×

zbMATH — the first resource for mathematics

Orthonormal bases of compactly supported wavelets. (English) Zbl 0644.42026
We construct compactly supported functions \(_ N\psi\) \((N\in\mathbb N)\) such that the family of “wavelets” \(\{_ N\psi _{mn};\, m,n\in\mathbb Z\}\), where \(_ N\psi _{mn}(x)=2^{-m/2}\psi (2^{-m}x-n),\) constitutes an orthonormal basis of \(L^ 2(\mathbb R)\). These functions \({}_ N\psi\) generalize the Haar basis; the case \(N=1\) reduces to the Haar basis. Typically support \((_ N\psi)=[-N+1,N]\); exists \(\mu >0\) so that \(_ N\psi \in C^{\mu N}\) for all \(N\in\mathbb N\). The function \(_ N\psi\) can thus be chosen to be of arbitrarily high regularity, at the price of increasing the support width of \(_ N\psi\). The paper starts by reviewing the concept of multiresolution analysis, as well as several algorithms used in vision analysis. The construction of the present orthonormal bases of compactly supported wavelets then follows from a synthesis of these different approaches.
Reviewer: Ingrid Daubechies

MSC:
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
65T60 Numerical methods for wavelets
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Grossman, SIAM J. Math. Anal. 15 pp 723– (1984)
[2] Goupillaud, Geoexploration 23 pp 85– (1984)
[3] Calderón, Studia Math. 24 pp 113– (1964)
[4] Calderón, Adv. Math. 16 pp 1– (1974)
[5] and , Physics Reports 12C, 1974, p. 77.
[6] and , Quantum Physics: a Functional Integral Point of View, Springer, New York, 1981.
[7] Principe d’incertitude, bases hilbertiennes et algèbres d’opérateurs, Séminaire Bourbaki, 1985-1986, nr. 662.
[8] Tchamitchian, C. R. Acad. Sc. Paris. 303, série 1 pp 215– (1986)
[9] Biorthogonalité et théorie des opérateurs, to be published in Rev. Mat. Iberoamericana.
[10] and , Ondelettes and phase cell cluster expansions; a vindication, Comm. Math. Phys., 1987.
[11] Federbush, Bull. Am. Math. Soc. 17 pp 93– (1987)
[12] , and , Analysis of sound patterns through wavelet transforms, to be published in International Journal on Pattern Analysis and Artificial Intelligence.
[13] A theory for multiresolution signal decomposition: the wavelet transform, Preprint GRASP Lab, Dept. of Computer and Information Science, Univ. of Pennsylvania, to be published.
[14] Wavelet transforms and edge detection, to be published in Stochastic Processes in Physics and Engineering (Ph. Blanchard, L. Streit and M. Hasewinkel, eds.),
[15] Grossmann, I. J. Math. Phys. 26 pp 2473– (1985)
[16] Ann. Inst. H. Poincaré 45 pp 293– (1986)
[17] Aslaksen, J. Math. Phys. 9 pp 206– (1968)
[18] J. Math. Phys. 10 pp 2267– (1969)
[19] Paul, J. Math. Phys. 25 (1985)
[20] pp. 3252–3263. Affine coherent states and the radial Schrödinger equation. I. Radial harmonic oscillator and hydrogen atom, to be published.
[21] and , Private demonstration.
[22] Daubechies, J. Math. Phys. 27 pp 1271– (1986)
[23] The wavelet transform, time-frequency localization and signal analysis, Preprint AT&T Bell Laboratories; to be published.
[24] Ondelettes et functions splines, Séminaire EDP, Ecole Polytechnique, Paris, France, December, 1986,.
[25] Ondelettes à localisation exponentielle, to be published, in Journ. de MaTh. Pures et Appl.
[26] A block spin construction of ondelettes, Part I: Lemarié functions, Comm. Math. Phys. 1987.
[27] Multiresolution approximation and wavelets. Preprint GRASP Lab., Dept. of Computer and Information Science, Univ. of Pennsylvania, to be published.
[28] Burt, IEEE Trans. Comm. 31 pp 482– (1983)
[29] Burt, ACM Trans. on Graphics 2 pp 217– (1983)
[30] , , and , Multiscale analysis, unpublished memorandum.
[31] and , The discrete wavelet transform, preprint. Dept. of Math., Yale University.
[32] Lemarié, Rev. Mat. Iberoamericana 2 pp 1– (1986) · doi:10.4171/RMI/22
[33] Smith, IEEE Trans. on ASSP 34 pp 434– (1986)
[34] Thèse de Doctorat, Université de Paris-Dauphine 1988.
[35] private communication.
[36] and , Aufgaben and Lehrsätze aus der Analysis, Vol. II, Springer, Berlin, 1971. · doi:10.1007/978-3-642-61987-8
[37] Wavelets with compact support, Zygmund Lectures (University of Chicago) May 1987, and private communication.
[38] and , Interpolation dyadique, in Fractals; Dimensions non Entières et Applications, ed. G. Cherbit, Masson (Paris), 1987, pp. 44–45. See also other references listed there.
[39] and , Two-scale difference equations. I. Global regularity of solutions, and –. II. Infinite matrix products, local regularity and fractals, Preprints AT&T Bell Laboratories; to be published.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.