] 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 [40, 8, 12, 24, 33, 14]. 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 ....

....However, as f(x) 2 V j is decomposed into orthonormal wavelet packets using the best basis algorithm of Coifman and Wickerhauser [18] the often crucial property of shift invariance is no longer valid. One way to achieve shift invariance is to adjust the time localization of the basis functions [40, 8, 12, 32]. That is, when an analyzed signal is translated in time by , a new best basis is selected whose elements are also translated by compared to the former best basis. Consequently, the expansion coefficients, that are now associated with translated basis functions, stay unchanged and the ....

[Article contains additional citation context not shown here]

I. Cohen, S. Raz and D. Malah "Orthonormal shift-invariant adaptive local trigonometric decomposition", to appear in Signal Processing, Vol. 57, No. 1.

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