R. Coifman and Y. Meyer, "Remarques sur l'analyse de Fourier a fen etre," C. R. Acad. Sci. Paris, vol. I, pp. 259--261, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....L intervals. Let l p = a p 1 a p , and min p l p =2. One can construct for each p 2 [1; L] a regular window g p [n] with support [a p ; a p 1 ] such that the local cosine family g p;q [n] g p [n] h (q 1 2 n a p io 1 p L ; 0 q lp (18) is an orthonormal basis of C [5, 15, 14]. Each local cosine vector g p;q has a support nearly localized in the interval [a p ; a p 1 ] and its Fourier transform has its energy mostly localized in the interval [q =l p ; q 1) l p ] It can thus be represented by the Heisenberg rectangle [a p ; a p 1 ] q =l p ; q 1) l p ] in a ....

....recording x[n] Darker tiles correspond to high amplitude coecients jhx ; g p;q ij whereas white tiles coecients jhx ; g p;q ij that are nearly zero. A dicult issue is to nd a time partition which de nes a local cosine 10 basis which is well adapted to an estimation problem. Coifman and Meyer [5, 14] have thus constructed a dyadic dictionary of local cosine bases, which includes orthonormal bases de ned over partitions whose intervals have sizes which are powers of 2. The dictionary is organized as a tree. For a p = pN2 1=2 and a p 1 = p 1)N2 1=2, we denote by g p the window ....

R.R. Coifman and Y. Meyer, Remarques sur l'analyse de Fourier a fen^etre,Compt. Rend. Acad. Sci. Paris Ser.A, vol 312, pp 259-261,1991. 28

....confined to finite size libraries. Such libraries are not only required to be flexible and versatile enough to describe various local features of signals, but also need to be aptly organized in a structure that facilitates 8 a fast search algorithm for the best basis . Coifman and Meyer [42, 46, 102] were the first to introduce libraries of orthonormal bases whose elements are localized in timefrequency plane and structured into a binary tree where the best basis can be e#ciently searched for. One of the libraries, a library of local trigonometric bases, consists of sines or cosines ....

R. R. Coifman and Y. Meyer, "Remarques sur l'analyse de Fourier a fenetre", Comptes Rendus de l'Academie des Sciences, Vol. 312, pp. 259--261, 1991.

....transform defined on the real line R. The periodic and reflection infinite extension rules can also be viewed as wavelet transforms restricted to a compact interval. 2 Cosine packet analysis is commonly known as local cosine analysis, and was first introduced by Ronald Coifman and Yves Meyer [CM91]. The term cosine packets was coined by David Donoho, another leading wavelets researcher, because cosine packet analysis is a mirror image of wavelet packet analysis [CMQW90] The difference is that localized cosine functions are used instead of wavelet packet functions. The Fourier cosine ....

....valuable for coding and data compression applications. One drawback of the DCT is the abrupt cutoff implicit in dividing the signal into disjoint blocks. This can cause undesirable block effects, such as Gibbs phenomena. To avoid the problems caused by the abrupt cutoff, Coifman and Meyer [CM91] introduced a new type of localized cosine transform with smooth cutoffs (tapers) A cosine packet function is obtained by damping a cosine function down to zero on an interval I using a taper function or bell function fi I . Type II and type IV cosine packet functions with frequency k defined on ....

R. Coifman and Y. Meyer. Remarques sur l'analyse de Fourier `a fenetre. C. R. Acad. Sci. Paris, 312:259--261, 1991.

....local trigonometric bases were introduced by Malvar [11] Here the so called two overlapping setting is considered, where the window functions have compact support such that w(x r)w(x s) # 0 if s r 1. The Malvar bases were independently discovered by Coifman and Meyer [7] in a generalized nonuniform setting, where the uniform spacing is replaced by an arbitrary partition . a 1 a 0 a 1 . of R with a j # # for j # #. An expository representation of these results can be found in [2] As another example let us mention the Wilson bases described ....

Coifman, R. R. and Meyer, Y., Remarques sur l'analyse de Fourier a fenetre, C. R. Acad. Sci. Paris 312 (1991), 259--261.

....as frame and Riesz basis conditions are given explicitly. Our methods are based on the properties of bivariate total folding and unfolding operators. 1. Introduction Recently, local trigonometric bases have been investigated by many authors. Let us mention here only Malvar [8] Coifman and Meyer [5], Daubechies, Ja#ard and Journee [6] Auscher, Weiss and Wickerhauser [1] Wickerhauser [10] Jawerth and Sweldens [7] Matviyenko [9] Xia and Suter [11] Chui and Shi [3,4] and the literature cited there. In the meantime the idea to use windowed sinusoids as an orthonormal basis for L 2 (IR) ....

Coifman R. R. and Meyer Y., Remarques sur l'analyse de Fourier a fenetre, C. R. Acad. Sci. Paris 312 (1991), 259--261.

....the window functions do not need to be symmetric and even two di#erent window functions are possible. In particular, an explicit formula for the computation of Riesz bounds is given which uses the Zak transform of the window function. Connections between Wilson bases and local trigonometric bases [2, 10, 11, 17, 20] were pointed out in [1, 7] It has turned out that Wilson bases can also be constructed with anti periodic cosines C k j : 2 cos # (2k 1)#( j 2 ) # ore sines S k j : 2 sin # (2k 1)#( j 2 ) # (cf. 10, 12] instead of the mixture of 1 periodic cosines and sines D k j ....

R. R. Coifman and Y. Meyer. Remarques sur l'analyse de Fourier a fenetre. C. R. Acad. Sci. Paris, 312:259--261, 1991.

....1 2 )#( j) j # Z, k # N 0 , in a so called two overlapping setting. Here, two overlapping means that the window function w is compactly supported such that only consecutive translates of w are allowed to have overlapping supports. Based on the results from [26, 37, 40] Coifman and Meyer [21] investigated more general orthonormal local trigonometric bases, where the translates w( j) are replaced by a sequence of window functions w j with variable support length which satisfy a two overlapping condition (see section 1.3) A detailed description of these Coifman Meyer bases with ....

....We will establish the results which we will need for our studies in the following sections. In particular, the application of folding operators is based on these results. In section 1.3 1.5, we consider the two overlapping setting. We start with the orthonormal bases of Coifman and Meyer [21] in section 1.3. Furthermore, the folding operators of Wickerhauser [47] are described. Section 1.4 gives a short overview of the biorthogonal bases of Jawerth and Sweldens [36] The general theory of Chui and Shi [17] is presented in section 1.5. Here, the theorems about Riesz stability and dual ....

[Article contains additional citation context not shown here]

R. R. Coifman and Y. Meyer. Remarques sur l'analyse de Fourier a fenetre. C. R. Acad. Sci. Paris, 312:259--261, 1991.

....is the classical Discrete Cosine Transform (DCT) Nevertheless, because of the periodicity of the cosine, the DCT basis has the disadvantage of having discontinuities at the boundaries. A more recent approach was devised by H. Malvar, 16, 17] in the discrete case and, R.R. Coiffmann, Y. Meyer, [5], for continuous signals: Lapped block transforms. In this transform the cosine basis are windowed 3 J. GOMES, L. VELHO, F.W. DASILVA and S.K. GOLDENSTEIN in such a way to avoid boundary discontinuities, and the intervals of the time partition overlap so as to maintain the orthogonality of the ....

R.R. Coifman and Y. Meyer. Remarques sur l'analyse de fourier a fen etre. C.R. Acad. Sci., pages 259--261, 1991.

....on popular signal processing tools like z transform and polyphase components [2] The continuous time case has received less attention in the signal processing literature. The continuous time counterpart of CMFB is known as local cosine bases (LCB) and it has been introduced by Coifman and Meyer [10]. Such a device has been used by Auscher, Weiss and Wickerhauser in [11] to construct the Lemari e and Meyer wavelet [12] Recently, Matviyenko [13] introduced biorthogonal LCB, showing that the dual is still an LCB, but with a different window. All the cited works consider only the single ....

....requires always the same set of values of ff(t 2i) f(t 2i 1)g i2Z , for every n. Such a fact will be exploited in Section 4.2 and 5.2 to write (27) as a PR filter bank and prove the completeness. 4 Single overlapping revisited In this section we briefly revisit the single overlapping case [10] to show how the framework presented in Section 3 can be used. 4.1 External orthogonality We need to check V j V k , j 6= k, j; k 2 Z. Because of the support restriction, only V j V j 1 needs to be checked, which by translation invariance reduces 10 to V 0 V 1 . We will prove that ....

R. Coifman and Y. Meyer, "Remarques sur l'analyse de Fourier `a fenetre," C.R. Acad. Sci. Paris, vol. I, pp. 259--261, 1991.

....Introduction Since Daubechies, Jaffard, and Journ e in [6] gave a method to construct an orthonormal basis of L 2 : L 2 (IR) consisting of windowed trigonometric functions, local trigonometric bases have been investigated by many authors. In particular, the approach of Coifman and Meyer [5] to consider two overlapping window functions turned out to be useful in many applications. A detailed study of two overlapping , orthonormal local trigonometric bases can be found in [1,2] To include various desirable features in [7] and [8] orthogonality is replaced by bi orthogonality. In ....

Coifman, R. R. and Y. Meyer, Remarques sur l'analyse de Fourier 'a fenetre, C. R. Acad. Sci. Paris 312 (1991), 259--261.

....In this paper we study a set of vocal command signals recorded in a noisy environment. We describe and use Fang s segmentation algorithm [6] to isolate near phonemes. A piecewise constant time frequency spectrum for the near phonems is then computed using the smooth cosine4 orthonormal basis [3,7,8,13] defined over the segmented time axis. This paper proposes a criterion to distinguish phonemes using this smooth local spectrum. Keywords: signal processing, wavelets, time frequency representation. AMS Subject classification: 94A11, 94A12 1. Introduction A signal can be decomposed into a ....

....combination of elementary waveforms, where each waveform is essentially supported by a rectangle R = a; b] Theta [ff; fi] in the time frequency plane. One now has available a large selection of waveforms or time frequency atoms (for example, windowed Fourier functions, Malvar Wilson bases [3,7,8] , and wavelet packets [9,10] Since the choice of time frequency atoms is not unique, the decomposition can be adapted to the analysed signal. Speech signals may be regarded as a sequence of overlapping phonemes, and one goal of time frequency representation is to isolate and analyze these ....

R.R. Coifman and Y. Meyer, Remarques sur l'analyse de Fourier `a fenetre, C. R. Acad. Sci. Paris 312, pp. 259-261, 1991.

....as frame and Riesz basis conditions are given explicitly. Our methods are based on the properties of bivariate total folding and unfolding operators. 1. Introduction Recently, local trigonometric bases have been investigated by many authors. Let us mention here only Malvar [8] Coifman and Meyer [5], Daubechies, Jaffard and Journee [6] Auscher, Weiss and Wickerhauser [1] Wickerhauser [10] Jawerth and Sweldens [7] Matviyenko [9] Xia and Suter [11] Chui and Shi [3,4] and the literature cited there. In the meantime the idea to use windowed sinusoids as an orthonormal basis for L 2 (IR) ....

Coifman R. R. and Meyer Y., Remarques sur l'analyse de Fourier 'a fenetre, C. R. Acad. Sci. Paris 312 (1991), 259--261.

....as frame and Riesz basis conditions are given explicitly. Our methods are based on the properties of bivariate total folding and unfolding operators. 1. Introduction Recently, local trigonometric bases have been investigated by many authors. Let us mention here only Malvar [8] Coifman and Meyer [5], Daubechies, Jaffard and Journee [6] Auscher, Weiss and Wickerhauser [1] Wickerhauser [10] Jawerth and Sweldens [7] Matviyenko [9] Xia and Suter [11] Chui and Shi [3,4] and the literature cited there. In the meantime the idea to use windowed sinusoids as an orthonormal basis for L 2 (IR) ....

Coifman R. R. and Meyer Y., Remarques sur l'analyse de Fourier 'a fenetre, C. R. Acad. Sci. Paris 312 (1991), 259--261.

....local trigonometric bases were introduced by Malvar [11] Here the so called two overlapping setting is considered, where the window functions have compact support such that w(x Gamma r)w(x Gamma s) j 0 if js Gamma rj 1. The Malvar bases were independently discovered by Coifman and Meyer [7] in a generalized nonuniform setting, where the uniform spacing is replaced by an arbitrary partition : a Gamma1 a 0 a 1 : of R with a j Sigma1 for j Sigma1. An expository representation of these results can be found in [2] As another example let us mention the Wilson ....

Coifman, R. R. and Meyer, Y., Remarques sur l'analyse de Fourier 'a fenetre, C. R. Acad. Sci. Paris 312 (1991), 259--261.

.... we can think of notes as time frequency atoms [1] Currently, two classes of time frequency atoms are in use, corresponding to two types of analyses that Ville proposed: The first, wavelet packets [2] splits the signal first in frequency and then in time, while the second, local cosine bases [3, 4, 5], does the opposite, that is, it slices first in time and then in frequency. Our approach in this work produces a mixture of these two classes leading to filters corresponding to more general tilings. Refer also to [6, 7] for some interesting works dealing with time frequency tilings. In discrete ....

R. Coifman and Y. Meyer, "Remarques sur l'analyse de Fourier `a fenetre," C.R. Acad. Sci. Paris, vol. I, pp. 259--261, 1991.

....class of bases for a best basis, to compress the covariance operator. This search is done using data provided by a few realizations of the process. For locally stationary processes, we have a fast implementation of the search for a best local cosine basis based on the local cosine trees of Meyer [8] and Coifman and Wickerhauser [5] In section 2 we study the properties of locally stationary processes and in section 3 we analyze the estimation of covariance operators with a best basis search. Fast numerical algorithms and their application to examples of locally stationary processes are ....

.... x; t) g x (t) cos( t ) 1 2 (e i OE x; t) e Gammai OE x; Gamma (t) and is also an approximate eigenvector. Indeed 0 (x; Gamma ) 0 (x; so (6) implies that T OE x; t) 0 (x; OE x; t) 9) Let us now explain the construction of Coifman, Malvar and Meyer [6, 8, 9] that defines an orthonormal basis of local cosine vectors. The real line R is partitioned into intervals [a p ; a p 1 ] of size l p = a p 1 Gamma a p : We suppose that the sequence a p is increasing and that lim p Gamma1 a p = Gamma1 ; lim p 1 a p = 1 so that the whole line is ....

[Article contains additional citation context not shown here]

R. Coifman and Y. Meyer, "Remarques sur l'analyse de Fourier `a fenetre", C. R. Acad. Sci. Paris S'er. I, pp. 259-261, 1991.

....1. INTRODUCTION Since Daubechies, Jaffard, and Journ e in [5] gave a method to construct an orthonormal basis of L 2 : L 2 (R) consisting of windowed trigonometric functions, local trigonometric bases have been investigated by many authors. In particular, the approach of Coifman and Meyer [4] to consider two overlapping window functions turned out to be useful for applications. A detailed study of two overlapping local trigonometric bases can be found in [1] To include various desirable features in [6] and [7] orthogonality is replaced by bi orthogonality. In particular, to ....

Coifman, R. R. and Meyer, Y., Remarques sur l'analyse de Fourier 'a fenetre, C. R. Acad. Sci. Paris 312 (1991), 259--261.

....algorithms. Both of these algorithms exhibit similar savings in exposure and similar quality of the reconstructed image in the region of interest. Recently, Olson [11] has improved his algorithm by replacing the usual wavelet transform with the local trigonometric transform of Coifman and Meyer [14] and has reduced the exposure still further. In this paper, we implement a wavelet based algorithm to reconstruct a good approximation of the low resolution parts of the image as well as the high resolution parts using only local measurements. The algorithm is based on the observation that in ....

R. R. Coifman, and Y. Meyer, "Remarques sur'l analyse de Fourier `a fenetre," serie I. C.R. Acad. Sci. Paris, 312:259-261, 1991.

.... the Ballian Low theorem [5] that we cannot use windowed exponentials of the form g n,m (x) imv 0 x g(x nt 0 ) 1) In order to circumvent the obstacle raised by the Ballian Low theorem various Wilson bases have been constructed that use sines and cosines rather than exponential [6] 7] [8]. We are interested in using exponentials, because the phase of the exponential will provide information about the direction of the pattern when describing images in two dimensions. Therefore we will use the construction described in [4] and more precisely we will use the smooth local ....

.... all the u j,n associated with all the intervals [a n , a n 1 ] we obtain an orthonormal basis of L (R) 4, 9] THEOREM 1 [4, 9] The collection u j,n j, n (R) We note that this basis uses exponentials; other smooth local bases that use sines or cosines only can be constructed also [8]. Figure 4 shows the real and imaginary parts of the function u j,n with a n 256, l n 256, and j 5. We note in Fig. 4 that u j,n is locally even around a n and locally odd around a n 1 . For this reason, the inner product of two contiguous basis functions u j,n and u j,n 1 will be zero. ....

R. R. Coifman and Y. Meyer, Remarques sur l'analyse de Fourier a fene tre, C. R. Acad. Sci. Paris I (1991), 259 -- 261.

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R.R. Coifman and Y. Meyer, Remarques sur l'analyse de Fourier a fenetre, C.R. Acad. Sci. Paris I (1991), 259--261.

....adaptive segmentation of the spatial domain in terms of oscillating patterns. An image is decomposed into blocks of different sizes within which a local Fourier expansion, or a DCT, is performed. Instead of abruptly cutting blocks in the image, we use a new family of smooth orthogonal projectors [3, 9, 29]. wavelet packets [10] Loosely speaking, wavelet packets make it possible to adaptively tile the frequency domain into different bands of arbitrary size. Wavelet packets have been used to characterize textures in images [6, 16, 20, 24] However, an elementary 2 D wavelet packet al..ways displays ....

....function of each interval, that abruptly cuts off a block in the image, by a new family of smooth projections. Combining these two ideas result in a new library of trigonometric bases, that can be tailored to any image. First, we review the construction of smooth Localized Cosine Transforms (LCT) [3, 9, 11] in one dimension. We consider a cover of R = S n= 1 n=1 [a n ; a n 1 [ and we write l n = a n 1 a n , and c n = a n a n 1 ) 2. We then perform a local Fourier analysis inside each block. To obtain a better frequency localization, we do not cut abruptly the signal, but we use a smooth ....

[Article contains additional citation context not shown here]

R.R. Coifman and Y. Meyer, Remarques sur l'analyse de Fourier a fenetre, C.R. Acad. Sci. Paris I (1991), 259--261.

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R.R. Coifman and Y. Meyer, Remarques sur l'analyse de Fourier a fenetre, C.R. Acad. Sci. Paris I (1991), 259--261.

.... know from the Ballian Low theorem [4] that we cannot use windowed exponentials of the form gn;m(x) e im 0 x g(x Gamma nt0 ) 1) In order to circumvent the obstacle raised by the BallianLow theorem various Wilson bases have been constructed that use sines and cosines rather than exponential [5, 6]. We are interested in using exponentials, because the phase of the exponential will provide information about the direction of the pattern when describing images in two dimensions. Therefore we will use the smooth localized orthonormal exponential bases defined in [3] These functions are ....

R.R. Coifman and Y. Meyer. Remarques sur l'analyse de fourier `a fenetre. C.R. Acad. Sci. Paris I, pages pp. 259--261, 1991.

No context found.

R. Coifman and Y. Meyer, "Remarques sur l'analyse de Fourier a fen etre," C. R. Acad. Sci. Paris, vol. I, pp. 259--261, 1991.

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R. R. Coifman and Y. Meyer, "Remarques sur l'analyse de Fourier a fenetre," C. R. Acad. Sci. Paris 312, pp. 259--261, 1991.

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