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Fundamental Papers in Wavelet Theory

Fundamental Papers in Wavelet Theory

Christopher Heil
David F. Walnut
Copyright Date: 2006
Pages: 912
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  • Book Info
    Fundamental Papers in Wavelet Theory
    Book Description:

    This book traces the prehistory and initial development of wavelet theory, a discipline that has had a profound impact on mathematics, physics, and engineering. Interchanges between these fields during the last fifteen years have led to a number of advances in applications such as image compression, turbulence, machine vision, radar, and earthquake prediction.

    This book contains the seminal papers that presented the ideas from which wavelet theory evolved, as well as those major papers that developed the theory into its current form. These papers originated in a variety of journals from different disciplines, making it difficult for the researcher to obtain a complete view of wavelet theory and its origins. Additionally, some of the most significant papers have heretofore been available only in French or German.

    Heil and Walnut bring together these documents in a book that allows researchers a complete view of wavelet theory's origins and development.

    eISBN: 978-1-4008-2726-8
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. I-IV)
  2. Table of Contents
    (pp. V-VIII)
  3. Contributor Affiliations
    (pp. IX-XII)
  4. Preface:
    (pp. XIII-XIV)
    Christopher Heil and David F. Walnut
  5. Acknowledgments
    (pp. XIV-XIV)
  6. Foreword:
    (pp. XV-XVIII)
    Ingrid Daubechies

    This volume collects early wavelet papers and precursors. It is a delight to find these diverse papers in one volume, instead of having to comb libraries of the different disciplines to find them. I am sure that many researchers working on, using, or just generally interested in wavelets will welcome this collection.

    In the late 1970s and a good part of the 1980s, “wavelet theory” (as I will call it, for want of a better name) emerged as the synthesis of ideas from many sources. In this book, one can trace how mathematics (Littlewood-Paley theory and its developments in harmonic...

  7. Introduction:
    (pp. 1-20)
    John J. Benedetto

    If ever there was a collection of articles that needed no introduction, this is it. Undaunted, I shall fulfill my charge as introducer by describing some of the intellectual background of wavelet theory and relating this background to the articles in this volume and to their expert introductions by Jelena Kovačević, Jean-Pierre Antoine, Hans Feichtinger, Yves Meyer, Guido Weiss, and Victor Wickerhauser.

    I was not a contributor to wavelet theory, but was close enough in the mid-1980s to hear the commotion. I was in the enviable position of having talented graduate students (including the editors of this volume), and so...

  8. Section I. Precursors in Signal Processing

    • Introduction:
      (pp. 23-27)
      Jelena Kovačević

      What a treat to look back and observe what happened within the field of wavelets over the past twenty years. We witnessed tremendous advances, wavelets became a commonplace technique, several standards are wavelet-based, wavelet software packages such as Matlab are used regularly. As icing on the cake, wavelets brought people from many disparate areas together. It is commonplace today to attend a wavelet meeting and sit in a room with mathematicians, engineers, statisticians, and physicists.

      Where did it all start? This volume gives the answer: many places. One of those many is signal processing. If you are not familiar with...

    • 1. The Laplacian Pyramid as a Compact Image Code
      (pp. 28-36)

      A common characteristic of images is that neighboring pixels are highly correlated. To represent the image directly in terms of the pixel values is therefore inefficient: most of the encoded information is redundant. The first task in designing an efficient, compressed code is to find a representation which, in effect, decorrelates the image pixels. This has been achieved through predictive and through transform techniques (cf. [9], [10] for recent reviews).

      In predictive coding, pixels are encoded sequentially in a raster format. However, prior to encoding each pixel, its value is predicted from previously coded pixels in the same and preceding...

    • 2. Digital Coding of Speech in Sub-bands
      (pp. 37-53)

      For digital transmission a signal must be sampled and quantized. Quantization is a nonlinear operation and produces distortion products that are typically broad in spectrum. Because of the characteristics of the speech spectrum, quantizing distortion is not equally detectable at all frequencies. Coding the signal in narrower sub-bands offers one possibility for controlling the distribution of quantizing noise across the signal spectrum and, hence, for realizing an improvement in signal quality. In earlier work, splitting of the spectrum by high-pass and low-pass filtering has been used advantageously for video and speech transmission.1,2

      A question, then, is what design of sub-bands...

      (pp. 54-58)
      D. Esteban and C. Galand

      Decomposition of the voice spectrum in sub-bands has been proposed by R. Crochiere et al. /1/ as a means to reduce the effect of quantizing noise due to coding. The main advantages of this approach are the following :

      first, to localize the quantizing noise in narrow frequency sub-bands, thus preventing noise interference between these sub-bands,

      second, to enable the attribution of bit resources to the various frequency bands according to perceptual criteria.

      As a result, the quantizing noise is perceptually more acceptable, and the signal to noise ratio is improved.

      The implementation proposed in /1/ is straightforward and takes...

      (pp. 59-62)
      M. J. T. Smith and T. P. Barnwell III

      In a recent paper by Barnwell [1], it was shown that alias-free reconstruction using recursive and non-recursive filters was possible where the analysis/reconstruction section had no frequency distortion or no phase distortion, but not both. In the development that follows, the coefficient symmetry condition on the analysis filters is lifted and exact reconstruction free of aliasing, phase distortion and frequency distortion is shown to be possible using FIR filters. In addition, the filter constraints that enable perfect reconstruction are discussed and an easily implementable design procedure providing high quality filters is presented.

      The frequency division of the subband coder is...

    • 5. Filters for Distortion-Free Two-Band Multirate Filter Banks
      (pp. 63-67)

      Multirate filter banks are used in a number of digital signal processing applications. In speech applications, for example, they are found in subband coding, adaptive transform coding, and some noise reduction techniques. With a multirate filter bank, the incoming signal is filtered into several narrow-band components. Each narrow-band component is decimated, modified, and interpolated to the original sampling frequency. Then the narrow-band components are summed to form a modified version of the input signal. The presence of this opcration is not always apparent since it often appears in a different form: the reconstruction of a signal from its modified short-time...

      (pp. 68-93)
      Martin VETTERLI

      Let us first briefly state the basic problem we want to solve. Suppose an infinite sequence of samples$x(n)$. This sequence is filtered into$N$sequences${y_0}(n)...{y_{N - 1}}(n)$(with linear, time invariant filters). The sequences${y_i}(n)$are subsampled by a factor$N$that is only every$N$th sample is kept, or${{y'}_i}(n)\; = \;{y_i}(Nn)$. Now, the problem is to recover$x(n)$from the subsampled sequences${{y'}_i}(n)$(see Fig. 1).

      Obviously, there is the same number of samples per unit of time in$x(n)$and in all the${{y'}_i}(n)$together, thus a solution should exist. Nevertheless, there were not many...

    • 7. Theory and Design of M-Channel Maximally Decimated Quadrature Mirror Filters with Arbitrary M, Having the Perfect-Reconstruction Property
      (pp. 94-110)

      Quadrature mirror filter (QMF) banks have rcceived considerable attention during thc past several years because of a wide variety of engineering applications [1]-[13]. An$M$-channel QMF bank is shown in Fig.1, where${H_0}(z)$,${H_1}(z)$, ... ,${H_{M - 1}}(z)$are the transfer functions of analysis bank filters, and${F_0}(z)$,${F_1}(z),...,{F_{M - 1}}(z)$represent the synthesis filters. In the analysis bank, the incoming signal$x(n)$is split into$M$frequency bands by filtering, and eaeh subband signal is maximally decimated, i. e., decimated by a factor of$M$. The$M$decimated signals are then processed in the synthesis bank by interpolating each signal,...

  9. Section II. Precursors in Physics:: Affine Coherent States

    • Introduction:
      (pp. 113-116)
      Jean-Pierre Antoine

      The story of the origin of wavelet analysis has been told many times (see the vivid account of B. B. Hubbard [Hub98]). The geophysicist Jean Morlet was analyzing microseismic data in the context of oil exploration. The technique consists in sending short impulsions (called wavelets by geophysicists) into the ground and analyzing the signals which have been reflected on density discontinuities. The result is usually a mess, a very noisy and confusing signal. Fourier methods are normally used for unraveling it, more precisely the windowed Fourier transform (WFT), but with mixed results. Then Morlet had the idea of exchanging roles...

    • 1. Continuous Representation Theory Using the Affine Group
      (pp. 117-125)
      Erik W. Aslaksen and John R. Klauder

      The general concepts and properties of continuous representation theory (CRT) have been developed Klauder¹; for convenience we briefly recall them here: Let$K$denote abstract Hilbert space and let$U[l]$be a family of unitary operators on$K$. If we now choose an arbitrary but fixed unit vector${\Phi _0}\; \in \;k$, called thefiducial vector,then we can generate a subset of$K$by operating on${\varphi _0}$with$U[l]$. Denote this subset by$\mathfrak{S}$; then

      $\mathfrak{S} = \{ U[l]{\Phi _0}:\;l\; \in \;\mathcal{L}\} $,

      where$\mathcal{L}$is some label space. With any vector$\Psi \; \in \;k$we can now associate the complex, bounded, continuous function

      $\varphi (l)\; = \;(U[l]{{\dot \Phi }_0},\;\Psi )$,

      and the set$ \equiv \;\{ \varphi (l):\;\Psi \; \in \;\} $is called...

      (pp. 126-139)

      It is well known that an arbitrary complex-valued square integrable function$\Psi (t)$admits a representation by Gaussians, shifted in direct and Fourier transformed space. If$g(t)\; = \;{2^{ - 1/2}}{\pi ^{ - 3/4}}{e^{ - {t^2}/2}}$and${t_0},{\omega _0}$are arbitrary real, consider

      (1.1)${g^{({t_0},\;{\omega _0})}}(t)\; = \;{e^{ - i{\omega _0}{t_0}/2}}{e^{i{\omega _0}t}}g(t\; - \;{t_0})$

      and form the inner product

      (1.2)$\Psi ({t_0},\;{\omega _0})\; = \;\int {{{\bar g}^{({t_0},\,{\omega _0})}}} (t)\psi (t)\,dt$


      (1.3)$\int {\int {|\,\Psi ({t_0},\,{\omega _0})\,{|^2}d{t_0}d{\omega _0}} = \int {|\,\psi (t)\,{|^2}dt} } $.

      The function$\Psi (t)$can be recovered from the function$\Psi ({t_0},{\omega _0})$through

      (1.4)$\psi (t) = \int {\int {{g^{({t_0},\,{\omega _0})}}} (t)} \Psi ({t_0},\;{\omega _0})\,d{t_0}\,d{\omega _0}$.

      The above statements remain true if the Gaussian$g$is replaced by an arbitrary square integrable function. The advantages of the Gaussian are (i) maximal concentration in direct and Fourier transformed space and (ii) the possibility of a simple intrinsic characterization...

    • 3. Transforms associated to square Integrable group representations. I. General results
      (pp. 140-146)
      A. Grossmann, J. Morlet and T. Paul

      This paper is the first of a series concerned with applications of various families of “generalized coherent states” to quantum mechanics, wave propagation, and signal analysis.

      Many properties of the classical (canonical) coherent states1.2are closely tied to the Weyl-Heisenberg group. In particular, the fundamental formula

      $1 = \int {|\,z\rangle {d^2}z\langle z\,|} $(1.1)

      is a way of writing the orthogonality relations3.4for the irreducible representation of that group.

      Aslaksen and Klauder⁵ have considered the analogous states for the two-parameter group of shifts and dilations and found that the “fiducial vector” (“analyzing wavelet” in our terminology) cannot be arbitrary, in contrast to the Weyl-Heisenberg case.


  10. Section III. Precursors in Mathematics:: Early Wavelet Bases

    • Introduction:
      (pp. 149-154)
      Hans G. Feichtinger

      The plain fact that wavelet families are very interesting orthonormal systems for$L^2(R)$makes it natural to view them as an important contribution to the field of orthogonal expansions of functions. This classical field of mathematical analysis was particularly flourishing in the first thirty years of the twentieth century, when detailed discussions of the convergence of orthogonal series, in particular of trigonometric series, were undertaken.

      Alfred Haar describes the situation in his 1910 paper inMath. Annalenappropriately as follows: for any given (family of) orthonormal system(s) of functions on the unit interval [0, 1] one has to ask...

    • 1. On the Theory of Orthogonal Function Systems
      (pp. 155-188)
      Alfred Haar

      In the theory of series expansions of real functions, the so-calledorthogonal function systemsplay a major rôle. By this, we mean a system of infinitely many functions${\varphi _1}(s),{\varphi _2}(s),...$which have, with respect to an arbitrary, measurable set$M$of points, theorthogonality property

      $\int_M {{\varphi _p}(s)\,{\varphi _q}(s)\,ds\; = \;0} {\rm{ (}}p \ne q,\;p,\,q\; = \;1,\;2,\; \ldots )$,

      $\int_M {{{({\varphi _p}(s))}^2}ds = 1{\rm{ (}}p = 1,\;2,\; \ldots )} $,

      where the integrals are taken in the Lebesgue sense; if they furthermore satisfy the so-calledcompleteness relation

      $\int_M {{{(u(s))}^2}ds = {{\left\{ {\int_M {u(s)\;{\varphi _1}(s)\;ds} } \right\}}^2} + \;{{\left\{ {\int_M {u(s)\;{\varphi _2}(s)\;ds} } \right\}}^2} + \; \cdots } $

      for all functions$u(s)$which together with their squares are integrable over the set$M$, then, following Hilbert, we denote the system acomplete orthogonal function system, or, for short, acomplete...

    • 2. A Set of Continuous Orthogonal Functions.
      (pp. 189-196)
      Philip Franklin

      There exist continuous functions for which, at some points of the interval of orthogonality the classical Fourier series fails to converge. The analogous expansions in orthogonal functions arising from the simpler boundary value problems seem to share this property with the Fourier expansion¹). This led A. Haar to ask if the property was common to all sets of orthogonal functions. He showed that it was not by exhibiting a set of orthogonal functions giving, as the expansion of any continuous function, a series converging uniformly to the function throughout the fundamental interval. The individual functions of his set, however, are...

      (pp. 197-215)
      Jan-Olov Strömberg

      Let$H^p(I)$be the subspace of distributions in the Hardy space$H^p(R)$that are supported in$I=[0, 1]$. The Franklin system$(m=0)$and higher-order spline system$(m>0)$have been studied and used as unconditional basis for${H^p}(I),p > {\text{1/ (m + 2)}}$in [1, 4, 7, and 9-11]. The existence of an unconditional basis for$H^1$was first shown by B. Maurey [8] and an explicit construction was made by L. Carleson [3]. The spline system as unconditional basis for the bi-Hardy space${H^1}(I\; \times \;I)$has been studied by S. Y. A. Chang.

      The purpose of...

    • 4. Uncertainty Principle, Hilbert Bases and Algebras of Operators
      (pp. 216-228)
      Yves Meyer

      The search for Hilbert bases connected to the uncertainly principle is motivated in the following way by R. Balian [1].

      One can be interested, in the theory of communications, to represent an oscillating signal as a superposition of elementary wavelets, each of which possesses at the same time a sufficiently well-defined frequency and a localization in time. The useful information is, in fact, often carried at the same time by the emitted frequencies and by the structure of the signal in time (the example of music is characteristic). The representation of a signal as a function of time exhibits badly...

    • 5. Wavelets and Hilbert Bases
      (pp. 229-244)
      P. G. Lemarié and Y. Meyer

      Wavelet transforms appear implicitly in a famous work of A. P. Calderón [2]. It was rediscovered and made explicit by J. Morlet and his collaborators [4], [7], [8] as an efficient technique of numerical analysis allowing signal processing in connection with oil prospecting.

      Wavelet transforms are similar to the Fourier transformation but the imaginary exponentials exp$(ix.\xi )$indexed by the frequencies$\xi \; \in \;{{\bf{R}}^n}$are replaced by the “wavelets”${\psi _Q}$indexed by the collection of all the cubes$Q\; \subset \;{{\bf{R}}^n}$. These “wavelets”${\psi _Q}$are all copies (by translation and change of scale) of the same regular function ψ, decreasing at infinity as...

    • 6. A Block Spin Construction of Ondelettes. Part I: Lemarié Functions
      (pp. 245-260)
      Guy Battle

      Quite recently Y. Meyer et al. [1, 2, 3] have constructed very useful bases of ondelettes (wavelets) to solve certain problems in functional analysis. These new functions are now expected to have applications to several areas of physics. They have already had an impact on constructive quantum field theory [4, 5, 6].

      A basis of ondelettes is defined to be an orthonormal basis—say for${L^2}({\mathbb{R}^d})$—whose functions are dyadic scalings (from${2^{ - \infty }}$to${2^\infty }$) and translates of just a finite number of them. The most familiar example is the standard basis of Haar functions on${\mathbb{R}^d}$. Indeed, Battle and...

  11. Section IV. Precursors and Development in Mathematics:: Atom and Frame Decompositions

    • Introduction:
      (pp. 263-268)
      Yves Meyer

      There are no doubts. We owewavelet analysisto Jean Morlet. How did Morlet discover wavelets? Here is the story. During the late seventies Morlet was working as a geophysicist for the Elf-Aquitaine company. He had to process the backscattered seismic signals which carry the information related to the geological layers. These seismic signals present transient patterns. Processing such signals with conventional tools such aswindowed Fourier analysisorGabor waveletscreates numerical artifacts. Morlet elaborated wavelet analysis to overcome this problem. Quoting the geophysicist Pierre Goupillaud,

      A product of the renowned Ecole Polytechnique, Morlet performed the exceptional feat of...

      (pp. 269-294)
      R. J. DUFFIN and A. C. SCHAEFFER

      A sequence$\{ {\lambda _n}\} ,\;n\; = \;0,\; \pm 1,\; \pm 2,\; \cdots $, of real or complex numbers we shall say hasuniform density1 if there are constants$L$and δ such that$|\,{\lambda _n} - n\,|\;\underline \le \;L$and$|{\lambda _n} - {\lambda _m}|\;\underline \ge \;\delta \; > \;0$for$n \ne \;m$. This is a more restrictive notion than density, for, considering only those${\lambda _n}$for which$n > 0$, it is clear that a sequence of uniform density 1 has a density as defined by Pólya equal to 1, but the converse is not true. Sequences of uniform density$d$are defined in a later part of the present paper for any$d > 0.$. If$f(z)$is an entire function of...

      (pp. 295-371)

      It is well known that the theory of functions plays an important role in the classical theory of Fourier series. Because of this certain function spaces, the$H^p$spaces, have been studied extensively in harmonic analysis. When$p > 1$,$L^p$> and$H^p$are essentially the same; however, when$p \le 1$the space$H^p$is much better adapted to problems arising in the theory of Fourier series. We shall examine some of the properties of$H^p$for$p \le 1$and describe ways in which these spaces have been characterized recently. These characterizations enable us to extend their definition to a very general...

    • 3. Painless nonorthogonal expansions
      (pp. 372-384)
      Ingrid Daubechies, A. Grossmann and Y. Meyer

      A classical procedure of applied mathematics is to store some incoming information, given by a function$f(x)$(where$x$is a continuous variable, which may be, e.g., the time) as a discrete table of numbers$\langle {g_j}\,|\,f\rangle = \smallint dx\,{g_j}(x)f(x)$rather than in its original (sampled) form. In order to have a mathematical framework for all this, we shall assume that the possible functions$f$are elements of a Hilbert space$H$[we take here$H = {L^2}(R)$]; the functions$g_j$are also assumed to be elements of this Hilbert space.

      One can, of course, choose the functions$g_j$so that the family$\{ {g_j}\} $...

    • 4. Decomposition of Besov Spaces
      (pp. 385-407)

      In the last decade, many function or distribution spaces have been found to admit a decomposition, in the sense that every member of the space is a linear combination of basic functions of a particularly elementary form. Such decompositions simplify the analysis of the spaces and the operators acting on them. Here we obtain two types of decompositions for distributions in the homogeneous Besov spaces$\dot B_p^{\alpha q},\; - \infty \; < \;\alpha \; < \; + \infty ,\;0\; < \;p,\;q\; \leqq + \infty $, and present some applications of these results.

      Defining the Fourier transform by$\hat f(\xi ) = \int {f(x){e^{ - ix\cdot\xi }}dx} $, let${{\text{\{ }}{\varphi _v}\} _{v \in {\mathbf{Z}}}}$be a family of functions on$R^n$satisfying

      (1.1)${\varphi _v}\: \in S$

      (1.2) supp${\text{supp}}\;{{\hat \varphi }_v} \subseteq \;\left\{ {\xi \; \in \;{{\mathbf{R}}^n}:\frac{1} {2}\; \leqq {2^{ - v}}|\,\xi \,|\; \leqq \;2} \right\}$,

      (1.3)$\,{{\hat \varphi }_v}(\xi )\,|\; \geqq \;c\; > \;0\;{\text{if}}\;\frac{3} {5}\; \leqq \;{2^{ - v}}|\,\xi \,|\; \leqq \;\frac{5} {3}$,


      (1.4)$|{\partial ^\gamma }{{\hat \varphi }_v}(\xi )| \leqq \;{c_\gamma }{2^{ - v|\gamma |}}$for every multi-index$\gamma $.


    • 5. Banach Spaces Related to Integrable Group Representations and Their Atomic Decompositions, I
      (pp. 408-441)
      Hans G. Feichtinger and K. H. Gröchenig

      The aim of an atomic decomposition for a space of functions or distributions is to represent every element as a sum of “simple functions,” usually called atoms. If this is possible, properties of these function spaces, such as duality, interpolation, or operator theory for them, can be understood better by means of the atomic decomposition. Of course, the meaning of “simple function” depends on the point of view. Thus, for example, the atoms in the decomposition of Hardy spaces are subject to support and moment conditions (cf. [CW]). The atoms for the spaces of Besov-Triebel-Lizorkin type are transforms of a...

    • 6. The Wavelet Transform, Time-Frequency Localization and Signal Analysis
      (pp. 442-486)

      In signal analysis one often encounters the socalled short-time Fourier transform, or windowed Fourier transform. This consists of multiplying the signal$f(t)$with a usually compactly supported window function$g$, centered around 0, and of computing the Fourier coefficients of the product$gf$. These coefficients give an indication of the frequency content of the signal$f$in a neighborhood of$t = 0$. This procedure is then repeated with translated versions of the window function (i.e.,$g(t)$is replaced by$g(t \pm {t_0}),g(t \pm 2{t_0}),...$, where$t_0$is a suitably chosen time translation step). This results in...

  12. Section V. Multiresolution Analysis

    • Introduction:
      (pp. 489-493)
      Guido Weiss

      The “classical” one-dimensionalorthonormal waveletsare those functions$\psi \; \in \;{L^2}(\mathbb{R})$for which the system$\{ {\psi _{j,k}}(x)\; = \;{2^{j/2}}\psi ({2^j}x - k)\} ,\;j,\;k\; \in \;\mathbb{Z}$, is an orthonormal basis for${L^2}(\mathbb{R})$. There is a relatively simple characterization of these functions:

      Theorem 1$\psi \; \in \;{L^2}(\mathbb{R})$is an orthonormal wavelet if and only if$\parallel \,\psi \,{\parallel _2} = 1$,

      $\sum\limits_{j \in \mathbb{Z}} {|\,\hat \psi ({2^j}\xi )\,{|^2}} = 1\;\;for\;a.e.\;\xi \; \in \;\mathbb{R}$(I)


      ${t_q}(\xi ) \equiv \sum\limits_{j = 0}^\infty {\hat \psi ({2^j}\xi )\overline {\hat \psi ({2^j}(\xi \; + \;2m\pi ))} \; = \;0} $(II)

      for$a,e,$$\xi \; \in \;\mathbb{R},\;m\; \in \;2\mathbb{Z} + 1$.

      We are using theFourier transform${\hat f}$of$f\; \in \;{L^2}(\mathbb{R})$that has the form

      $\hat f(\xi ) = \int_\mathbb{R} {f(t){e^{ - i\xi t}}} dt$.

      Theorem 1 was obtained independently by G. Gripenberg and X. Wang ([Gri95], [Wan95]). Equalities (I) and (II) were known since the beginning of the development of wavelet theory. In particular, (I) is a variant of the Calderon...

    • 1. A Theory for Multiresolution Signal Decomposition: The Wavelet Representation
      (pp. 494-513)

      In computer vision, it is difficult to analyze the information content of an image directly from the gray-level intensity of the image pixels. Indeed, this value depends upon the lighting conditions. More important are the local variations of the image intensity. The size of the neighborhood where the contrast is computed must be adapted to the size of the objects that we want to analyze [41]. This size defines a resolution of reference for measuring the local variations of the image. Generally, the structures we want to recognize have very different sizes. Hence, it is not possible to define a...

    • 2. Wavelets with Compact Support
      (pp. 514-523)
      Yves Meyer

      The aim of this paper is to give a new proof of a recent and still unpublished result by I. Daubechies. I. Daubechies’ theorem is the following.

      Theorem 1For any$r \geqslant 1$there exists a function$\psi \; = \;{\psi _r}$of the real variable$x$with the following properties:

      (1.1)$\psi (x)$is of class$C^r$on the real line,

      (1.2)$\psi (x)$is compactly supported,

      (1.3)the collection${2^{j/2}}\psi ({2^j}x - k),\;j\; \in \;\mathbb{Z},\;k\; \in \;\mathbb{Z}$,is an orthonormal basis for${L^2}(\mathbb{R})$.

      The only orthonormal basis with the structure given by (1.2) and (1.3) which has been known before is the Haar system for which$\psi (x)\; = \;1$on [0, 1/2),$\psi (x) = - 1$...

      (pp. 524-542)

      In this article, we study the properties of the multiresolution approximations of L²(R). We show how they relate to wavelet orthonormal bases of L²(R). Wavelets have been introduced by A. Grossmann and J. Morlet [7] as functions whose translations and dilations could be used for expansions in L²(R). J. Stromberg [16] and Y. Meyer [14] have proved independently that there exists some particular wavelets$\psi (x)$such that${(\sqrt {{2^j}} \psi ({2^j}x - k))_{(j,\,k) \in {{\text{Z}}^2}}}$is an orthonormal basis of L²(R); these bases generalize the Haar basis. If$\psi (x)$is regular enough, a remarkable property of these bases is to provide an unconditional basis of most classical...

    • 4. Wavelets, Multiresolution Analysis, and Quadrature Mirror Filters
      (pp. 543-559)
      A. Cohen

      The purpose of this article is to elucidate the mathematical relations between the theory of wavelets and the theory of quadrature mirror filters (QMF).

      Since the work of J. O. Stromberg (1981), we have known how to construct orthonormal bases for${L^2}(\mathbb{R})$of the form${\{ \psi ({2^j}x - k)\} _{j \in \mathbb{Z},\,k \in \mathbb{Z}}}$. This discovery was foreshadowed by the Haar system (1909) and later by Franklin’s system (1927).

      Several years later, Y. Meyer ([1], [2], [3]) discovered a basis of the type${\{ \psi ({2^j}x - k)\} _{j \in \mathbb{Z},\,k \in \mathbb{Z}}}$where the wavelet$\psi $belongs to the Schwartz class. He then introduced with S. Mallat ([4], [5]) the algebraic framework for multiresolution analyses, which...

    • 5. Tight frames of compactly supported affine wavelets
      (pp. 560-563)
      Wayne M. Lawton

      This paper discusses families of functions${h_{a,b}}(x)\; = \;{a^{ - 1/2}}h((x - b)\,/\,a),\;a \ne 0$, calledwavelets,¹ which are generated from a single function$h\, \in \,L\,{}^2(R)$by dilation, translation, and possibly reflection. Wavelets provide a means of representing functions as a continuous linear superposition that is analogous to the Fourier transform. This representation is related to group representations. Let$G$denote theaffine group$\{ x \to ax + b|a \ne 0\} $and let$U$denote the unitary representation of$G$on${L^2}(R)$defined by$U(a,b)h = {h_{a.b}}$. In this framework wavelets are subsets of${L^2}(R)$having the form$D(h)\; = \;\{ {h_{a,b}}\,|(a,b)\, \in \,D\} $for some subset$D$of$G$.

      Let$G$, denote either$G$or the connected affine...

    • 6. Orthonormal Bases of Compactly Supported Wavelets
      (pp. 564-652)

      In recent years, families of functions${h_{a,b}}$,

      (1.1)${h_{a,\,b}}(x)\; = \;|\,a\,{|^{ - 1/2}}h\left( {\frac{{x\; - \;b}} {a}} \right)$,$a,\;b\; \in \;\mathbb{R},\;a \ne 0$,

      generated from one single function$h$by the operation of dilations and translations, have turned out to be a useful tool in many different fields of mathematics, pure as well as applied. Following Grossmann and Morlet [1], we shall call such families “wavelets”.

      Techniques based on the use of translations and dilations are certainly not new, They can be traced back to the work of A. Calderón [2] on singular integral operators, or to renormalization group ideas (see [3]) in quantum field theory and statistical mechanics, Even in these two disciplines, however,...

  13. Section VI. Multidimensional Wavelets

    • Introduction:
      (pp. 655-658)
      Guido Weiss

      We have considered affine systems$\{ {\psi _{j,k}}\} = \{ {2^{ - j/2}}\psi ({2^{ - j}}x - k)\} ,\;j,\;k\; \in \;\mathbb{Z},$, obtained by first translating (by$k$) and then dilating (by${2^j}$) a function$\psi \; \in \;{L^2}(\mathbb{R})$. In the introduction to Section V, we have seen that there exists a characterization of those ψ for which$\{ {\psi _{j,k}}\} $is an orthonormal basis for${L^2}(\mathbb{R})$(or, more generally, a Parseval frame). We also described the important MRA method for constructing such systems. We shall now turn our attention to functions in${L^2}({\mathbb{R}^d}),\;d\; \geqslant \;1$, and explore some of the natural extensions of these notions.

      A natural extension of the notion of a wavelet to higher dimensions is to consider tensor...

    • 1. Wavelets, Spline Functions, and Multiresolution Analysis
      (pp. 659-689)
      Yves Meyer

      The first orthonormal basis of wavelets was obtained during the summer of 1985, and this accidental discovery remained mysterious. Nowadays the works of P. G. Lemarié and of S. Mallat have proved that the “historical basis” and its variants result simply from natural algorithms on nested spaces of spline functions.

      We propose to describe this new approach to series of wavelets.

      Let us recall the general characteristics we expect from what we will call anorthonormal basis of wavelets. In this account we shall deal with an orthonormal basis of the reference space${L^2}({\mathbb{R}^n})$, which, however, remains efficient when one...

    • 2. Multiscale Analyses and Wavelet Bases
      (pp. 690-693)
      Karlheinz Gröchenig

      A wavelet basis is an orthogonal basis for${L^2}({\mathbb{R}^n})$of the form${2^{nj/2}}{\psi _\varepsilon }({2^j}x - k)$, where$j \in \mathbb{Z},\;k \in {\mathbb{Z}^n}$, where ε ranges over a finite set$E$, and where${\psi _\varepsilon }$enjoys good smoothness properties and good decay properties at infinity. We often require, for an integer$r \geqslant 2$, that

      $\mathop {\sup }\limits_{x \in {\mathbb{R}^n}} {(1\: + |\,x\,|)^{n + r}}|\,{\partial ^\alpha }{\psi _\varepsilon }(x)\,|\: \leqslant \:C$(1)

      for all multi-indices α of length$\,\alpha \,|\; \leqslant \;r$.

      Wavelet bases are used for signal and image processing [6] and are unconditional bases for many of the classic function spaces [1], [2].

      At this time, we do not know how to construct all of the wavelet bases of${L^2}({\mathbb{R}^n})$. The several examples that we do know...

    • 3. Nonseparable Multidimensional Perfect Reconstruction Filter Banks and Wavelet Bases for ${\Re ^n}$
      (pp. 694-716)
      Jelena Kovačević and Martin Vetterli

      Since the introduction of digital multirate filter banks for the compression of speech signals 15 years ago [8], they have been widely used mainly for subband coding of speech, still images, and video [2], [37], [43], [47]. The underlying theory progressed from cancellation of aliasing (or repeated spectra), to building systems achieving exact reconstruction of the signal, and from two-channel orthogonal banks [26], [30], to general multichannel systems [31], [33], [38], [3 9], [44]. For implementational reasons all of these efforts concentrated on filters having rational transfer functions.

      Independent of this work, the theory of wavelets was developed in applied...

    • 4. Multiresolution Analysis, Haar Bases and Self-Similar Tilings of ${R^n}$
      (pp. 717-730)
      K. Gröchenig and W. R. Madych

      Recall that the Haar system on${L^2}(R)$is the collection of functions

      ${2^{k/2}}\psi ({2^k}x - j),{\text{ }}j,\;k \in Z,$(1)


      $\psi (x) = \left\{ {\begin{array}{*{20}{c}} {1,} & {{\text{if}}\;{\text{0}} \leqslant x < 1/2,} \\ { - 1,} & {{\text{if}}\;{\text{1/2}}\; \leqslant x < 1,} \\ {0,} & {{\text{otherwise,}}} \\ \end{array} } \right.$

      where$Z$denotes the set of integers. Note the role played by the dilation$x \to 2x$and the translations$x \to x - j$. It is well known that this collection is a complete orthonormal system for${L^2}(R)$.

      The point o f this paper is to construct analogous systems for${L^2}({R^n}),\;n \geqslant 2$, where the dilation noted above is replaced by appropriate linear transformations of${R^n}$and the integers$Z$are replaced by an appropriate lattice in${R^n}$. The motivation and framework for our construction is outlined in Section II....

  14. Section VII. Selected Applications

    • Introduction:
      (pp. 733-740)
      Mladen Victor Wickerhauser

      Over the past decade, wavelet transforms have been widely applied. Good implementations of the discrete wavelet transform (DWT) were built into software systems such as Matlab and S-Plus, and the DWT became a frequently used tool for data analysis and signal processing. There are certain problems, though, on which this tool works particularly well. The most common ingredient in those problems is some complicated object that can be closely approximated by a few superposed wavelets. This compilation includes four seminal articles that introduced some of these stand out DWT applications. I have taken a random and sparse sampling of relevant...

    • 1. Fast Wavelet Transforms and Numerical Algorithms I
      (pp. 741-783)

      The purpose of this paper is to introduce a class of numerical algorithms designed for rapid application of dense matrices (or integral operators) to vectors. As is well known, applying directly a dense$N \times N$-matrix to a vector requires roughly$N^2$operations, and this simple fact is a cause of serious difficulties encountered in large-scale computations. For example, the main reason for the limited use of integral equations as a numerical tool in large-scale computations is that they normally lead to dense systems of linear algebraic equations, and the latter have to be solved, either directly or iteratively. Most iterative methods...

      (pp. 784-832)
      Ronald A. DeVore, Björn Jawerth and Vasil Popov

      We characterize functions with a given degree of nonlinear approximation by linear combinations with$n$terms of a function$φ$, its dilates and their translates. This gives a unified viewpoint of recent results on nonlinear approximation by spline functions and give their extension to functions of several variables. Our approach is formulated in terms of wavelet decompositions.

      1. Introduction. There has recently been great interest in the numerical applications of wavelet decompositions. Such applications call for the efficient recovery of a function$f$from the coefficients in such a decomposition. We shall discuss in this paper nonlinear methods for accomplishing...

    • 3. Adapting to Unknown Smoothness via Wavelet Shrinkage
      (pp. 833-857)
      David L. Donoho and Iain M. Johnstone

      Suppose that we are given$N$noisy samples of a function$f$

      ${y_i} = f({t_i})\; + \;{z_i},{\text{ }}i = 1,\; \ldots ,\;N$, (1)

      with${t_i} = (i - 1)/N,\;{z_i}$iid$N(0,\;{\sigma ^2})$. Our goal is to estimate the vector${\text{f}} = (f({t_i}))_{i = 1}^N$with small mean squared error (MSE); that is, to find an estimate${\hat f}$depending on${y_1},...,{y_N}$with small risk$R({\text{\hat f,}}\;{\text{f)}} = {N^{ - 1}}.\;E\,\parallel \,{\text{\hat f}} - {\text{f}}\,\parallel _2^2 = E\;{\text{Av}}{{\text{e}}_i}{(\hat f({t_i})\; - \;f({t_i}))^2}$.

      To develop a non trivial theory, one usually specifies some fixed class$\mathcal{F}$functions to which$f$is supposed to be long. Then one may seek an estimator$f$attaining the minimax risk$R(N,\;\mathcal{F}) = {\text{in}}{{\text{f}}_{{\text{\hat f}}}}\,{\text{su}}{{\text{p}}_f}\;R({\text{\hat f,}}\;{\text{f)}}$.

      This approach has led to many theoretical developments of considerable interest (see, for example, Stone 1982,...

    • 4. Hölder Exponents at Given Points and Wavelet Coefficients
      (pp. 858-860)
      Stéphane Jaffard

      For ease of notation, we will limit the discussion to the one-dimensional case; however, the statements and proofs extend immediately to the general case.

      Let$\psi (x)$be a Lipschitz function of the real variable$x$that also satisfies the following conditions:

      $|\,\psi (x)\,|\; \leqslant \;C{(1\; + |\,x\,|)^{ - 3}},{\text{ |}}\,\psi '{\text{(}}x)\,|\; \leqslant \;C{(1\; + |\,x\,|)^{ - 3}}$; (1)

      the set${\psi _{(j,k}}) = {2^{j/2}}\psi ({2^j}x - k),\;j,\;k \in \mathbb{Z}$, is an orthonormal basis for${L^2}(\mathbb{R})$. (2)


      $\int_{ - \infty }^\infty {|\,f(x)\,|\,{{(1\; + \;|\,x\,|)}^{ - 3}}dx\; < \;\infty } $,

      then the wavelet coefficients

      $c(j,\;k) = \int_{ - \infty }^\infty {f(x)} {{\bar \psi }_{(j,k)}}(x)\;dx$

      make sense. If, in addition,$f(x)$satisfies$\,f(x)\; - \;f({x_0})\,|\; \leqslant \;C|\,x\; - \;{x_0}\,{|^\alpha }$for some exponent$\[\alpha \in (0,1]\]$, then this implies immediately that the condition

      $|\,c(j,\,k)\,|\; \leqslant \;C'{2^{ - j(\alpha + 1/2)}}(1\; + |\,{2^j}{x_0} - \;k\,{|^\alpha })$(3)

      is satisfied for all$j\; \in \;\mathbb{R}$and all$k\; \in \;\mathbb{Z}$.

      We propose to examine, conversely, if (3) implies...

    • 5. Embedded Image Coding Using Zerotrees of Wavelet Coefficients
      (pp. 861-878)
      Jerome M. Shapiro

      This paper addresses the two-fold problem of 1) obtaining the best image quality for a given bit rate, and 2) accomplishing this task in an embedded fashion, i.e., in such away that all encodings of the same image at lower bit rates are embedded in the beginning of the bit stream for the target bit rate.

      Theproblem is important in many applications,particularly for progressive transmission, image browsing [25]. multimedia applications, and compatible transcoding in a digital hierarchy ofmultiple bit rates.It is also applicable to transmission over a noisy channel in the sense that the ordering of the bits in order...