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Journal ArticleDOI

Multiresolution approximations and wavelet orthonormal bases of L^2(R)

01 Jan 1989-Transactions of the American Mathematical Society (American Mathematical Society (AMS))-Vol. 315, Iss: 1, pp 69-87
TL;DR: In this paper, the authors study the properties of multiresolution approximation and prove that it is characterized by a 2π periodic function, which is further described in terms of wavelet orthonormal bases.
Abstract: A multiresolution approximation is a sequence of embedded vector spaces   V j  jmember Z for approximating L 2 (R) functions. We study the properties of a multiresolution approximation and prove that it is characterized by a 2π periodic function which is further described. From any multiresolution approximation, we can derive a function ψ(x) called a wavelet such that   √  2 j ψ(2 j x −k)   (k ,j)member Z 2 is an orthonormal basis of L 2 (R). This provides a new approach for understanding and computing wavelet orthonormal bases. Finally, we characterize the asymptotic decay rate of multiresolution approximation errors for functions in a Sobolev space H s .

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Citations
More filters
Journal ArticleDOI
TL;DR: In this paper, it is shown that the difference of information between the approximation of a signal at the resolutions 2/sup j+1/ and 2 /sup j/ (where j is an integer) can be extracted by decomposing this signal on a wavelet orthonormal basis of L/sup 2/(R/sup n/), the vector space of measurable, square-integrable n-dimensional functions.
Abstract: Multiresolution representations are effective for analyzing the information content of images. The properties of the operator which approximates a signal at a given resolution were studied. It is shown that the difference of information between the approximation of a signal at the resolutions 2/sup j+1/ and 2/sup j/ (where j is an integer) can be extracted by decomposing this signal on a wavelet orthonormal basis of L/sup 2/(R/sup n/), the vector space of measurable, square-integrable n-dimensional functions. In L/sup 2/(R), a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi (x). This decomposition defines an orthogonal multiresolution representation called a wavelet representation. It is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters. Wavelet representation lies between the spatial and Fourier domains. For images, the wavelet representation differentiates several spatial orientations. The application of this representation to data compression in image coding, texture discrimination and fractal analysis is discussed. >

20,028 citations

Journal ArticleDOI
Ingrid Daubechies1
TL;DR: This work construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity, by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction.
Abstract: We construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity. The order of regularity increases linearly with the support width. We start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction. The construction then follows from a synthesis of these different approaches.

8,588 citations

Journal ArticleDOI
Olivier Rioul1, Martin Vetterli
TL;DR: A simple, nonrigorous, synthetic view of wavelet theory is presented for both review and tutorial purposes, which includes nonstationary signal analysis, scale versus frequency,Wavelet analysis and synthesis, scalograms, wavelet frames and orthonormal bases, the discrete-time case, and applications of wavelets in signal processing.
Abstract: A simple, nonrigorous, synthetic view of wavelet theory is presented for both review and tutorial purposes. The discussion includes nonstationary signal analysis, scale versus frequency, wavelet analysis and synthesis, scalograms, wavelet frames and orthonormal bases, the discrete-time case, and applications of wavelets in signal processing. The main definitions and properties of wavelet transforms are covered, and connections among the various fields where results have been developed are shown. >

2,945 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that under fairly general conditions, exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual Riesz bases of compactly supported wavelets.
Abstract: Orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide. We show here that under fairly general conditions, exact reconstruction schemes with synthesis filters different from the analysis filters give rise: to two dual Riesz bases of compactly supported wavelets. We give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions, and we present a sufficient condition for the decay of their Fourier transforms. We study the regularity of these biorthogonal bases. We provide several families of examples, all symmetric (corresponding to “linear phase” filters). In particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity; we also show how to construct symmetric biorthogonal wavelet bases “close” to a (nonsymmetric) orthonormal basis.

2,854 citations

Book
01 Feb 2006
TL;DR: Wavelet analysis of finite energy signals and random variables and stochastic processes, analysis and synthesis of long memory processes, and the wavelet variance.
Abstract: 1. Introduction to wavelets 2. Review of Fourier theory and filters 3. Orthonormal transforms of time series 4. The discrete wavelet transform 5. The maximal overlap discrete wavelet transform 6. The discrete wavelet packet transform 7. Random variables and stochastic processes 8. The wavelet variance 9. Analysis and synthesis of long memory processes 10. Wavelet-based signal estimation 11. Wavelet analysis of finite energy signals Appendix. Answers to embedded exercises References Author index Subject index.

2,734 citations

References
More filters
Journal ArticleDOI
TL;DR: In this paper, it is shown that the difference of information between the approximation of a signal at the resolutions 2/sup j+1/ and 2 /sup j/ (where j is an integer) can be extracted by decomposing this signal on a wavelet orthonormal basis of L/sup 2/(R/sup n/), the vector space of measurable, square-integrable n-dimensional functions.
Abstract: Multiresolution representations are effective for analyzing the information content of images. The properties of the operator which approximates a signal at a given resolution were studied. It is shown that the difference of information between the approximation of a signal at the resolutions 2/sup j+1/ and 2/sup j/ (where j is an integer) can be extracted by decomposing this signal on a wavelet orthonormal basis of L/sup 2/(R/sup n/), the vector space of measurable, square-integrable n-dimensional functions. In L/sup 2/(R), a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi (x). This decomposition defines an orthogonal multiresolution representation called a wavelet representation. It is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters. Wavelet representation lies between the spatial and Fourier domains. For images, the wavelet representation differentiates several spatial orientations. The application of this representation to data compression in image coding, texture discrimination and fractal analysis is discussed. >

20,028 citations

Journal ArticleDOI
Ingrid Daubechies1
TL;DR: This work construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity, by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction.
Abstract: We construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity. The order of regularity increases linearly with the support width. We start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction. The construction then follows from a synthesis of these different approaches.

8,588 citations

Journal ArticleDOI
TL;DR: In this article, the authors studied square integrable coefficients of an irreducible representation of the non-unimodular $ax + b$-group and obtained explicit expressions in the case of a particular analyzing family that plays a role analogous to coherent states (Gabor wavelets) in the usual $L_2 $ -theory.
Abstract: An arbitrary square integrable real-valued function (or, equivalently, the associated Hardy function) can be conveniently analyzed into a suitable family of square integrable wavelets of constant shape, (i.e. obtained by shifts and dilations from any one of them.) The resulting integral transform is isometric and self-reciprocal if the wavelets satisfy an “admissibility condition” given here. Explicit expressions are obtained in the case of a particular analyzing family that plays a role analogous to that of coherent states (Gabor wavelets) in the usual $L_2 $ -theory. They are written in terms of a modified $\Gamma $-function that is introduced and studied. From the point of view of group theory, this paper is concerned with square integrable coefficients of an irreducible representation of the nonunimodular $ax + b$-group.

3,423 citations

Journal ArticleDOI
TL;DR: It is shown that it is possible to design tree-structured analysis/reconstruction systems which meet the sampling rate condition and which result in exact reconstruction of the input signal.
Abstract: In recent years, tree-structured analysis/reconstruction systems have been extensively studied for use in subband coders for speech. In such systems, it is imperative that the individual channel signals be decimated in such a way that the number of samples coded and transmitted do not exceed the number of samples in the original speech signal. Under this constraint, the systems presented in the past have sought to remove the aliasing distortion while minimizing the overall analysis/reconstruction distortion. In this paper, it is shown that it is possible to design tree-structured analysis/reconstruction systems which meet the sampling rate condition and which result in exact reconstruction of the input signal. The conditions for exact reconstruction are developed and presented. Furthermore, it is shown that these conditions are not overly restrictive and high-quality frequency division may be performed in the analysis section. A filter design procedure is presented which allows high-quality filters to be easily designed.

785 citations

Journal ArticleDOI
TL;DR: The main features of so-called wavelet transforms are illustrated through simple mathematical examples and the first applications of the method to the recognition and visualisation of characteristic features of speech and of musical sounds are presented.
Abstract: This paper starts with a brief discussion of so-called wavelet transforms, i.e., decompositions of arbitrary signals into localized contributions labelled by a scale parameter. The main features of the method are first illustrated through simple mathematical examples. Then we present the first applications of the method to the recognition and visualisation of characteristic features of speech and of musical sounds.

622 citations