Fundamental Papers in Wavelet Theory

Chris Heil is Professor of Mathematics at the Georgia Institute of Technology. His research interests are in harmonic analysis, especially time-frequency and time-scale methods and their applications. David Walnut is Professor of Mathematics at George Mason University. His research interests are also in harmonic analysis, especially sampling theory, Radon transforms, and tomography. He is the author of "Introduction to Wavelet Analysis". Ingrid Daubechies is the author of "Ten Lectures on Wavelets", which won the American Mathematical Society's 1994 Leroy P. Steele Prize for exposition.

• Contributor Affiliations ix Preface: Christopher Heil and David F. Walnut xiii Acknowledgments xiv Foreword: Ingrid Daubechies xv Introduction: John J. Benedetto 1 Section I. Precursors in Signal Processing Introduction: Jelena Kovacevic 23 1.Peter J. Burt and Edward H. Adelson, The Laplacian pyramid as a compact image code, IEEE Trans. Commun., 31 (1983), 532-540. 28 2.R. E. Crochiere, S. A. Webber, and J. L. Flanagan, Digital coding of speech in sub-bands, Bell System Technical J., 55 (1976), 1069-1085. 37 3.D. Esteban and C. Galand, Application of quadrature mirror filters to split-band voice coding schemes, ICASSP'77, IEEE Internat. Conf. on Acoustics, Speech, and Signal Processing, 2, April 1977, 191-195. 54 4.M.J.T. Smith and T. P. Barnwell III, A procedure for designing exact reconstruction filter banks for tree-structured subband coders, ICASSP'84, IEEE Internat. Conf. on Acoustics, Speech, and Signal Processing, 9, March 1984, 421-424. 59 5.Fred Mintzer, Filters for distortion-free two-band multirate filter banks, IEEE Trans. Acoust., Speech, and Signal Proc., 33 (1985), 626-630. 63 6.Martin Vetterli, Filter banks allowing perfect reconstruction, Signal Processing, 10 (1986), 219-244. 68 7.P. P. Vaidyanathan, Theory and design of M-channel maximally decimated quadrature mirror filters with arbitrary M, having the perfect-reconstruction property, IEEE Trans. Acoust., Speech, and Signal Proc., 35 (1987), 476-492. 94 Section II. Precursors in Physics: Affine Coherent States Introduction: Jean-Pierre Antoine 113 1.Erik W. Aslaksen and John R. Klauder, Continuous representation theory using the affine group, J. Math. Physics, 10 (1969), 2267-2275. 117 2.A. Grossmann, and J. Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. Math. Anal., 15 (1984), 723-736. 126 3.A. Grossmann, J. Morlet, and T. Paul, Transforms associated to square integrable group representations I, J. Math. Physics, 26 (1985), 2473-2479. 140 Section III. Precursors in Mathematics: Early Wavelet Bases Introduction: Hans G. Feichtinger 149 1.Alfred Haar, Zur Theorie der orthogonalen Funktionensysteme [On the theory of orthogonal function systems], Mathematische Annalen, 69 (1910), 331-371. Translated by Georg Zimmermann. 155 2.Philip Franklin, A set of continuous orthogonal functions, Mathematische Annalen, 100 (1928), 522-529. 189 3.Jan-Olov Stromberg, A modified Franklin system and higher-order spline systems on Rn as unconditional bases for Hardy spaces, Conf. on Harmonic Analysis in Honor of A. Zygmund, Vol. II, W. Beckner et al., eds., Wadsworth (Belmont, CA), (1983), 475-494. 197 4.Yves Meyer, Principe d'incertitude, bases hilbertiennes et algebres d'operateurs [Uncertainty principle, Hilbert bases, and algebras of operators], Seminaire Bourbaki, 1985/86. Asterisque No. 145-146 (1987), 209-223. Translated by John Horvath. 216 5.P. G. Lemarie and Y. Meyer, Ondelettes et bases hilbertiennes [Wavelets and Hilbert bases], Revista Matematica Iberoamericana, 2 (1986), 1-18. Translated by John Horvath. 229 6.Guy Battle, A block spin construction of ondelettes I, Comm. Math. Physics, 110 (1987), 601-615. 245 Section IV. Precursors and Development in Mathematics: Atom and Frame Decompositions Introduction: Yves Meyer 263 1.R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), 341-365. 269 2.Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83 (1977), 569-645. 295 3.Ingrid Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Physics, 27 (1986), 1271-1283. 372 4.Michael Frazier and Bjorn Jawerth, Decompositions of Besov spaces, Indiana Univ. Math. J., 34 (1985), 777-799. 385 5.Hans G. Feichtinger and K. H. Grochenig, Banach spaces related to integrable group representations and their atomic decompositions I, J. Funct. Anal., 86 (1989), 307-340. 408 6.Ingrid Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory, 39 (1990), 961-1005. 442 Section V. Multiresolution Analysis Introduction: Guido Weiss 489 1.Stephane G. Mallat, A theory for multiresolution signal decomposition: The wavelet representation, IEEE Trans. Pattern Anal. Machine Intell., 11 (1989), 674-693. 494 2.Yves Meyer, Wavelets with compact support, Zygmund Lectures, U. Chicago (1987). 514 3.Stephane G. Mallat, Multiresolution approximations and wavelet orthonormal bases for L2(R), Trans. Amer. Math. Soc., 315 (1989), 69-87. 524 4.A. Cohen, Ondelettes, analysis multiresolutions et filtres mirroirs en quadrature [Wavelets, multiresolution analysis, and quadrature mirror filters], Ann. Inst. H. Poincare, Anal. Non Lineaire, 7 (1990), 439-459. Translated by Robert D. Ryan. 543 5.Wayne M. Lawton, Tight frames of compactly supported affine wavelets, J. Math. Phys., 31 (1990), 1898-1901. 560 6.Ingrid Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math., 41 (1988), 909-996. 564 Section VI. Multidimensional Wavelets Introduction: Guido Weiss 655 1.Yves Meyer, Ondelettes, fonctions splines et analyses graduees [Wavelets, spline functions, and multiresolution analysis], Rend. Sem. Mat. Univ. Politec. Torino, 45 (1987), 1-42. Translated by John Horvath. 659 2.Karlheinz Grochenig, Analyse multi-echelle et bases d'ondelettes [Multiscale analyses and wavelet bases], C. R. Acad. Sci. Paris Serie I, 305 (1987), 13-17. Translated by Robert D. Ryan. 690 3.Jelena Kovacevic and Martin Vetterli, Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for Rn, IEEE Trans. Inform. Theory, 38 (1992), 533-555. 694 4.K. Grochenig and W. R. Madych, Multiresolution analysis, Haar bases and self-similar tilings of Rn, IEEE Trans. Inform. Theory, 38 (1992), 556-568. 717 Section VII. Selected Applications Introduction: Mladen Victor Wickerhauser 733 1.G. Beylkin, R. Coifman, and V. Rokhlin, Fast wavelet transforms and numerical algorithms, I, Comm. Pure Appl. Math., 44 (1991), 141-183. 741 2.Ronald A. DeVore, Bjorn Jawerth, Vasil Popov, Compression of wavelet decompositions, Amer. J. Math., 114 (1992), 737-785. 784 3.David L. Donoho and Iain M. Johnstone, Adapting to unknown smoothness by wavelet shrinkage, J. Amer. Statist. Assoc., 90 (1995), 1200-1224. 833 4.Stephane Jaffard, Exposants de Holder en des points donne et coefficients d'ondelettes [Holder exponents at given points and wavelet coefficients], C. R. Acad. Sci. Paris Serie I, 308 (1989), 79-81. Translated by Robert D. Ryan. 858 5.Jerome M. Shapiro, Embedded image coding using zerotrees of wavelet coefficients, IEEE Trans. Signal Processing, 41 (1993), 3445-3462. 861

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