I. Daubechies and T. Paul, "Time-frequency localization operators --- a geometric phase space approach: II. The use of dilations," Inverse Problems, no. 4, pp. 661--680, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....g[n] n] then (4.45) simply means that [n] is normalized, i.e. k k j [n]j = 1. Note that (4.45) can always be achieved by a simple scaling of either window unless hg; i = 0. STFT Filter. The STFT lter procedure consists of the following three steps that are illustrated in Fig. 4. 12 [8, 9, 15, 16, 18 20]. 1. Analysis: The STFT of the input signal x[n] is calculated according to (4.43) 2. Weighting : The STFT of x[n] is multiplied by the prescribed TF weight function M(n; F (n; M(n; STFT x (n; The resulting TF function F (n; corresponds in a certain sense to the desired ....

I. Daubechies, \Time-frequency localization operators: A geometric phase space approach," IEEE Trans. Inf. Theory, vol. 34, pp. 605-612, July 1988.

....The signal g(t) Fig. 1) comprises a short pulse, a tone and a component with nonlinear frequency modulation. The spectrogram has no interference terms, at the expense of comparatively poor energy concentration. The optimal expansions of g(t) obtained by the method of frames (minimum l# norm) [16], Matching Pursuit [26] basis pursuit (minimum l# norm) 2] and WPD are illustrated in Fig. 5. While these algorithms use the conventional library of wavelet packets and fail to represent the signal e#ciently, the SIWPD (Fig. 5(f ) facilitates an e#cient representation by a small number of ....

I. Daubechies, Time---frequency localization operators: a geometric phase space approach, IEEE Trans. Inform. Theory 34 (1988) 605---612.

....nonlinear frequency modulation. The corresponding Wigner distribution and spectrogram are displayed in Fig. 4.2. The spectrogram has no interference terms, at the expense of comparatively poor energy concentration. The optimal expansions of g(t) obtained by the Method of Frames (minimum l norm) [52], Matching Pursuit [96] Basis Pursuit (minimum l norm) 14] and WPD are illustrated in Fig. 4.3. While these algorithms use the conventional library of wavelet packets and fail to represent the signal 0 0.2 0.4 0.6 0.8 1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 Figure 4.1: Test ....

I. Daubechies, "Time-frequency localization operators: a geometric phase space approach", IEEE Trans. Inform. Theory, Vol. 34, 1988, pp. 605-612.

....the U.S. Army Research Laboratory under the Federated Laboratory Program, Cooperative Agreement DAAL0l 96 2 0002. 1 INTRODUCTION We will discuss the advantages and disadvantages of using four methods of decomposition for image compression and restoration. The methods are Method of Frames (MOF) [1], Best Orthogonal Basis (BOB) 3] Matching Pursuit (MP) 4] and Basis Pursuit (BP) 2] What these methods have in common is a requirement to use waveforms from a dictionary to represent an image. A dictionary, F , is simply a collection of parameterized waveforms, f g , used as a basis for ....

Ingrid Daubechies, Time-Frequency Localization Operators: A Geometric Phase Space Approach, IEEE Transaction on Information Theory, Vol.36, No. 3, September 1990 pp. 961-1005

....Implementation Following [8, 20] we now discuss a TF implementation of the Wiener and projection lters that is based on the multiwindow short time Fourier transform (STFT) and that is valid in the underspread case. A simple TF lter method is an STFT lter that consists of the following steps [8, 13, 20, 52 57]: STFT analysis : Calculation of the STFT [19, 54, 55, 58] of the input signal x(t) STFT (g) x (t; f) Z t 0 x(t 0 ) g t;f (t 0 ) dt 0 ; where g t;f (t 0 ) g(t 0 t) e j2 ft 0 with g(t) a normalized window. STFT weighting : Multiplication of the STFT by a weighting ....

I. Daubechies, \Time-frequency localization operators: A geometric phase space approach," IEEE Trans. Inf. Theory, vol. 34, pp. 605-612, July 1988.

....in term of sparsity. Basis pursuit method [12] BP) consists in minimizing the functional: J( k s k 2 2 k k 1 (52) Between all possible solutions, the chosen one has the minimum l 1 norm. This choice of l 1 norm is very important. A l 2 norm, as used in the method of frames [17], does not preserve the sparsity [12] The minimization of equation 52, which involves linear programming, is however much more time consuming than MP, and remains prohibitive for large signals [31] We present in the following an alternative approach, that we call Combined Transforms Method ....

I. Daubechies. Time-frequency localization operators: A geometric phase space approach. IEEE Transactions on Information Theory, 34(1):605-612, 1988.

....the goals, follows. Speed. It should be possible to obtain a representation in order O(n) or O(n log(n) time. 1.2. Finding a representation. Several methods have been proposed for obtaining signal representations in overcomplete dictionaries. These range from general approaches, like the MOF [9] and the method of MP [25] to clever schemes derived for specialized dictionaries, like the method of BOB [7] These methods are described briefly in section 2.3. In our view, these methods have both advantages and shortcomings. The principal emphasis of the proposers of these methods is in ....

....indeed consist of frequencies near # and essentially vanish far away from # . For fixed #t, discrete dictionaries can be built from time frequency lattices, # k = k## and # # = ### , and # # 0, # 2 ; with ## and ## chosen su#ciently fine these are complete. For further discussions see, e.g. [9]. Recently, Coifman and Meyer [6] developed the wavelet packet and cosine packet dictionaries especially to meet the computational demands of discrete time signal processing. For 1 d discrete time signals of length n, these dictionaries each contain about n log 2 (n) waveforms. A wavelet packet ....

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I. Daubechies, Time-frequency localization operators: A geometric phase space approach, IEEE Trans. Inform. Theory, 34 (1988), pp. 605--612.

....s, in x = As: This has an innite number of solutions. The classical way is to select the solution s with minimum norm, which is given by s = A y x; where A y is the pseudo inverse of A: A y = A T (AA T ) Gamma1 This solution is often referred to as the method of frames solution [14]. Now consider each basis window in each possible window position of the image as an overcomplete basis for the whole image. Then, if the transform we use is orthogonal, the sliding window algorithm is equivalent to calculating the method of frames decomposition, shrinking each component, and ....

I. Daubechies. Time-frequency localization operators: a geometric phase space approach. IEEE Transactions on Information Theory, 34:605612, 1988.

....domain of localization is expanded by dilations (see [Da2] LW] RT] The distribution function, indicating how many eigenvalues are bigger than , 1 0, is the principal object in this study. The distribution of the eigenvalues of Gabor Toeplitz operators was rst studied by Daubechies in [Da1]. She observed that if the symbol is a characteristic function of a disk centered at the origin and the function de ning the reproducing formula, i.e. the window, is a standard Gaussian, then the corresponding Gabor Toeplitz localization operator is diagonalized by Hermite functions. She obtained ....

I. Daubechies, Time-frequency localization operators: A geometric phase space approach, IEEE Trans. Inform. Theory 34, 605-612, (1988).

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I. Daubechies and T. Paul, "Time-frequency localization operators --- a geometric phase space approach: II. The use of dilations," Inverse Problems, no. 4, pp. 661--680, 1988.

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I. Daubechies, "Time-frequency localization operators: A geometric phase space approach," IEEE Trans. Inform. Theory, vol. 34, pp. 605--612, July 1988.

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I. Daubechies and T. Paul, "Time-frequency localization operators --- a geometric phase space approach: II. The use of dilations," Inverse Problems, no. 4, pp. 661-- 680, 1988. Printed in the UK.

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I. Daubechies, "Time-frequency localization operators: A geometric phase space approach," IEEE Trans. Inform. Theory, vol. 34, pp. 605--612, July 1988.

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I. Daubechies and T. Paul, "Time-frequency localization operators --- a geometric phase space approach: II. The use of dilations," Inverse Problems, no. 4, pp. 661--680, 1988.

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I. Daubechies, "Time-frequency localization operators: A geometric phase space approach," IEEE Trans. Inform. Theory, vol. 34, pp. 605--612, July 1988.

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I. Daubechies and T. Paul, "Time-frequency localization operators --- a geometric phase space approach: II. The use of dilations," Inverse Problems, no. 4, pp. 661-- 680, 1988.

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I. Daubechies, "Time-frequency localization operators: A geometric phase space approach," IEEE Trans. Inform. Theory, vol. 34, pp. 605--612, July 1988. 49

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I. Daubechies, "Time-frequency localization operators: A geometric phase space approach," IEEE Trans. Inform. Theory, vol. 34, pp. 605--612, July 1988.

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I. Daubechies. Time-frequency localization operators: a geometric phase space approach. IEEE Trans. Inform. Theory, 34(4):605-612, 1988.

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I. Daubechies. Time-frequency localization operators: a geometric phase space approach. IEEE Transactions on Information Theory, vol. 34, pp. 605--612, 1988.

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I. Daubechies. Time-frequency localization operators: a geometric phase space approach. IEEE Trans. Inform. Theory, 34(4):605-612, 1988.

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Daubechies, I., Time-frequency localization operators: a geometric phase space approach, IEEE Trans. Inf. Theory 34 (1988), 605--612.

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]I. Daubechies and T. Paul, Time-frequency localization operators-a geometric phase space approach: II The use of dilations, Inverse Problems 4 (1988), 661--680.

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]I. Daubechies, Time-frequency localization operators: a geometric phase space approach, IEEE Trans. Inform. Theory 34 (1988), 605--612.

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I. Daubechies, "Time-frequency localization operators: A geometric phase space approach," IEEE Trans. Inf. Theory, Vol. 34, No. 4, pp. 605-612, 1988.

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