certains résultats présentés ici s'´ etendent aisémentaisémentà des cas plus généraux, et il est donc possible que la démarche entreprise dans cette thèse, qui donne les outils de base ,
Wavelets and signal processing, IEEE Signal Processing Magazine, vol.8, issue.4, pp.14-38, 1991. ,
Cycle-octave and related transforms in seismic signal analysis, Geoexploration, vol.23, issue.1, pp.85-10285, 1984. ,
DOI : 10.1016/0016-7142(84)90025-5
Wide-band ambiguity function and the ax + b group, Signal Processing, Part I : Signal Processing Theory, pp.1-12, 1990. ,
Constant-Q signal analysis and synthesis, Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, pp.375-378, 1978. ,
A unified approach to short-time Fourier analysis and synthesis, Proc. IEEE, pp.1558-1564, 1977. ,
DOI : 10.1109/PROC.1977.10770
Biorthogonal bases of compactly supported wavelets, Communications on Pure and Applied Mathematics, vol.10, issue.5 ,
DOI : 10.1002/cpa.3160450502
The ?-transform and decomposition of distributions, Proc. Conf. Fucntion Spaces and Appl, 1986. ,
New technique stores images more efficiently, 12 Novembre, pp.1-12, 1991. ,
Time-scale energy distributions: a general class extending wavelet transforms, IEEE Transactions on Signal Processing, vol.40, issue.7, pp.1746-1757, 1992. ,
DOI : 10.1109/78.143446
Acoustic Signal Compression with Wavelet Packets, Wavelets : A Tutorial in Theory and Applications, pp.679-700, 1992. ,
DOI : 10.1016/B978-0-12-174590-5.50026-5
Perfect reconstruction filter banks with rational sampling factors, IEEE Trans. Signal Processing ,
Iterated filter banks with rational factors : Links with discrete wavelet transforms, IEEE Trans. Signal Processing. Special issue on wavelets ,
Digital Coding of Speech in Sub-bands, Bell System Technical Journal, vol.55, issue.8, pp.1069-1085, 1976. ,
DOI : 10.1002/j.1538-7305.1976.tb02929.x
Application of quadrature mirror filters to split-band voice coding schemes, Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, pp.191-195, 1977. ,
Perfect reconstruction filter banks for HDTV representation and coding Image coding using lattice vector quantization of wavelet coefficients, Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, pp.349-364, 1990. ,
Subband image coding with biorthogonal wavelets, IEICE Trans. Fondamentals, issue.7, pp.871-881, 1992. ,
Applications of wavelet transform to image compression and texture analysis, Proc. Int. Colloqueum " Wavelets and Applications, 1993. ,
Compact image coding from edges with wavelets, [Proceedings] ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing, pp.2745-2748, 1991. ,
DOI : 10.1109/ICASSP.1991.150970
Application of compactly supported wavelets to image compression, Image Processing Algorithms and Techniques, pp.150-160, 1990. ,
A theory for multiresolution signal decomposition: the wavelet representation, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.11, issue.7, pp.674-693, 1989. ,
DOI : 10.1109/34.192463
Best wavelet packet bases in a rate-distortion sense, IEEE Transactions on Image Processing, vol.2, issue.2, 1993. ,
DOI : 10.1109/83.217221
Design of multidimensional non-separable regular filter banks and wavelets, Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, pp.389-392, 1992. ,
Quadrature mirror filter banks, M-band extensions and perfect-reconstruction techniques, IEEE ASSP Magazine, vol.4, issue.3, pp.4-20, 1987. ,
DOI : 10.1109/MASSP.1987.1165589
Perfect reconstruction FIR filter banks: some properties and factorizations, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.37, issue.7, pp.1057-1071, 1989. ,
DOI : 10.1109/29.32283
URL : http://infoscience.epfl.ch/record/33918
The role of lossless systems in modern digital signal processing: a tutorial, IEEE Transactions on Education, vol.32, issue.3, pp.181-197, 1989. ,
DOI : 10.1109/13.34150
Biorthogonal bases of compactly supported wavelets, Communications on Pure and Applied Mathematics, vol.10, issue.5 ,
DOI : 10.1002/cpa.3160450502
Filters for distortion-free two-band multirate filter banks, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.33, issue.3, pp.626-630, 1985. ,
DOI : 10.1109/TASSP.1985.1164587
Exact reconstruction techniques for tree-structured subband coders, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.34, issue.3, pp.434-441, 1986. ,
DOI : 10.1109/TASSP.1986.1164832
Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math, vol.XLI, issue.7, pp.909-996, 1988. ,
The discrete wavelet transform: wedding the a trous and Mallat algorithms, IEEE Transactions on Signal Processing, vol.40, issue.10, pp.2464-2482 ,
DOI : 10.1109/78.157290
The Laplacian Pyramid as a Compact Image Code, IEEE Transactions on Communications, vol.31, issue.4, pp.532-540, 1983. ,
DOI : 10.1109/TCOM.1983.1095851
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.136.5161
Fast algorithms for discrete and continuous wavelet transforms, IEEE Transactions on Information Theory, vol.38, issue.2, pp.569-586, 1992. ,
DOI : 10.1109/18.119724
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.598.7428
Wavelet Analysis and its applications, vol. I. An Introduction to Wavelets, 1992. ,
Ten lectures on wavelets, Philadelphia : CBMS-NSF Series in Appl, Math, 1992. ,
DOI : 10.1137/1.9781611970104
Biorthogonal bases of compactly supported wavelets, Communications on Pure and Applied Mathematics, vol.10, issue.5 ,
DOI : 10.1002/cpa.3160450502
A theory for multiresolution signal decomposition: the wavelet representation, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.11, issue.7, pp.674-693, 1989. ,
DOI : 10.1109/34.192463
Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math, vol.XLI, issue.7, pp.909-996, 1988. ,
A filter family designed for use in quadrature mirror filter banks, ICASSP '80. IEEE International Conference on Acoustics, Speech, and Signal Processing, pp.291-294, 1980. ,
DOI : 10.1109/ICASSP.1980.1171025
Necessary and sufficient conditions for constructing orthonormal wavelet bases, Tech. Rep. AD900402, Aware, inc, 1990. ,
DOI : 10.1063/1.529093
Two-Scale Difference Equations II. Local Regularity, Infinite Products of Matrices and Fractals, SIAM Journal on Mathematical Analysis, vol.23, issue.4, pp.1031-1079, 1992. ,
DOI : 10.1137/0523059
Non-separable bidimensional wavelet bases, Revista Matem??tica Iberoamericana ,
DOI : 10.4171/RMI/133
Sobolev regularity of wavelets and stability of iterated filter banks, Proc. Int. Colloqueum " Wavelets and Applications, 1992. ,
On the regularity of wavelets, IEEE Transactions on Information Theory, vol.38, issue.2, pp.872-876, 1992. ,
DOI : 10.1109/18.119743
Exact reconstruction techniques for tree-structured subband coders, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.34, issue.3, pp.434-441, 1986. ,
DOI : 10.1109/TASSP.1986.1164832
Subdivision schemes in CADG Advances in numerical analysis II : Wavelets, subdivision algorithms and radial functions, pp.36-104, 1991. ,
Multi-dimensional sub-band coding: Some theory and algorithms, Signal Processing, vol.6, issue.2, pp.97-112, 1984. ,
DOI : 10.1016/0165-1684(84)90012-4
URL : http://infoscience.epfl.ch/record/33932
Wavelet regularity of iterated filter banks with rational sampling changes, IEEE International Conference on Acoustics Speech and Signal Processing, 1993. ,
DOI : 10.1109/ICASSP.1993.319473
u L est pair, il existe un complément bi-orthogonal ,
En effet, puisque la longueur des filtres (symétriques) est paire, ces filtres admettent nécessairement chacun un zérò a z = ?1 On peut doncécriredoncécrire G(z) = (1 + z ?1 )F (z) et G ? (z) = (1 + z ?1 )F ? (z) En observant (5.4) on remarque qu'alors le couple formé des filtres de longueur impaire, (1 + z ?1 ) 2 F (z) et F ? (z) est solution, o` u F ? (z) se déterminè a partir de (1 + z ?1 ) 2 F (z) par l'algorithme CB 0 . Il suffit donc, appelé CB 1 dans la suite, se déduit immédiatement du précédent ,
partir d'un filtre passe-bas G(z) de longueur L ayant K zéroszérosà z = ?1, de déterminer un complément biorthogonal G ? (z) ayantégalementayantégalement K zéroszérosà z = ?1. Pour ce faire, il suffit d'ajouter K tels zéroszérosà G(z), d'appliquer CB 0, ) est alors L ? = L + 2(K ? 1). Imposer une forte contrainte de platitude augmente donc la dissymétrie des longueurs ,
on imposerait un nombre différent de zéros zérosà z = ?1 dans les deux filtres. On a préféré cependant utiliser le même K, afin de simplifier leprobì eme et de s'approcher du cas orthogonal en ,
z) (cf. § 5.2.2) pour obtenir son complément birothogonal G ? (z) ,
Multiresolution signal decomposition, 1992. ,
A Fast Karhunen-Loeve Transform for a Class of Random Processes, IEEE Transactions on Communications, vol.24, issue.9, pp.1023-1029, 1976. ,
DOI : 10.1109/TCOM.1976.1093409
On the Application of Haar Functions, IEEE Transactions on Communications, vol.21, issue.3, pp.209-216, 1973. ,
DOI : 10.1109/TCOM.1973.1091637
Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math, vol.XLI, issue.7, pp.909-996, 1988. ,
Orthonormal Bases of Compactly Supported Wavelets II. Variations on a Theme, SIAM Journal on Mathematical Analysis, vol.24, issue.2, 1993. ,
DOI : 10.1137/0524031
Biorthogonal bases of compactly supported wavelets, Communications on Pure and Applied Mathematics, vol.10, issue.5 ,
DOI : 10.1002/cpa.3160450502
Wavelets and filter banks: theory and design, IEEE Transactions on Signal Processing, vol.40, issue.9, 1992. ,
DOI : 10.1109/78.157221
URL : http://infoscience.epfl.ch/record/33904
Filters for distortion-free two-band multirate filter banks, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.33, issue.3, pp.626-630, 1985. ,
DOI : 10.1109/TASSP.1985.1164587
Exact reconstruction techniques for tree-structured subband coders, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.34, issue.3, pp.434-441, 1986. ,
DOI : 10.1109/TASSP.1986.1164832
Lattice structures for optimal design and robust implementation of two-channel perfect-reconstruction QMF banks, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.36, issue.1, pp.81-94, 1988. ,
DOI : 10.1109/29.1491
On the approximation problem in nonrecursive digital filter design, IEEE Transactions on Circuit Theory, vol.18, issue.3, pp.411-413, 1971. ,
DOI : 10.1109/TCT.1971.1083275
Explicit formula for the coefficients of maximally flat nonrecursive digital filter transfer functions expressed in powers of cos w, Proc. IEEE, pp.1135-1136, 1985. ,
DOI : 10.1109/PROC.1985.13244
The discrete wavelet transform: wedding the a trous and Mallat algorithms, IEEE Transactions on Signal Processing, vol.40, issue.10, pp.2464-2482, 1992. ,
DOI : 10.1109/78.157290
Chebyshev Approximation for Nonrecursive Digital Filters with Linear Phase, IEEE Transactions on Circuit Theory, vol.19, issue.2, pp.189-194, 1972. ,
DOI : 10.1109/TCT.1972.1083419
On optimum nonrecursive digital filters, Proc. 9th Allerton Conf. Circuit System Theory, 1971. ,
Une méthode simple de calcul de bancs de filtres/ondelettes bi-orthogonales, Proc. European Signal Processing Conf. (EUSIPCO), 1993. ,
Image coding using lattice vector quantization of wavelet coefficients, [Proceedings] ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing, pp.2273-2276, 1991. ,
DOI : 10.1109/ICASSP.1991.150745
Two-channel perfect-reconstruction FIR QMF structures which yield linear-phase analysis and synthesis filters, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.37, issue.5, pp.676-690, 1989. ,
DOI : 10.1109/29.17560
URL : http://authors.library.caltech.edu/6339/1/NGUieeetassp89.pdf
Lagrange multiplier approaches to the design of two-channel perfect reconstruction linear phase FIR filter banks, Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, pp.1731-1734, 1990. ,
Fast algorithms for discrete and continuous wavelet transforms, IEEE Transactions on Information Theory, vol.38, issue.2, pp.569-586, 1992. ,
DOI : 10.1109/18.119724
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.598.7428
Efficient multirate realization for narrow transition-band FIR filters, 1988., IEEE International Symposium on Circuits and Systems, pp.2019-2022, 1988. ,
DOI : 10.1109/ISCAS.1988.15338
Analyse, synthèse et complexité de calcul de bancs de filtres numériques, 1986. ,
Short-length FIR filters and their use in fast nonrecursive filtering, IEEE Transactions on Signal Processing, vol.39, issue.6, pp.1322-1332, 1991. ,
DOI : 10.1109/78.136539
Multirate digital filters, filter banks, polyphase networks, and applications: a tutorial, Proc. IEEE, pp.56-93, 1990. ,
DOI : 10.1109/5.52200
Perfect reconstruction FIR filter banks: some properties and factorizations, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.37, issue.7, pp.1057-1071, 1989. ,
DOI : 10.1109/29.32283
URL : http://infoscience.epfl.ch/record/33918
Image coding using wavelet transform, Speech, Signal Processing ,
DOI : 10.1109/83.136597
URL : https://hal.archives-ouvertes.fr/hal-01322224
Multifrequency channel decompositions of images and wavelet models, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.37, issue.12, pp.2091-2110, 1989. ,
DOI : 10.1109/29.45554
Extension of finite length signals for sub-band coding, Signal Processing, vol.17, issue.2, pp.161-168, 1989. ,
DOI : 10.1016/0165-1684(89)90019-4
Compression d'images par ondelettes, 1992. ,
Transformée en ondelettes et compression numérique des images, 1991. ,
Wavelets on the interval, Proc. Int. Colloqueum " Wavelets and Applications, 1992. ,
URL : https://hal.archives-ouvertes.fr/hal-01311753
Time-varying orthonormal tilings of the time-frequency plane, IEEE International Conference on Acoustics Speech and Signal Processing ,
DOI : 10.1109/ICASSP.1993.319471
Multiresolution signal decomposition, 1992. ,
Signal-adapted multiresolution transform for image coding, IEEE Transactions on Information Theory, vol.38, issue.2, pp.897-904, 1992. ,
DOI : 10.1109/18.119749
Pyramidal lattice vector quantization for multiscale image coding, IEEE Transactions on Image Processing, vol.3, issue.4 ,
DOI : 10.1109/83.298393
Sphere packings, lattices and groups, 1988. ,
Asymptotic quantization error of continuous signals and the quantization dimension, IEEE Transactions on Information Theory, vol.28, issue.2, pp.139-149, 1982. ,
DOI : 10.1109/TIT.1982.1056490
An Algorithm for Vector Quantizer Design, IEEE Transactions on Communications, vol.28, issue.1, pp.84-95, 1980. ,
DOI : 10.1109/TCOM.1980.1094577
Asymptotically optimal block quantization, IEEE Transactions on Information Theory, vol.25, issue.4, 1979. ,
DOI : 10.1109/TIT.1979.1056067
Efficient bit allocation for an arbitrary set of quantizers (speech coding), IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.36, issue.9, pp.1445-1453, 1988. ,
DOI : 10.1109/29.90373
Efficient quadree coding of images and video, Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, pp.2661-2664, 1991. ,
DOI : 10.1109/icassp.1991.150949
Best wavelet packet bases in a rate-distortion sense, IEEE Transactions on Image Processing, vol.2, issue.2, 1993. ,
DOI : 10.1109/83.217221
On the choice of " wavelet " filters for still image compression, Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, 1993. ,
A filter family designed for use in quadrature mirror filter banks, ICASSP '80. IEEE International Conference on Acoustics, Speech, and Signal Processing, pp.291-294, 1980. ,
DOI : 10.1109/ICASSP.1980.1171025
Transformée en ondelettes et compression numérique des images, 1991. ,
Some aspects of perception-based image coding, 1989. ,
L'analyse par ondelettes, Pour La Science, issue.119, pp.28-37, 1987. ,
Wavelets and signal processing, IEEE Signal Processing Magazine, vol.8, issue.4, pp.14-38, 1991. ,
DOI : 10.1109/79.91217
Fast algorithms for discrete and continuous wavelet transforms, IEEE Transactions on Information Theory, vol.38, issue.2, pp.569-586, 1992. ,
DOI : 10.1109/18.119724
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.598.7428
A Remez exchange algorithm for orthonormal wavelets, IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, vol.41, issue.8 ,
DOI : 10.1109/82.318943
Analysis and synthesis of sound signals using a discrete wavelet transform (DWT), 13th International Congress on Acoustics, 1989. ,
Fourier and wavelet spectrograms seen as smoothed Wigner-Ville distributions, Proc. Int. Colloqueum " Wavelets and Applications, pp.93-103, 1989. ,
Wavelets and affine smoothing of the Wigner-Ville distribution, Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, pp.2455-2458, 1990. ,
Transformées en ondelettes discrètes et continues?Comparaison et algorithmes rapides, inTreizì eme Colloque GRETSI, vol.1, pp.193-196, 1991. ,
Wavelet regularity of iterated filter banks with rational sampling changes, IEEE International Conference on Acoustics Speech and Signal Processing, 1993. ,
DOI : 10.1109/ICASSP.1993.319473
Une méthode simple de calcul de bancs de filtres/ondelettes bi-orthogonales, Proc. European Signal Processing Conf. (EUSIPCO), 1993. ,
Articles sur les transformées en ondelettes soumis par le département ETP Rapport Technique CRPE, CRPE (CNET/CNRS), vol.192, pp.38-40, 1991. ,
H) plots magnitude (frequency) response of FIR filter described by vector H. By default, a linear scale is assumed, MAGNITUDE, issue.1 ,
gives stop-band attenuation, in dB, of half-band FIR filter with transition band (0.25 ? B/2, 0.25 + B/2), where B is the normalized transition bandwidth) assumes H corresponds to a square magnitude, =ATTENUATION(. . . ) also gives the tolerance ? in the stop band ,
This is a measure of phase distortion : If it is close to zero, then H is close to being linear phase. If it is zero, then the filter is linear phase CAUTION : This measure is generally spoiled by zeros in the stop band's magnitude response, causing diracs to appear in the group delay. However, phase transitions by ? should not be a problem. [var1,var2]=PHASEVAR(H) gives group delay variations in the first (0, ?/2) and second (?/2, ?) half band, respectively. This measure is more significant for half-band filters (var1 for low-pass filters, var2 for half-band filters) and avoids the problem of Diracs ,
Zeroes on the unit circle are not selectable : They are retained with twice less multiplicity.) ZP can be obtained from design routines like REMEZWAV, etc. The resulting filter is normalized such that NORM(H)=1, but this may be inappropriate for e.g. MAGNITUDE (gain problem). (Note that checking the result with ZEROES will add computational round-off errors ,
describe filters H n of the same magnitude response (as given by FACTORALL), returns this matrix, sorted from the closest to the farthest from linear phase. A filter is closer to linear phase if its group delay deviation in the pass-band, as given by PHASEVAR, is smaller, HSORTED,INDEX]=PHASESORT(H) also gives the corresponding indexes of filters n = 0, 1, . . . (=phase codes as described in FACTORALL). [HSORTED,INDEX,VAR]=PHASESORT(H) also gives the group delay variations ,
gives optimum low-pass filter described by vector H, of length L, with K zeroes at the Nyquist frequency, normalized transition bandwidth B, normalized frequency offset ? = (? p +? s )/2?0.25, and (optional) weight coefficient C = ? 1 /? 2 (pass-band attenuation is greater as C is larger). [H,? 1 ]=REMEZZ(. . . ) also gives the maximum deviation ? 1 in the passband (? 2 = ? 1 /C, attenuation is ?20 log 10,0) uses global variable glob as initial guess of alternations and set glob for next call (to minimize the number of iterations) ,
the nth derivative of the scaling function (father wavelet) associated to low-pass FIR filter described by vector H. DERIV(H) sets n to 1. DERIV(H,G,n) plots the nth derivative of the (mother) wavelet associated to low-pass and high-pass filters described by vectors H and G, respectively. More generally, DERIV(H,G,n) plots the nth derivative of the limit function of an iterative " subdivision " procedure whose initial sequence is G, DERIV(H,n) sets G = H. By default, the number of iterations is the maximum permissible on this computer ,
Hölder Regularity of filter described by vector H, as estimated by a sharp upper bound. (The regularity may be negative.) Removal of zeroes in H at half the sampling frequency is done assuming remainder values < 10 ?3, REG(H,Z) forces the number Z of such zeros ,
If L =length(H) is even, L ? =length(H ? )=length(H) If L =length(H) is odd ,