A. N. Akansu, R. A. Haddad, and H. Caglar, The Binomial QMF-Wavelet Transform for Multiresolution Signal Décomposition, submitted IEEE Trans. Signal Proc, 1990.

J. B. Auen and L. R. Rabiner, A Unified Approach to Short-Time Fourier Analysis and Synthesis, Proc. IEEE, vol.65, issue.11, pp.1558-1564, 1977.

M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, Image Coding Using Vector Quantization in the Wavelet Transform Domain, Proc, pp.2297-2300, 1990.

L. Auslander and I. Germer, Wide-Band Ambiguity Function and ajc+b group, Institute for Mathematies and its Applications, vol.1, pp.1-12, 1990.

G. F. Boudreaux-bartels, Time-Varying Signal Processing Using the Wigner Distribution Time-Frequency Signal Représentation, Adv. in Geopkysical Data Proc, vol.2, pp.33-79, 1985.

. Pj, . Burt, . Eji, and . Adelson, The Laplacian Pyramid as a Compact Image Code, * IEEE Trans. on Com, vol.31, issue.4, pp.532-540, 1983.

. Pj and . Burt, Multiresolution Techniques for Image Représentation, Analysis, and 'Smart' Transmission, Proc. SPIE Conf. on Visual Communication and Image Processing, pp.2-15, 1989.

A. Calderôn, Intermediate Spaces and Interpolation, the Complex Melhod, Studia Math, vol.24, pp.113-190, 1964.

B. A. Cipra, A New Wave in Appbed Mathematics, vol.249, 1990.

T. A. Claasen and W. G. Mecklenbrfluker, The Wigner Distribution -A Tool for Time-Frequency Signal

, Continuous-Time Signais, vol.1, pp.217-250, 1980.

L. Cohen, Generalized Phase-Space Distribution Functions, J. Math. Phys, vol.7, issue.5, pp.781-786, 1966.

A. Cohen, I. Daubechies, and J. C. Feauvean, Bionhogonal Bases of Compactly Supported Wavelets, 1990.

A. Croisier and C. Galand, Perfect Channel Splitting by Use of Interpolation, Decimation, Tree Décomposition Techniques, Int. Conf. on Information Sciences/Systems, pp.443-446, 1976.

R. E. Crochiere, S. Weber, and J. , Digital Coding of Speech in Subbands, Bell SYSL Tech. J, vol.55, pp.1069-1085

R. E. Crochiere and L. , Multirate Digital Signal Processing, 1983.

I. Daubechies, Onhononnal Bases of Compactly Supported Wavelets, Comm. in Pure and Applied Math, vol.1, issue.7, pp.909-996, 1988.

I. Daubechies, The Wavelet Transform, Time-Frequency Localization and Signal Analysis, IEEE Trans. on Info. Theory, vol.36, issue.5, pp.961-1005, 1990.

O. Rkxjl-and-m and £. Veiteri±wavel, TRANSFORMS, vol.29

I. Daubechies, Orthonormal Bases of Compaaly Supported Wavelets II. Variations on a Thème, 1990.

I. Daubechies and J. C. Lagarias, Two-Scale Différence Equations EL Local Regularity, Infinité Products of Matrices and Fractals, submitted SIAM J. Math. Anal, 1990.

R. J. Duftin and A. C. Schaeffer, A Class of Nonharmonic Fourier Séries, Trans. Am. Math. Soc, vol.72, pp.341-366, 1952.

D. Esteban and C. Galand, Application of Quadrature Mirror Filters to Split-Band Voice Codir g Schemes, Int. Conf. Acoust., Speech, Signal Proc, pp.191-195, 1977.

. Pjlandrin, Some Aspects of Non-Stationary Signal Processing with Emphasis on Time-Fiequency and Time-Scale Methods, pp.68-98, 1989.

P. Flandrin, Wavelets and Affine Smootfaing of the Wigner-Vnie Distribution, Proc. 1990 IEEE lut. Conf. Acoust., Speech, Signal Proc, vol.4, p.58, 1990.

J. Fourier, Théorie Analytique de la Chaleur, Oeuvres de Fourier, tome premier, CDarboux, 1888.

. Pjfranklin, A Set of Continuons Orthogonal Functions, Math. Annal, vol.100, pp.522-529, 1928.

M. Frazier and B. Jawerth, The ^-Transform and Décomposition of Distributions, Proc. Conf. Fonction Spaces and Appi, 1986.

D. Gabor, Theory of Communication, J. of me 8E, vol.93, pp.429-457, 1946.

P. Goupillaud, A. Grossmann, and J. Modet, Cycle-Octave and Related Transforms in Seismic Signal Analysis, Ceoexploration, vol.23, p.85, 1984.

A. Grossmann and J. Modet, Décomposition of Hardy Fonctions into Square Integrable Wavelets of Constant Shape, SIAM JMathAnal, vol.15, pp.723-736, 1984.

A. Grossmann, R. Kronland-martinet, and J. Morlet, Reading and Understanding Continuous Wavelet Transforms, pp.2-20, 1989.

A. Haar, Zur Théorie der Orthogonalen Funktionensysteme, Math. Ann, vol.69, pp.331-371, 1910.

W. ;. Ce and . Hefl, Wavelets and Frames, Signal Processing. Part 1: Signal Processing Theory, L Auslander, 1901.

, Institute for Mathematics and its Applications, vol.22, pp.147-160, 1990.

. Herrmann, On the Approximation Probem in NonRecursive Digital FDter Design, IEEE Trans. Circuit Theory, vol.18, issue.3, pp.41-42, 1971.

J. Littlewood and R. Paley, Theorems on Fourier Series and Power Series, vol.42, pp.52-89, 1937.

S. Mallat, Theory for Multiresolution Signal Décomposition: the Wavelet Représentation, IEEE Trans. on Patlern Analysis and Machine Intell, vol.1, issue.7, pp.674-693, 1989.

. S-mallat, Multifrequency Channel Décompositions of Images and Wavelet Models, IEEE Trans

A. .. Speech, Signal Proc, vol.37, issue.12, pp.2091-2110, 1989.

M. Rioul, . Veïterll, and . Transforms, , p.30

S. Mallat, Multiresolution Approximations and Wavelet Orthonormal Bases of L2(R), Trans. Amer. Math. Soc, vol.15, issue.l, pp.69-87, 1989.

Y. Meyer, Onhonormal Wavelets, pp.21-37, 1989.

[. Mey90]-y-meyer, O. Opérateurs, and T. I. , , 1990.

. Fmintza, Tilters for Distortion-Free Two-Band Multirate Filter Banks, IEEE Trans. on Acoust.. Speech, Signal Proc, vol.33, pp.626-630, 1985.

. Tpaul, Affine Cohérent States and the Radial SchrOdinger Equation 1. Radial Harmonie Oscillator and the Hydrogen Alom

P. Rioui and . Flandrin, Time-Scale Energy Distributions: A General Class Extending Wavelet Transforms, IEEE Trans. Signal Proc, 1990.

, A Unifying Multiresolution Theory for the Discrète Wavelet Transform, Regular Filter Banks and Pyramid Transforms, submitted to IEEE Trans. Signal Proc, 1990.

P. Rioul and . Duhamel, Structures and Fast Algorithms for Implementing Wavelet Transforms

E. Ajlosenfeld, Multiresolution Techniques in Computer Vision, 1984.

M. and T. P. Bamwell, Exact Reconstruction for Tree-Structured Subband Codera, IEEE Trans. on Acoust., Speech and Signal Proc, vol.34, pp.434-441, 1986.

. T. Mj and . Smith, A New Filter Bank Theory for Time-Frequency Représentation, IEEE Trans. on Acoust.. Speech and Signal Proc, issue.3, pp.314-327, 1987.

J. M. Speiser, Wide-Band Ambiguity Functions, IEEE Trans. on Info. Theory, pp.122-123, 1967.

. Pp, P. Vaidyanathan, and . Hoang, Lattice Structures for Optimal Design and Robust Implementation of Two-Band Perfect Reconstruction QMF Banks, IEEE Trans. on Acoust., Speech and Signal Proc, vol.36, issue.1, pp.81-94, 1988.

P. P. Vaidyanathan and Z. Doganata, The Role of Lossless Systems in Modem Digital Signal Processing, IEEE Trans. Education, Spécial issue on Circuits and Systems, vol.32, issue.3, pp.181-197, 1989.

M. Vetterii and D. L. Gall, Perfect Reconstruction FIR FUter Banks: Some Properties and Factorizations, IEEE Trans. on Acoust., Speech Signal Prpc, vol.37, issue.7, pp.1057-107, 1989.

M. Veuerli and C. Heriey, Wavelets and Filter Banks: Relationships and New Results, Proc. 1990 IEEE Int. Conf. Acoust.. Speech, Signal Proc, pp.1723-1726, 1990.

M. Veoerli and C. Herley, Wavelets and Filter Banks: Theory and Design: submitted, IEEE Trans. on Signal Proc, 1990.

. Wavelets, Time-Frequency Methods and Phase Space, Proc. Int. Conf. Marseille, Françe, 1987.

P. M. Woodward, Probability and Information Theory with Application to Radar, p.33, 1953.

M. Ricul, . Veiterll, and . Transforms, , p.31

J. E. Younberg and . Boll, Constant-Q Signal Analysis and Synthesis, IEEE Int. Conf. on Acoust

, Speech, and Signal Proc. ICASSP-78, pp.375-378, 1978.

, Books on Wavelets: see [WAV89], [MEYSO] and I. Daubechies, Wavelets, Lectun Notes

. Wavelets, R. R. Applications, and . Coifman,

W. Tutorials-on,

R. R. Coifman, Wavelet Analysis and Signal Processing, Signal Processing, Part 1: Signal Processing Theory, vol.22, 1990.

C. E. Heil and D. F. Walnut, Continuous and Discrète Wavelet Transforms, SIAM Review, vol.31, issue.4, pp.628-666, 1989.

Y. Meyer and S. Jaffard, L'Analyse par Ondeleoes, pp.28-37, 1987.

G. Strang, Wavelets and Diladon Equations: A Brief Introduction, SIAM Review, vol.31, issue.4, pp.614-627, 1989.

, Mathematics. Mathematical Physics

P. G. Lemarié and Y. Meyer, Ondelettes et Bases Hilbertiennes, Revista Matematica Iberoamericana, vol.2, issue.142, pp.1-18, 1986.

G. Battle, A Block Spin Construction of Ondelettes, IL The Quantum Field Theory (QFT) Connection, Comm. Math. Phys, vol.114, pp.93-102, 1988.

S. Dubuc, Interpolation Through an Itérative Scheme, J. Math. Analysis Appl, vol.114, pp.185-204, 1986.

M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, Image Coding Using Vector Quantization in the Wavelet Transform Domain, Proc. 1990 IEEE InL Conf. AcousL, Speech, Signal Proc, pp.2297-2300, 1990.

T. Blu,

. Idaubechies, Orthonormal Bases of Compactly Supportée Wavelets, Comm. in Pure and Applied Math, vol.41, issue.7, pp.909-996, 1988.

I. Daubechies, Orthonormal Bases of Compactly Supported Wavelets II. Variations on a Thème

I. Daubechies and J. C. Lagarias, Two-Scale Difference Equations IJZxistence and Global Regularity of Solutions, to appear in SIAM, J.Math. Anal

I. Daubechies and J. C. Lagarias, Two-Scale Difference Equations HLocal Regularity, Infinite Products of Matrices and Fractals, to appear in SIAM, J.Math. Anal

I. Daubechies,

G. Deslauriers and S. Dubuc, Interpolation Dyadique et Fractals, Dimensions non entières et applications, G, pp.44-56, 1987.

G. Deslauriers and S. Dubuc, Symmetric Itérative Interpolation Processes, Constructive Approximation, vol.S, pp.49-68, 1989.

S. Dubuc, Interpolation Through an Itérative Scheme, J. Math. Anal. Appl, vol.1, issue.14, pp.185-204, 1986.

N. Dyn, D. Levin, and J. A. Gregory, A 4-point Interpolatory Subdivision Scheme for Curve Design, Computer Aided Géométrie Design, vol.4, pp.2-7, 1987.

C. A. Micchelli and H. Prautzsch, Refinement and Subdivision for Spaces of Integer Translates of a Compactly Supported Function, pp.192-222, 1987.

. Rioul, A Unifying Discrete-Time Multiresolution Theory that unifies Octave-Bond Filter Banks, Wavelet and Pyramid Transforms, submitted to IEEE Trans, Signal Proc, 1990.

. Mj and . Shensa, Affine Wavelets: Wedding the Atrous and Mallat Algorithms

M. Veuerli and C. Herley, Wavelets and Filter Banks: Relationships and New Results, Proc. 1990 IEEE Int. Conf. Acoust., Speech, Signal Proc, pp.1723-1726, 1990.

, The minimal value of C(N) is obtained for N = N* such that the first derivative of C(N) vanishes. One has CQf)=minN C(N) =

. Where-n*-satisfies-n*-=, U2-1)(1.ntr + 1 -ln2/4) + 31n2. For large filter lengths L this gives IniV* = InL + 0

, for which the total number of operations (mults+adds) is 2L-1. The FFT-based DWT algorithm therefore a significantly improves the direct method

M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, Image Coding Using Vector Quantization in the

. Wavelet-transfonn-domain, Proc. 1990 IEEE Int. Conf. Acoust., Speech, Signal Proc, pp.2297-2300, 1990.

J. Bertrand, P. Bertrand, and J. P. Ovariez, Discrète Mellin Transfonn for Signal Analysis, Proc. 1990 IEEE Int. Conf. Acoust., Speech, Signal Proc, pp.1603-1606, 1990.

A. Cohen, I. Daubechies, and J. C. Feauveau, Biorthogonal Bases of Compactly Supported Wavelets

I. Daubechies, Orthonormal Bases of Compactly Supported Wavelets, Comm. in Pure and Applied Math, vol.41, issue.7, pp.909-996, 1988.

. Daubechies, The Wavelet Transfonn, Time-Frequency Localization and Signal Analysis, IEEE Trans. on Info. Theory, vol.36, issue.5, pp.961-1005, 1990.

I. Daubechies and J. C. Lagarias, Two-Scale Différence Equations LExistence and Global Regularity of Solutions, 1990.

G. Deslauriers and S. Dubuc, Symmetric Itérative Interpolation Processes, Constructive Approximation, vol.5, pp.49-68, 1989.

P. Duhamel, Implementation of Split-Radix FFT Algorithms for Complex, Real, and Real-Symmetric Data, IEEE Trans. Acoust., Speech, Signal Proc, vol.34, issue.2, pp.285-295, 1986.

P. Dutilleux, An Implementation of the 'Algorithme à Trous' to Compute the Wavelet Transform, pp.298-304, 1989.

P. Flandrin, Some Aspects of Non-Stationary Signal Processing with Emphasis on Tune-Frequency and TimeScale Methods, pp.68-98, 1989.

. A. I-i]-r, C. S. Gopinath, and . Burrus, Efficient Computation of the Wavelet Transforms, Proc. 1990 IEEE Int. Conf. Acoust., Speech, Signal Proc, pp.1599-1601, 1990.

P. Goupillaud, A. Grossmann, and J. Morlet, Cycle-Octave and Related Transforms in Seismic Signal Analysis, Geoexploration, vol.23, p.85, 1984.

A. Grossmann and R. Kronland-martinet, Time and Scale Représentations Obtained Through Continuous Wavelet Transforms, Proc. Int. Conf. EUSIPCO'88, Signal Processing IV: Théories and Applications, pp.475-482, 1988.

C. E. Heil and D. F. Walnut, Continuous and Discrète Wavelet Transfonns, SIAM Review, vol.31, issue.4, pp.628-666, 1989.

. Cje and . Heil, Wavelets and Frames, Signal Processing, Part 1: Signal Processing Theory, L. Auslander, T

, Institute for Mathematics and its Applications, vol.22, 1990.

A. Grossmann, J. Morlet, and T. Paul, Transfonns Associated to Square Integrable Group Représentations. I. General Results, /. Math. Phys, vol.26, issue.10, pp.2473-2479, 1985.

. Rioul and . Duhamel,

M. Holschneider, R. Kronland-martinet, J. Morlet, and P. Tchamitchian, A Real-Time Algorithm for Signal Analysis with the Help of the Wavelet Transform, pp.286-288, 1989.

R. Kronland-martinet, J. Morlet, and A. Grossmann, Analysis of Sound Patterns Through Wavelet Transforms, Int. J. Pattern Récognition and Araficial Intelligence, issue.2, pp.97-126, 1987.

S. Mallat, A Theory for Multiresolution Signal Décomposition: the Wavelet Représentation, IEEE Trans. on Pattern Analysis and Machine Intell, issue.7, pp.674-693, 1989.

S. Mallat, Multifrequency Channel Décompositions of Images and Wavelet Models, IEEE Trans. Acoust., Speech, Signal Proc, vol.37, issue.12, pp.2091-2110, 1989.

S. Mallat and S. Zhong, Signal Characterization from Muldscale Edges, Proc. lOth Int. Conf. Pattern Recognition, Pattern Récognition, Systems and Applications, pp.16-21, 1990.

Y. Meyer, Ondelettes et Opérateurs, [in French] Tome I, Hemnann éd, 1990.

. Zj, P. Mou, and . Duhamel, Short-Length FIR Filters and Their Use in Fast Nonrecursive Filtering, IEEE Trans. Signal Proc, 1991.

. Hj and . Nussbawner, Fast Fourier Transform and Convolution Algorithms, 1981.

. Rioul, Structures and Algorithms for the Orthonormal Discrète Wavelet Transform, Proc. 1990 Digital Signal Processing Workshop, pp.3-3, 1990.

. Rioul, Fast Algorithms for the Continuous Wavelet Transform, IEEE Int. Conf. Acoust., Speech, Signal Proc, 1991.

P. Rioul and . Flandrin, Time-Scale Energy Distributions: A New Class Extending Wavelet Transforms, Speech, Signal Proc, p.69288, 1990.

, Lyon Cedex 02

. Rioul, A Discrete-Time Multiresolution Theory that Unififes Octave-Band Filter Banks, Pyramid and Wavelet Transforms, IEEE Trans. Acoust., Speech, Signal Proc, 1990.

M. Vetterli, Wavelet Transforms in Signal Processing, 1991.

. Mj and . Shensa, Affine Wavelets: Wedding the Atrous and Mallat Algorithms, submitted to IEEE Trans. on Acoust.. Speech, and Signal Proc, 1990.

. T. Mj, T. P. Smith, and . Barnwell, Exact Reconstruction for Tree-Structured Subband Coders, IEEE Trans. Acoust., Speech, Signal Proc, vol.34, pp.434-441, 1986.

M. Vetterli and A. , Synthèse et Complexité de Calcul de Bancs de Filtres Numériques, 1986.

M. Vetterli, Running FIR and IIR Filtering Using Multirate Filter Banks, IEEE Trans. Acoust., Speech, Signal Proc, vol.36, issue.5, pp.730-738, 1988.

M. Vetterli and C. Herley, Wavelets and Filter Banks: Relationships and New Results, Proc. 1990 IEEE Int. Conf. Acoust., Speech, Signal Proc, p.172, 1990.

. Rioul and . Duhamel,

M. Vetterli and C. Herley, Wavelets and Filter Banks: Theory and Design, IEEE Trans. Acoust., Speech, Signal Proc, 1990.

. Wavelets, Time-Frequency Methods and Phase Space, Proc. Int. Conf. Marseille, France, 1987.