Wvelets and Sianal U Processin

OLlVlER RlOUL and MARTIN VETTERLI

avelet theory provides a unified framework for a number of techniques which had been developed inde- pendently for various signal processing applications. For ex- ample, multiresolution signal processing, used in computer vision; subband coding, developed for speech and image compression; and wavelet series expansions, developed in applied mathematics, have been recently recognized as different views of a single theory. In fact, wavelet theory covers quite a large area. It treats both the continuous and the discrete-time cases. It provides very general techniques that can be applied to many tasks in signal processing, and therefore has numerous potential applications. In particular, the Wavelet Transform (WT) is of inter- est for the analysis of non-stationary signals, because it provides an alternative to the classical Short-Time Fourier Transform (STFT) or Gabor transform [GAE346, ALL77, POR801. The basic difference is as follows. In contrast to the STET, which uses a single analysis window, the WT uses short windows at high frequencies and long windows at low frequencies. This is in the spirit of so-called “constant-Q” or constant relative bandwidth frequency analysis. The WT is also related to time-frequency analysis based on the Wigner-Ville distribution [FLA89, FLASO, RIOSOal. For some applications it is desirable to see the WT as a signal decomposition onto a set of basis functions. In fact, basis functions called wauelets always underlie the wavelet analysis. They are obtained from a single prototype wavelet by dilations and contractions (scal-

14 IEEE SP MAGAZINE OClOBER 1991

Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 29,2010 at 16:23:44 EDT from IEEE Xplore. Restrictions apply. ings) as well as shifts. The prototype wavelet can be thought of as a bandpass filter, and the constant-Q NON-STATIONARY SIGNAL property of the other bandpass filters (wavelets) follows ANALYSIS because they are scaled versions of the prototype. Therefore, in a WT, the notion of scale is introduced The aim of signal analysis is to extract relevant as an alternative to frequency, leading to a so-called information from a signal by transforming it. Some This means that a signal is time-scale representation. methods make a priori assumptions on the signal to be mapped into a time-scale plane [the equivalent of the analyzed: this may yield sharp results if these assump- time-frequency plane used in the STFT). tions are valid, but is obviously not of general ap- There are several types of wavelet transforms, and, plicability. In this paper we focus on methods that are depending on the application, one may be preferred to applicable to any general signal. In addition, we con- the others. For a continuous input signal, the time and sider invertible transformations. The analysis thus un- scale parameters can be continuous [GR089], leading ambiguously represents the signal, and more involved to the Continuous Wavelet Transform (CWT).They may operations such as parameter estimation, coding and as well be discrete (DAU88, MAL89b, MEY89, DAUSOa], pattern recognition can be performed on the “transform leading to Wavelet Series expansion. Finally, the a side,” where relevant properties may be more evident. wavelet transform can be defined for discrete-time sig- Such transforms have been applied to stationay nals [DAU88, RI090b. VETSOb], leading to a Discrete signals, that is, signals whose properties do not evolve Wavelet Transform (DWT). In the latter case it uses in time (the notion of stationarity is formalized precisely multirate signal processing techniques [CRO83] and is in the statistical signal processing literature). For such related to subband coding schemes used in speech and signals dt),the natural “stationary transform” is the image compression. Notice the analogy with the (Con- well-known Fourier transform (FOU88l: tinuous) Fourier Transform, Fourier Series, and the Discrete Fourier Transform. Wavelet theory has been developed as a unifying framework only recently, although similar ideas and constructions took place as early as the beginning of the century [HAAlO, FRA28, LIT37, CAL641. The idea of The analysis coefficients Xu define the notion of looking at a signal at various scales and analyzing it global frequencyfin a signal. As shown in (l),they are with various resolutions has in fact emerged inde- computed as inner products of the signal with sinewave pendently in many different fields of mathematics, basis functions of infinite duration. As a result, Fourier physics and engineering. In the mid-eighties, re- analysis works well if x( t) is composed of a few stationary searchers of the “French school,”lead by a geophysicist, components (e.g., sinewaves). However, any abrupt a theoretical physicist and a mathematician (namely, change in time in a non-stationary signal x(t) is spread Morlet, Grossmann, and Meyer), built strong mathe- out over the whole frequency axis in xyl. Therefore, an matical foundations around the subject and named analysis adapted to nonstationay signals requires more their work “Ondelettes”(Wavelets). They also interacted than the Fourier Transform. considerably with other fields. The usual approach is to introduce time dependency The attention of the signal processing community in the Fourier analysis while preserving linearity. The was soon caught when Daubechies and Mallat, in ad- idea is to introduce a “local frequency” parameter (local dition to their contribution to the theory of wavelets, in time) so that the ‘local” Fourier Transform looks at established connections to discrete signal processing the signal through a window over which the signal is results [DAU88], [MAL89a]. Since then, a number of approximately stationary. Another, equivalent way is to theoretical, as well as practical contributions have been modify the sinewave basis functions used in the Fourier made on various aspects of WT’s, and the subject is Transform to basis functions which are more con- growing rapidly [wAV89], [IT92]. centrated in time (but less concentrated in frequency). The present paper is meant both as a review and as a tutorial. It covers the main definitions and properties of wavelet transforms, shows connections among the SCALE VERSUS FREQUENCY various fields where results have been developed, and focuses on signal processing applications. Its purpose The Short-Time Fourier Transform: is to present a simple, synthetic view of wavelet theory, Analysis with Fixed Resolution. with an easy-to-read, non-rigorous flavor. An extensive bibliography is provided for the reader who wants to go The “instantaneous frequency” [FLA89] has often into more detail on a particular subject. been considered as a way to introduce frequency de-

OClOBER 1991 IEEE SP MAGAZINE 15

Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 29,2010 at 16:23:44 EDT from IEEE Xplore. Restrictions apply. and assume it is stationary when seen through a win- Sliding dow g(t) of limited extent, centered at time location T. Window g(t) The Fourier Transform (1) of the windowed signals xft ) g*(t - T) yields the Short-Time Fourier Transform II I (STFT)

STFT(T,f) = Nt) g*(t- T) e-’Jrrftdt (2)

which maps the signal into a two-dimensional function in a time-frequency plane (TJ.Gabor originally only defined a synthesis formula, but the analysis given in (2) follows easily. The parameter f in (2) is similar to the Fourier frequency and many properties of the Fourier transform carry over to the STFT. However, the analysis here depends critically on the choice of the window g(t). Fig. 1. Timefrequency plane corresponding to the Short-Time Figure 1 shows vertical stripes in the time-frequency Fourier Transform. It can be seen either as a succession of plane, illustrating this “windowing of the signal” view of Fourier Transforms of a windowed segment of the signal (ver- the STFT. Given a version of the signal windowed tical stripes) or as a modulated analysis fdter bank (horizontal around time t, one computes all “frequencies” of the stripes). STFT. An alternative view is based on a filter bank inter- pendence on time. If the signal is not narrow-band, pretation of the same process. At a given frequencyf, however, the instantaneous frequency averages dif- (2) amounts to filtering the signal “at all times” with a ferent spectral components in time. To become accurate bandpass filter having as impulse response the window in time, we therefore need a two-dimensional time-fre- function modulated to that frequency. This is shown as quency representation S(tJ of the signal dt) composed the horizontal stripes in Fig. 1. Thus, the STFT may be of spectral characteristics depending on time, the local seen as a modulated filter bank [ALL77],[POR80]. frequency f being defined through an appropriate From this dual interpretation, a possible drawback definition of S(tJ. Such a representation is similar to related to the time and frequency resolution can be the notation used in a musical score, which also shows shown. Consider the ability of the STFT to discriminate “frequencies”played in time. between two pure sinusoids. Given a window function The Fourier Transform (1)was first adapted by Gabor g(t) and its Fourier transform m, define the [GAB461 to define S(tJ as follows. Consider a signal dt), “bandwidth Afof the filter as

a) frequency

I I !- time I I time

Fig. 2. Basis functions and timezfrequency resolution of the Short-TimeFourier Transform (STlT)and the Wavelet Transform (WlJ. The tiles represent the essential concentration in the time-frequencyplane of a given basis function. (a)Coverage of the timefrequency plane for the STFT, (b)for the WT. (c)Corresponding basis functions for the STFT, (d)for the WT (“wavelets”).

16 IEEE SP MAGAZINE OClOBER 1991

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Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 29,2010 at 16:23:44 EDT from IEEE Xplore. Restrictions apply. component can be analyzed with good time resolution (3) or frequency resolution, but not both.

where the denominator is the energy of g(8. Two The Continuous Wavelet Transform: sinusoids will be discriminated only if they are more A Multiresolution Analysis. than Af apart (This is an rms measure, and others are possible). Thus, the resolution in frequency of the STFT To overcome the resolution limitation of the STFT, analysis is given by A$ Similarly, the spread in time is one can imagine letting the resolution At and Af vary in given by At as the time-frequency plane in order to obtain a multi- resolution analysis. Intuitively, when the analysis is t21g(t)12dt viewed as a filter bank, the time resolution must in- At2 = I (4) crease with the central frequency of the analysis filters. I I g(t)I dt We therefore impose that Afis proportional to f, or

where the denominator is again the energy of g(t).Two pulses in time can be discriminated only if they are more than At apart. Now, resolution in time and frequency cannot be where c is a constant. The analysis filter bank is then arbitrarily small, because their product is lower composed of band-pass filters with constant relative bounded. bandwidth (so-called “constant-Q” analysis). Another way to say this is that instead of the frequency respon- 1 ses of the analysis filter being regularly spaced over the Time - Bandwidth product = At Aft ~ 4n (51 frequency axis (as for the STFT case), they are regularly spread in a logarithmic scale (see Fig. 3).This kind of This is referred to as the uncertainty principle, or filter bank is used, for example, for modeling the fre- Heisenberg inequality. It means that one can only trade quency response of the cochlea situated in the inner ear time resolution for frequency resolution, or vice versa. and is therefore adapted to auditory perception, e.g. of Gaussian windows are therefore often used since they music: filters satisfymg (6)are naturally distributed into meet the bound with equality IGAB461. octaves. More important is that once a window has been When (6)is satisfied, we see that Afand therefore also chosen for the STFT, then the time-frequency resolution At changes with the center frequency of the analysis given by (3),(4) is_fiued over the entire time-frequency filter. Of course, they still satisfy the Heisenberg ine- plane (since the same window is used at all frequencies). quality (5). but now, the time resolution becomes ar- This is shown in Fig. 2a, while Fig. 2c shows the bitrarily good at high frequencies, while the frequency associated basis functions of the STFT. For example, if resolution becomes arbitrarily good at low frequencies. the signal is composed of small bursts associated with For example, two very close short bursts can always be long quasi-stationary components, then each type of eventually separated in the analysis by going up to

a) . Constant Bandwidth (STFT Case)

I I\ I\ I\ I\ I\ I I I ,Frequency f I fo 2fo 3fo 41, 5fo 6fo 7fo Bfo 9fo

Constant Relative Bandwidth (Wr Case)

fo 2fo 410 8fo

~ ______~ Fg.3. Division of the frequency domain [a)for the STlT (uniformcoverage) and [b)for the WT (logarithmic coverage).

OClOBER 1991 IEEE SP MAGAZINE 17

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Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 29,2010 at 16:23:44 EDT from IEEE Xplore. Restrictions apply. Box 7: The Notion of Scale and Resolution where a is a scalefactor (the constant 1/m is used for First, recall that when a functionjt) is scaled: energy normalization). This results in the definition of the CWT: Jt) -+Jat). where a > 0,

then it is contracted if a > 1 and expanded if a < 1 (7) Now. the CWT can be written either as Since the same prototype Ut), called the basic 1 CWTx(z,a) = x(t) h*(%)dt (B1.l) wavelet, is used for all of the filter impulse responses, no specific scale is privileged, i.e. the wavelet analysis is self-similar at all scales. Moreover, this simplification or, by a change of variable, as is useful when deriving mathematical properties of the CWT. CWTx(z,a) = GfNat) h*(t- i)dt (B 1.2) To make the connection with the modulated window used in the STFT clearer, the basic wavelet h(t) in (7) could be chosen as a modulated window [GOU84, The interpretation of (B1.1) is that as the scale GR084, GR0891 f increases, the filter impulse response (5)be- comes spread out in time, and takes only long-time behavior into account. Equivalently, (B 1.2) indi- Then the frequency responses of the analysis filters cates that as the scale grows, an increasingly con- indeed satisfy (6)with the identification tracted version of the signal is seen through a constant length filter. That is, the scale factor a has the interpretation of the scale in maps. Very large a=-fo scales mean global views, while very small scales f mean detailed views. But more generally, h(t) can be any band-pass func- A related but different notion is that of resolu- tion and the scheme still works. In particular one can tion. The resolution of a signal is linked to its dispense with complex-valuedtransforms and deal only frequency content. For example, lowpass filtering a with real-valued ones. signal keeps its scale, but reduces its resolution. It is important to note that here, the local frequency Scale change of continuous time signals does not f = a has little to do with that described for the STFT: alter their resolution, since the scale change can be fo indeed, it is associated with the scaling scheme (see Box reversed. However, in discrete-time signals, in- 1). As a result, this local frequency, whose definition creasing the scale in the analysis involves subsam- depends on the basic wavelet, is no longer linked to pling, which automatically reduces the resolution. frequency modulation (aswas the case for the STFT) but Decreasing the scale (which involves upsampling is now related to time-scalings. This is the reason why can be undone, and does not change the resolution. the terminology “scale”is often preferred to “frequency” The interplay of scale and resolution changes in for the CWT, the word “frequency”being reserved for the discrete-time signals is illustrated in Fig. 9 and fully STFT. Note that we define scale in wavelet analysis like explained in [RI090b], mob]. the scale in geographical maps: since the filter bank impulse responses in (7)are dilated as scale increases, large scale corresponds to contracted signals, while small scale corresponds to dilated signals. higher analysis frequencies in order to increase time resolution (see Fig. 2b). This kind of analysis of course works best if the signal is composed of high frequency components of short duration plus low frequency com- WAVELET ANALYSIS AND SYNTHESIS ponents of long duration, which is often the case with signals encountered in practice. A generalization of the concept of changing resolu- Another way to introduce the CWT is to define tion at different frequencies is obtained with so-called wavelets as basis functions. In fact, basis functions “wavelet packets” [wIC89], where arbitrary time-fre- already appear in the preceding definition (7)when one quency resolutions (within the uncertainty bound (5)) sees it as an inner product of the form are chosen depending on the signal. The Continuous Wavelet Transform (CWT) exactly follows the above ideas while adding a simplification:all impulse responses of the filter bank are defined as which measures the “similarity”between the signal and scaled (i.e. stretched or compressed) versions of the the basis functions same prototype h(4. i.e.,

18 IEEESP MAGAZINE OOOBER 1991

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Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 29,2010 at 16:23:44 EDT from IEEE Xplore. Restrictions apply. particular basis function. Thus, we expect that any general signal can be represented as a decomposition 2: into wavelets, i.e. that the original waveform is syn- s and GWTS as thesized by adding elementary building blocks, of con- stant shape but different size and amplitude. Another The inner product is often used as a similarity way to say this is that we want the continuously labelled measurement, and because both STFT’s and CWT’s wavelets h,r(t)to behave just like an orthogonal basis are inner products, they appear in several detec- [MEYSO]. The analysis is done by computing inner tion/estimation problems. Consider, for example, products, and the synthesis consists of summing up all the problem of estimating the location and velocity the orthogonal projections of the signal onto the of some target in radar or sonar applications. The wavelets. estimation procedure consists in first emitting a known signal ut).In the presence of a target, this signal will return to the source (received signal At)) with a certain delay z, due to the target’s location, where c is a constant that depends only on Wt). The and a certain distortion (Doppler effect), due to the measure in this integration is formally equivalent to dt target‘s velocity. df[GOU84].We have assumed here that both signal and For narrow-band signals, the Doppler effect wavelets are either real-valued or complex analytic so amounts to a single frequency shift fo and the that only positive dilations a > 0 have to be taken into characteristics of the target will be determined by account. Otherwise (8)is more complicated [GR084]. maximizing the cross-correlation function (called Of course, the ha,r(t) are certainly not orthogonal “narrow-band cross-ambiguity function”) [WO0531 since they are very redundant (they are defined for continuously varying a and T). But surprisingly, the reconstruction formula (8)is indeed satisfied whenever x(t) h(t - z) dt = STJ!T(zJ) h(t)is of finite energy and band pass (whichimplies that it oscillates in time like a short wave, hence the name For wide-band signals, however, the Doppler “wavelet”).More precisely, if h(t) is assumed sufficiently frequency shift vanes in the signal’s spectrum, regular, then the reconstruction condition is causing a stretching or a compression in the signal. h(tj dt = 0. The estimator thus becomes the “wide-band cross- ambiguity function” lSPE671, IAUS901 Note that the reconstruction takes place only in the sense of the signal’s energy. For example, a signal may be reconstructed only with zero mean since h(t) dt = 0. In fact the type of convergence of (8)may be strengthened and is related to the numerical robust- As a result, in both cases, the “maximum ness of the reconstruction [DAUSOa]. likelihood”estimator takes the form of a STlT or a Similar reconstruction can be considered for the CWT, i.e. of an inner product between the received STFT, and the similarity is remarkable [DAUSOal. How- signal and either STFT or wavelet basis functions. ever, in the STFT case, the reconstruction condition is The basis function which best fits the signal is used less restrictive: only finite energy of the window is to estimate the parameters. required. Note that, although the wide-band cross-am- biguity function is a CWT, for physical reasons, the dilation parameter a stays on the order of mag- nitude of 1, whereas it may cover several octaves when used in signal analysis [FLA89].

The spectrogram, defined as the square modulus of the STFT, is a very common tool in signal analysis because it provides a distribution of the energy of the signal in the time-frequency plane. A similar distribu- called wavelets. The wavelets are scaled and translated tion can be defined in the wavelet case. Since the CWT versions of the basic wavelet prototype ut)(see Fig. 2d). behaves like an orthonormal basis decomposition, it Of course, basis functions can be considered for the can be shown that it is isometric [GR084],i.e., it STFT as well. For both the STFT and the CWT, the preserves energy. We have sinewaves basis functions of the Fourier Transform are replaced by more localized reference signals labelled by 2d~,da time and frequency (or scale) parameters. In fact both J jl CWT(~,a) I -a2 ~ = transforms may be interpreted as special cases of the cross-ambiguity function used in radar or sonar where Ex = I g t) I dt is the energy of the signal dt). processing (see Box 2). This leads us to define the wavelet spectrogram, or The wavelet analysis results in a set of wavelet scalogram, as the squared modulus of the CWT. It is a coefficients which indicate how close the signal is to a distribution of the energy of the signal in the time-scale

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Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 29,2010 at 16:23:44 EDT from IEEE Xplore. Restrictions apply. b) STFT tn

scale a

4fo I 4 210

I///////,/,/,/,,,/, fo

scale a = fo 1 f

~~ Fig. 4. Regions of influence of a Dirac pulse at t=b(a) for the CWT and (b)for the STFT: as well as of three sinusoids (of frequen- ciesfo, 2f0, 4fo) for ic) the CGand (dj the STFT.

dr da that the phase representation more accurately reveals plane, associated with measure A, and thus ex- a2 isolated, local bursts in a signal than the scalogram pressed in power per frequency unit, like the does (see Box 3). spectrogram. However, in contrast to the spectrogram, To illustrate the above points, Fig. 5 shows some the energy of the signal is here distributed with different examples of spectrograms and scalograms for synthetic resolutions according to Fig. 2b. signals and a speech signal (see Box 3). Figure 4 illustrates differences between a scalogram More involved energy representations can be and a spectrogram. Figure 4a shows that the influence developed for both time-frequency and time-scale of the signal’s behavior around t = in the analysis is [BER88, FLASO, RIOSOa], and a link between the limited to a cone in the time-scale plane; it is therefore spectrogram, the scalogram and the Wigner-Ville dis- very “localized” around to for small scales. In the STFT tribution can be established (see Box 4). case, the corresponding region of influence is as large as the extent of the analysis window over all frequen- cies, as shown in Fig. 4b. Moreover, since the time-scale WAVELET FRAMES AND analysis is logarithmic in frequency, the area of in- ORTH ON 0RMAL BASE S fluence of some pure frequencyfo in the signal increases with fo in a scalogram (Fig. 4c), whereas it remains constant in a spectrogram (Fig. 4d). Discretization of Time-Scale Parameters Both the spectrogram and the scalogram produce a more or less easily interpretable visual two-dimensional We have seen that the continuously labelled basis representation of signals [GR089],where each pattern functions (wavelets) h,r(t) behave in the wavelet in the time-frequency or time-scale plane contributes to analysis and synthesis just like an orthonormal basis. the global energy of the signal. However, such an energy The following natural question arises: if we appropriate- representation has some disadvantages, too. For ex- ly discretize the time-scale parameters a, t, can we ample, the spectrogram, as well as the scalogram, obtain a true orthonormal basis? The answer, as we cannot be inverted in general. Phase information is shall see, is that it depends on the choice of the basic necessary to reconstruct the signal. Also, since both the wavelet h(t). spectrogram and the scalogram are bilinear functions There is a natural way to discretize the time-scale of the analyzed signal, cross-terms appear as inter- parameters a, r [DAUSOa]: since two scales QJ < a1 ferences between patterns in the time-frequency or roughly correspond to two frequencies fo > fl, the time-scale plane [KADSl]and this may be undesirable. wavelet coefficients at scale a1 can be subsampled at In the wavelet case, it has been also shown [GR089] (fio/fi)‘h the rate of the coefficients at scale QJ, according

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Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 29,2010 at 16:23:44 EDT from IEEE Xplore. Restrictions apply. BOX 3: Spectrograms and Sca logra ms We present in Fig. 5 spectrograms and scalograms in the scalogram. Figure 5.2 shows the analysis of for some synthetic signals and a real signal. The three starting sinusoids with difTerent starting times signals are of length 384 samples, and the STFT uses (a low frequency starts fist, followed by a medium a Gaussian-like window of length L = 128 samples. and a high frequency sinewave). Figure 5.3 shows the The scalogram is obtained with a Morlet wavelet (a transforms of a chirp signal. Again, the transitions complex sinusoid windowed with a Gaussian en- are well resolved at high frequencies in the scalogram. velope) of length from 23 to 363 samples. Finally, Fig. 5.4 shows the analysis of a segment of The horizontal axis is time in both spectrograms speech signal, where the onset of voicing is seen in and scalograms. The signal is shown on the top. The both representations. vertical axis is frequency in the spectrogram (high Note that displaying scalograms is sometimes frequencies on top) and scale in the scalogram (small tricky, because parameters like display look-up tables scale at the top). Compare these figures with Fig. 4, (which map the scalogram value to a grey scale value which indicates the axis system used, and gives the on the screen) play an important but not always well rough behavior for Diracs and sinewaves. understood role in the visual impression. Such First, Fig. 5.1 shows the analysis of two Diracs and problems are common in spectrogram displays as two sinusoids with the STFT and the CWT. Note how well. the Diracs are well time-localized at high frequencies

Flg. 5.1. Spectrogram and scalogram for the STFT and CWT analysis of two Dirm pulses and huo sinusoids. 14 Magnitude of the m.fb) Phase of the SZT. IC)Amplitude of the WT. (4Phase of the WT.

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FIg. 5.2.Spectrogram and scalogramfor the STFTand CWTanalysis of three sinusoids with staggered starting times. The lowfrequency one comesfist,followed by the medium and highfrequency ones. (a)Magnitude of the STT.[b) Phase of the STT.(c) Amplitude of the WT. (dl Phase of the WT. E-

I

m

Fg.5.3. Spectrogram and scalogram for the SIFT and CWT analysis of a chirp signal. (a) Magnitude of the STlT (b)Phase of the SIFT. (c]Amplitude of the WT. (d) Phase of the WT.

I

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Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 29,2010 at 16:23:44 EDT from IEEE Xplore. Restrictions apply. BOX 3: Spectrograms and Scalograms (continued)

Flg. 5.4. Spectrogram and scalogramfor the STFT and CWT analysis of a segment of speech, including onset of voicing. (a) Magnitude of the STlT. (b) Phase of the STFL (c) Amplitude of the WT. (d) Phase of the WT.

to Nyquist’s rule. We therefore choose to discretize the are overcomplete. Equation (11) is then still very close time-scale parameters on the sampling grid drawn in to (8) and signal reconstruction takes place within Fig. 7. That is, we have a = a$ and b = k &’ T, where j non-restrictive conditions on h(t).On the other hand, if and k are integers. The corresponding wavelets are the sampling is sparse, e.g. the computation is done octave by octave (a0 = 2), a true orthonormal basis will be obtained only for very special choices of h(t) hj.k (t)= h(d’t - kT) d”2 (9) [DAUSOa. MEYSO]. resulting in wavelet coefficients Wavelet Frames

An analogy is the following: assume that the wavelet The theory of wavelet frames [DUF52, DAUSOa] analysis is like a microscope. First one chooses the provides a general framework which covers the two magnification, that is, Then one moves to the extreme situations just mentioned. It therefore permits chosen location. Now, if one looks at very small details, one to balance (i) redundancy, i.e. sampling density in then the chosen magnification is large and corresponds Fig. 7, and (ii)restrictions on h(t)for the reconstruction scheme (11) to work. The trade-off is the following: if the to j negative and large. Then, Tcorresponds to small d’ redundancy is large (high “oversampling”),then only steps, which are used to catch small details. This mild restrictions are put on the basis functions (9).But justifies the choice b = k a$ Tin (9). if the redundancy is small (i.e., close to “critical”sam- The reconstruction problem is to find ao, T, and h(t) pling), then the basis functions are very constrained. such that The idea behind frames [DUF52] is based on the as- sumption that the linear operator x(t)A cj,k, where cj,k is defined by (10). is bounded, with bounded inverse. The family of wavelet functions is then called a frame and is such that the energy of the wavelet coefficients cj.k (sum where c is a constant that does not depend on the signal of their square moduli) relative to that of the signal lies (compare with (8)).Evidently, if a0 is close enough to 1 between two positive “frames bounds” A and B, (and if T is small enough), then the wavelet functions A.E, I I 9.k I I B.E,v j.k

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Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 29,2010 at 16:23:44 EDT from IEEE Xplore. Restrictions apply. Box 4: Merging Spectrogram, Scalogram, and Wigner Distribution into a Common Class of Energy Representations There has been considerablework in extending the written as [FLA901, [RIO9Oal spectrogram into more general time-frequency energy distributions TF(z,jl. These all have the basic property of distributing the energy of the signal all over the time-frequency plane, i.e., i.e., as some 2D "affine" correlation between the signa I TF( zf) dz df = I I Nt)I 2dt and the "basic" wavelet's Wigner-Ville distribution This remarkable formula tells us that there exis1 strong links between Wavelet Transforms and Wig Among them, an alternative to the spectrogram for ner-Ville distributions. And, as a matter of fact, it car nonstationary signal analysis is the Wigner-Ville dis- be generalized to define the most general class o tribution [CLASO, BOU85, FLA891 time-scale energy distributions [BERSS, FLA9O RI09Oa1, just as in the time-frequency case. Figure 6 shows that it is even possible to go con tinuously from the spectrogram of a given signal to it: scalogram [FLASO, RI090al. More precisely, starting More generally, the whole class of time-frequency from the Wigner-Ville distribution, by progressivelj energy distributions has been fully described by controlling Gaussian smoothing functions, one goe: Cohen [COH66], [COH89]: they can all be seen as through a set of energy representations which eithei smoothed (or, more precisely, correlated) versions of tends to the spectrogram if regular two-dimensiona the Wigner-Ville distribution. The spectrogram is it- smoothing is used, or to the scalogram if "affine' self recovered when the "smoothing" function is the smoothing is used. This property may allow us tc Wigner-Ville distribution of the analysis window! decide whether or not we should choose time-scalf A similar situation appears for time-scale energy analysis tools, rather than time-frequency ones for distributions. For example, the scalogram can be i given problem.

WIGNER VlLLE v.0

p.06 SPECTRLXRAM

Ffg. 6. From spectrograms to scalcgrams via Wigner-ViUe. By controlling the parameter p [which is a measure of the time-& quency extent of the smoothinsfunction). it is possible to make afuu transition between time-scale and time-frequency analyses. Here seven analyses of the same signal [composed of three Gaussian packets) are shown. Note that the best joint time-frequency resolution is attained for the Wigner-Ville distribution, while both spectrogram and scalogram [which can be thought of as smoothed versions of Wigner-ViUe)provide reduced cross-tern eflects comped to Wigner-ViUe (after [FLASO,RM)sod).

24 IEEE SP MAGAZINE OCTOBER 1991 --

Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 29,2010 at 16:23:44 EDT from IEEE Xplore. Restrictions apply. where Ex is the energy of the signal dt). These frame bounds can be computed from m, Tand ut)using Daubechies’ formulae [DAUSOa].Interestingly enough, they govern the accuracy of signal reconstruc- tion by (11). More precisely, we have

with relative SNRgreater than (B/A+ l)/(B/A- 1) (see Fig. 8). The closer A and B, the more accurate the reconstruction. It may even happen that A=B (“tight frame”),in which case the wavelets behave exactly like an orthonormal basis, although they may not even be Fg. 7. Dyadic sampling grid in the time-scale plane. Each linearly independent [DAUSOa]!The reconstruction for- node corresponds to a wavelet basis function hj,k(t) with scale mula can also be made exact in the general case if one 2-Jand shvt 2-Jk. uses different synthesis functions h;.k(t) (which con- stitute the dualframe of the hjk(t)s [DAUSOal). scalings and shifts, but also they form an orthonormal Introduction to orthogonal wavelet bases basis. What is most interesting is that there do exist well-behaved functions h(t) that can be used as If a tight frame is such that all wavelets hj,k(t)(9) are prototype wavelets, as we shall see below. This is in necessary to reconstruct a general signal, then the sharp contrast with the STET, where, according to the wavelets form an orthonormal basis of the space of Balian-Low theorem [DAUSOa], it is impossible to have signals with finite energy [HEISO].Recall that orthonor- orthonormal bases with functions well localized in time mality means and frequency (that is, for which the time-bandwidth product At Afis a finite number). Recently, the wavelet orthonormal scheme has been extended to synthesis functions h’jk(t) # hjk(t),leading to so-called biorthogonal wavelet bases [COHSOal, An arbitrary signal can then be represented exactly [VETSOal, [VETSObl. as a weighted sum of basis functions, THE DISCRETE TIME CASE

j,k In this section, we first take a purely discrete-time That is, not only the basis functions hj,k(t) are ob- point of view. Then, through the construction of iterated tained from a single prototype function wt) by means of filter banks, we shall come back to the continuous-time

SNR (db)

Lt” 1 N=4 35

30

25 1 N=3 \ 20

15

10 5 T

Ftg. 8. Reconstruction SignallNoise Ratio (SNR)error afterframe decompositionfor dtfferent sampling densities an = 2IfN[N = number of voices per octave), b = ad k bo [aBer[DAUSOa]). The basic wavelet is the Morlet wauelet (modulated Gaussian) used in lGRO89l. The reconstruction is done “as if“ wauelets were orthogonal (see text), and its accuracy grows as N increases and bo decreases, i.e. as the density oJ the sampling gnd of rlg. / increases. 1 hereJOre, reaunaancy rejnes re ormnogona[-[uce reconstruction.

OCTOBER 1991 IEEE SP MAGAZINE 25

Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 29,2010 at 16:23:44 EDT from IEEE Xplore. Restrictions apply. case and show how to construct orthonormal bases of Three Dimensional Dis lays wavelets for continuous-time signals (DAU881. In the discrete time case, two methods were of Complex Wavelet fransforms developed independently in the late seventies and early As seen in Box 3, the wavelet transform using a eighties which lead naturally to discrete wavelet trans- complex wavelet like the Morlet wavelet (a complex forms, namely subband coding [CRI76], [CR076], sinusoid windowed by a Gaussian) leads to a com- [EST771 and pyramidal coding or multiresolution signal plex valued function on the plane. analysis [BUR83]. The methods were proposed for Phase information is also useful and thus, there coding, and thus, the notion of critical sampling (of is interest in a common display of magnitude and requiring a minimum number of samples) was of impor- phase. This is possible by using height as mag- tance. Pyramid coding actually uses some oversam- nitude and color as phase, leading to so-called pling, but because it has an easier intuitive explanation, “ phasemagrams”. we describe it first. Two examples are shown here: a synthetic chirp While the discrete-time case has been thoroughly in the upper figure (similar to the one in Fig. 5.3); studied in the filter bank literature in terms of frequency and a triangle function below. In both cases, the bands (see e.g. [VAI87]),we insist here on notions which discontinuous points are clearly identified at small are closer to the wavelet point of view, namely those of scales (top of the figure). The chirp has two such scale and resolution. Scale is related to the size of the points (beginning and end), while the triangle has signal. while resolution is linked to the amount of detail three. At large scales, these signals look just like a present in the signal (see Box 1 and Fig. 9). single discontinuity, which is what an observer Note that the scale parameter in discrete wavelet would indeed see from very far away. For the chirp, analysis is to be understood as follows: For large scales, the phase cycles with increasing speed, as ex- dilated wavelets take “global views” of a subsampled pected. signal, while for small scales, contracted wavelets analyze small “details” in the signal.

The Multiresolution Pyramid

Given an original sequence x(n),n E Z , we derive a lower resolution signal by lowpass filtering with a half- band low-pass filter having impulse response g(n).Fol- lowing Nyquist’s rule, we can subsample by two (drop every other sample), thus doubling the scale in the analysis. This results in a signal y(n) given by

The resolution change is obtained by the lowpass filter (loss of high frequency detail). The scale change is due to the subsampling by two, since a shift by two in the original signal x(n) results in a shift by one in y(n). Now, based on this lowpass and subsampled version of 3rd. we try to find an approximation, a(n).to the original. This is done by first upsampling y(n) by two (that is, inserting a zero between every sample) since we need a signal at the original scale for comparison.

y‘(2n) = y(n), y’(2n+l)= 0

Then, y’(n) is interpolated with a filter with impulse response g‘( n) to obtain the approximation 4n).

Signal analyses with a Morlet wavelet. The display shows magnitude as height and phase as color (phasemagrad. The horizontal axis is time. Above) a synthetic chirp signal, withfrequency increasing with Note that if g( n) and g‘(n) were perfect halfband filters time. Below) a trianglefunction. (having a frequency passband equal to 1 over the nor- malized frequency range -x/2, x/2 and equal to 0 elsewhere), then the Fourier transform of a(n)would be

26 IEEESPMAGAZINE OCTOBER 1991

Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 29,2010 at 16:23:44 EDT from IEEE Xplore. Restrictions apply. - -_ clear that d(n) contains exactly the frequencies above x/2 of 4n). and thus, d(n)can be subsampled by two as lowpass well without loss of information. This hints at the fact that critically sampled schemes must exist. resolution: halved scale: doubled The separation of the original signal xfn)into a coarse approximation a( n) plus some additional detail con- tained in d(n)is conceptually important. Because of the resolution change involved (lowpass filtering followed by subsampling by two produces a signal with half the resolution and at twice the scale of the original), the resolution: halved above method and related ones are part of what is called scale: doubled Multiresolution Signal Analysis [ROS84] in computer vision. The scheme can be iterated on y(n). creating a hierar- chy of lower resolution signals at lower scales. Because of that hierarchy and the fact that signals become shorter and shorter (or images become smaller and resolution: unchanged scale: halved smaller), such schemes are called signal or image pyramids lBUR831. Fig. 9. Resolution and scale changes in discrete time (by-fac tors oJ2). Note that the scale ofsgnals is defined as in Subband Coding Schemes geographical maps. (a)Halfband lowpassfiltering reduces the resolution by 2 (scale is unchanged). (b)Halfland 1owpass.fil- We have seen that the above system creates a redun- tering followed by subsampling by 2 doubles the scale (arid dant set of samples. More precisely, one stage of a halves the resolution as in (a)).(c) Upsampling by 2Jollowed pyramid decomposition leads to both a half rate low by haljband 1owpassJltering halves the scale (resolutionis resolution signal and a full rate difference signal, result- unchanged). ing in an increase in the number of samples by 50%. This oversampling can be avoided if the filters g(n) and equal to the Fourier transform of x(n)over the frequency g‘( n) meet certain conditions [VET9Ob]. range (-x/2, x/2J while being equal to zero elsewhere. We now look at a different scheme instead, where no That is, a(n)would be a perfect halfband lowpass ap- such redundancy appears. It is the so-called subband proximation to x(n). coding scheme first populariLed in speech compression Of course, in general, a(n)is not going to be equal to [CRI76, CR076, EST77J.The lowpass, subsampled ap- x(n) (in the previous example, x(n)would have to be a proximation is obtained exactly as explained above, but, halfband signal). Therefore, we compute the difference instead of a difference signal, we compute the “added between a(n)(our approximation based on y(n))and xfn), detail” as a highpass filtered version of x(n)(using a filter with impulse response h(n)),followed by subsampling by two. Intuitively, it is clear that the “added detail” to the lowpass approximation has to be a highpass signal, It is obvious that x(n)can be reconstructed by adding and it is obvious that if g(n) is an ideal halfband lowpass d(n) and a(n),and the whole process is shown in Fig. filter, then an ideal halfband highpass filter h(n)will lead 10. However, there has to be some redundancy, since a to a perfect representation of the original signal into two signal with sampling ratef, is mapped into two signals subsampled versions. d(n) and y(n) with sampling ratesf, and fs/2, respec- This is exactly one step of a wavelet decomposition tively. using sin(x)/xfilters, since the original signal is mapped In the case of a perfect halfband lowpass filter, it is into a lowpass approximation (at twice the scple) and

------______------Flg. 10. Pyramid scheme. Denuahon of a lowpass. subsampled appromrnation y(n) from which an approxmmation aln) to xln) E denued by upsampling and interpolation Then. the difference between the approxmnation a(rJ and the onginal x[n)is computed as d[n).Perfect reconstructionis simply obtained by adding a(n) back.

OCTOBER 1991 IEEE SP MAGAZINE 21

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Fg. 1 1. Subband Coding scheme. [a)Two subsampled approximations, one corresponding to low and the other to high frequen- cies, are computed. The reconstructed signal is obtained by re-interpolating the approximations and summing them The$lters on the le$ form an analysisPlter bank, while on the right is a synthesisJlter bank. (b)Block diagram (FilterBank tree) of the Discrete Wavelet Transform implemented with discrete-time filters and subsampling by two. The frequency resolution is given in Fg.3b.

an added detail signal (also at twice the scale). In convolutions followed by subsampling by two, evaluates particular, using these ideal filters, the discrete version inner products of the sequence x(n)and the sequences is identical to the continuous wavelet transform. {g(-n+2k),w-n+21)} (the time reversal comes from the What is more interesting is that it is not necessary convolution, which reverses one of the sequences). Thus to use ideal (that is, impractical) filters, and yet x(n)can be recovered from its two filtered and subsampled versions which we now call yo(n) and yl(n). To do so, both are upsampled and filtered by g(n) and h‘(n) respectively, and finally added together, as shown in Fig. lla. Now, unlike the pyramid case, the reconstructed signal (which we now call is not 2n)) Because the filter impulse responses form an or- identical to dn),unless the filters meet some specific thonormal set, it is very simple to reconstruct dn) as constraints. Filters that meet these constraints are said to have perfect reconstruction property, and there are a number of papers investigating the design of perfect reconstruction filter banks [MIN85, SMI86, VAI88, VET861. The easiest case to analyze appears when the that is, as a weighted sum of the orthogonal impulse analysis and synthesis filters in Fig. 1 la are identical responses, where the weights are the inner products of (within time-reversal) and when perfect reconstruction the signal with the impulse responses. This is of course is achieved (that is, 2n) = dn),within a possible shift). the standard expansion of a signal into an orthonormal Then it can be shown that the subband analysis/syn- basis, where the resynthesis is the sum of the or- thesis corresponds to a decomposition onto an or- thogonal projections (see Introduction to orthogonal thonormal basis, followed by a reconstruction which wavelet bases above). amounts to summing up the orthogonal projections. We From (12).(13) it is also clear that the synthesis filters will assume FIR filters in the following. Then, it turns are identical to the analysis filters within time reversal. out that the highpass and lowpass filters are related by Such orthogonal perfect reconstruction filter banks have been studied in the digital signal processing litera- ture, and the orthonormal decomposition we just indi- cated is usually referred to as a “paraunitary” or where L is the filter length (which has to be even). Note “lossless”filter bank [vAI89]. An interesting property of that the modulation by (-l)ntransforms indeed the such filter banks is that they can be written in lattice lowpass filter into a highpass one. form [vAI88], and that the structure and properties can Now, the filter bank in Fig. lla, which computes be extended to more than two channels [vAI87, VAI89,

28 IEEE SP MAGAZINE OClOBER I991

Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 29,2010 at 16:23:44 EDT from IEEE Xplore. Restrictions apply. VET89). More general perfect reconstruction (bior- wavelet transform. In the discrete time case, the role of thogonal) filter banks have also been studied (see e.g. the wavelet is played by the highpass filter h(n) and the m86,VETSOb, COHSOall. It has been also noticed cascade of subsampled lowpass filters followed by a [MAL89b, SHESO, RIOSOb] that filter banks arise highpass filter (which amounts to a bandpass filter). naturally when implementing the CWT. These filters, which correspond roughly to octave band Note that we have assumed linear processing filters, unlike in the continuous wavelet transform, are throughout. If non-linear processing is involved (like not exact scaled versions of each other. In particular, quantization), the oversampled nature of the pyramid since we are in discrete time, scaling is not as easily scheme described in the preceding section may actually defined, since it involves interpolation as well as time lead to greater robustness. expansion. Nonetheless, under certain conditions, the discrete The Discrete Wavelet Transform system converges (after a certain number of iterations) to a system where subsequent filters are scaled versions We have shown how to decompose a sequence dn) of each other. Actually, this convergence is the basis for into two subsequences at half rate, or half resolution, the construction of continuous time compactly sup- and this by means of "orthogonal" filters (orthogonal ported wavelet bases [DAU88]. with respect to even shifts). Obviously, this process can Now, we would like to find the equivalent filter that be iterated on either or both subsequences. In par- corresponds to the lower branch in Fig. 1 lb, that is the ticular, to achieve finer frequency resolution at lower iterated lowpass filter. It will be convenient to use frequencies (as obtained in the continuous wavelet z-transforms of filters, e.g. Qz) = g(n) z-~in the fol- transform), we iterate the scheme on the lower band n only. If g(n) is a good halfband lowpass filter, h(n)is a lowing. It can be easily checked that subsampling by good halfband highpass filter by (12). Then, one itera- two followed by filtering with G(z) is equivalent to filter- tion of the scheme on the first lowband creates a new ing with G(2)followed by the subsampling (3inserts lowband that corresponds to the lower quarter of the zeros between samples of the impulse response, which frequency spectrum. Each further iteration halves the are removed by the subsequent subsampling). That is, width of the lowband (increases its frequency resolution the first two steps of lowpass filtering can be replaced by two), but due to the subsampling by two, its time by a filter with z-transform G(z).G(3),followed by sub- resolution is halved as well. At each iteration, the sampling by 4. More generally, calling G'(z) the current high band portion corresponds to the difference equivalent filter to i stages of lowpass filtering and between the previous lowband portion and the current subsampling by two (that is, a total subsampling by 2'). one, that is, a passband. Schematically, this is we obtain equivalent to Fig. 1 lb, and the frequency resolution is as in Fig. 3b. i- 1 An important feature of this discrete algorithm is its C'(Z) = n Q3', relatively low complexity. Actually, the following some- 1=0 what surprising result holds: independent of the depth of the tree in Fig. llb, the Complexity is linear in the Call its impulse response g'(n). number of input samples, with a constant factor that As i infinitely increases, this filter becomes infinitely depends on the length of the filter. The proof is long. Instead, consider a function f'(x) which is straightforward. Assume the computation of the first piecewise constant on intervals of length 1/2' and has filter bank requires COoperations per input sample (CO value 2''2 g'(n) in the interval [n/2',(n+1)/2']. That is, is typically of the order of L). Then, the second stage requires also COoperations per sample of its input, but, f'(x)is a staircase function with the value given by the because of the subsampling by two, this amounts to samples of g'(n) and intervals which decrease as 2-i. It C0/2 operations per sample of the input signal. There- can be verified that the function is supported on the fore, the total complexity is bounded by interval [0, L-11, where L is the length of the filter g(n). NOW, for i going to infinity, ~'(x)can converge to a &tal = + -coco + - + ... < 2co co 24 continuous function g&),or a function with finitely many discontinuities, even a fractal function, or not which demonstrates the efficiency of the discrete converge at all (see Box 5). wavelet transform algorithm and shows that it is inde- A necessary condition for the iterated functions to pendent of the number of octaves that one computes. converge to a continuous limit is that the filter G(z) This bounded complexity had been noticed in the mul- should have a sufficient number of zeros at z = - 1, or tirate filtering context [RAM88].Further developments half sampling frequency, so as to attenuate repeat can be found in [RIO9 la].Note that a possible drawback spectra [DAU88, DAUSOb, RIO9 lb]. Using this condi- is that the delay associated with such an iterated filter tion, one can construct filters which are both orthogonal bank grows exponentially with the number of stages. and converge to continuous functions with compact support. Such filters are called regular, and examples Iterated Filters and Regularity can be found in [DAU88, COHSOa, DAUSOb, RI090b, VETSOb]. Note that the above condition can be inter- There is a major difference between the discrete preted as ajatness condition on the spectrum of G(z) scheme we have just seen and the continuous time at half sampling frequency. In fact, it can be shown

OClOBER 1991 IEEE SP MAGAZINE 29

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It is well known that the structure of computations A in a Discrete Wavelet Transform and in an octave- band fflter bank are identical. Therefore, besides the different views and interpretations that have been given to them, the main di@erence lies in the filter design. Wavelet fflters are chosen so as to be regular.

30 IEEE SPMAGAZINE OClOBER 1991

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Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 29,2010 at 16:23:44 EDT from IEEE Xplore. Restrictions apply. 0 14, . , , , , , ,

-05 0 05 1 15 2 25 3 35

..~~~~~..... ~..~~~~~~~~ ~~~~~ -..-..~-~~ -..~.~~~ .-.....~~ ~ ~ Fig. 13. Scaling functions satisfying two-scale difference equations. (a)the hat function. (b)the D4 wavelet obtained from a 4-tap regularjlter by Daubechies.

[AKASOJ, [SHESO]that the well-known Daubechies or- So far, we have only discussed the iterated lowpass thonormal filters [DAU88]are deduced from "maximally and its associated scaling function. However, from Fig. flat" low-pass filters [HER711. Note that there are many 1 lb, it is clear that a bandpass filter is obtained in the other choices that behave very differently in terms of same way, except for a final highpass filter. Therefore, phase, selectivity in frequency, and other criteria (see in a fashion similar to (15), the wavelet k(x)is obtained e.g. [DAUSObJ).An important issue related to regular as filter design is the derivation of simple estimates for the m regularity order (see Box 5). It is still not clear whether regular filters are most adapted to coding schemes [ANISO]. The minimal regularity order necessary for good coding performance that is, it also satisfies a two scale equation. of discrete wavelet transform schemes, if needed at all, Now, if the filters h(n) and g(n) form an orthonormal is also not known and remains a topic for future inves- set with respect to even shifts, then the functions gC(x-I) tigation. and h&k) form an orthonormal set (see Box 6). Be- cause they also satisfy two scale difference equations, Scaling Functions and Wavelets it can be shown [DAU88] that the set k(2-k-k),i,k E 2, Obtained from Iterated Filters forms an orthonormal basis for the set of square in- tegrable functions. Recall that gC(4is the final function to which f(x) Figure 14 shows two scales and shifts of the 4-tap converges. Because it is the product of lowpass filters, Daubechies wavelet [DAU88]. While it might not be the final function is itself lowpass and is called a "scaling obvious from the figure, these functions are orthogonal function" because it is used to go from a fine scale to a to each other, and together with all scaled and trans- coarser scale. Because of the product (14) from which lated versions, they form an orthonormal basis. the scaling function is derived, gc(x)satisfies the follow- Figure 15 shows an orthogonal wavelet based on a ing two scale difference equation [DAUSOc]: length- 18 regular filter. It is obviously a much smoother function (actually, it possesses 3 continuous deriva- tives). Finally, Fig. 16 shows a biorthogonal set of linear phase wavelets, where the analysis wavelets are or- thogonal to the synthesis wavelets. These were obtained Figure 13 shows two such examples. The second one from a biorthogonal linear phase filter bank with length- is based on the 4-tap Daubechies filter which is regular 18 regular filters moa,VET9ObI. and orthogonal to its even translates [DAU88]. We have shown how regular filters can be used to

OCTOBER 1991 IEEE SPMAGAZINE 31

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Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 29,2010 at 16:23:44 EDT from IEEE Xplore. Restrictions apply. 02

0 15

01

0 05

0 n behind the -0 05 spaces such that: 0 15

01 f JX)E Vi then Vci. Call Wi the 0 05 1. This is written 0

(B6.1) -0 05

* necessary to go -0 1 e has -1 0 123 4 5 6 7

363 ... (B6.2) Fig. 14. Two scales 01 the 04 wavelet and sht@s. This set of functions orthogonal. tion can be attained by a is

Now, assume we have an orthonormal basis for Vo made up of a function gdxj and its integer APPLICATIONS OF WAVELETS IN translates. Because Vo E V-1, gc(xj can be written in terms of the basis in V-1,i.e., (15)is satisfled: SIGNAL PROCESSING

From the derivation of the wavelet transform as an alternative to the STFT, it is clear that one of the main applications will be in non-stationary signal analysis. Then it can be +rifled that the function Mxj (16) While conceptually, the CWT is a classical constant-Q (with the relation (12))and its integer translates analysis, its simple definition (based on a single func- form an orthonormal basis for WO.And, because of tion rather than multiple filters) allows powerful (B6.2), h&j and its scaled and translated versions analytical derivations and has already lead both to new form a wavelet basis W9a,W9c, MEY901. insights and new theoretical results [wAV89]. . The multiresolution idea is now very intuitive. Applications of wavelet decompositions in numerical Assume we have an approximation of a signal at a analysis, e.g. for solving partial differential equations, resolution corresponding to Vi. Then a better ap- seem very promising because of the “zooming”property proximation is obtained by adding the details cor- which allows a very good representation of discon- responding to WO.that is, the projection of the tinuities, unlike the Fourier transform [BEY89]. signal in WO.This amounts to a weighted sum of Perhaps the biggest potential of wavelets has been wavelets at that scale. Thus. by iterating this idea, claimed for signal compression. Since discrete wavelet a square integrable signal can be seen as the suc- transforms are essentially subband coding systems, cessive approximation or weighted sum of wavelets and since subband coders have been successful in at herand Aner scale. speech and image compression, it is clear that wavelets will find immediate application in compression problems. The only difference with traditional subband generate wavelet bases. The converse is also true. That coders is the fact that filters are designed to be regular is, orthonormal sets of scaling functions and wavelets [that is, they have many zeroes at z = 0 or z = n). Note can be used to generate perfect reconstruction filter that although classical subband filters are not regular banks [DAU88, MAL89a, MAL89cl. (see Box 5 and Fig. 12).they have been designed to have Extension of the wavelet concept to multiple dimen- good stopbands and thus are close to being “regular”, sions, which is useful, e.g. for image coding, is shown at least for the first few octaves of subband decomposi- in Box 7. tion.

32 IEEE SP MAGAZINE OCTOBER 1991

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01

0 05

0

-0 05

-0 1

-0 15 2 4 6 8 10 12 14 16 18

Frg. 15. Orthonormal wavelet generated from a length-18 regular3lter [DAU88/.The time function is shown on the le3 and the spectrum is on the right.

It is therefore clear that drastic improvements of already appeared. For example, statistical signal compression will not be achieved so easily simply be- processing using wavelets is a promising field. Multi- cause wavelets are used. However, wavelets bring new scale models of stochastic processes [BAS89],[CH091], ideas and insights. In this respect, the use of wavelet and analysis and synthesis of l/f noise [GACSl], decompositions in connection with other techniques WOR901 are examples where wavelet analysis has been (like vector quantization [ANI901 or multiscale edges successful. “Wavelet packets” [WIC89], which cor- [MAL89d]) are promising compression techniques respond to arbitrary adaptive tree-structured filter which make use of the elegant theory of wavelets. banks, are another promising example. New developments, based on wavelet concepts, have

1 I”” 08-

06-

04-

02-

0

-0 2 -

-0 4 -

-0 6 -

-0 8 -

1 #I I I, 0 2 4 6 8 10 12 14 16 18

0 2 4 6 8 10 12 14 16 18

Q. 16. Biorthogonal wavelets generated from 18-tap regularPlters [VETSObI. (a)Analysis wavelet. (b)Synthesis wavelet. The timefunction is shown on the left and the spectrum is on the right.

OCTOBER 1991 IEEESP MAGAZINE

Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 29,2010 at 16:23:44 EDT from IEEE Xplore. Restrictions apply. Box 7:Multidimensional filter banks and wavelets In order to app€ywavelet decompositions to multi- pling by 2 in each dimension, that is, overall subsam- dimensional signals (e.g., images), multidimensional pling by 4 (see Fig. 17). extensions of wavelets are required. An obvious way More interesting (that is, non-trivial) multidimen- to do this is to use “separablewavelets” obtained from sional wavelet schemes are obtained when non- products of one-dimensional wavelets and scaling separable subsampling is used [KOV92]. For functions (MAL89a. MAL89c, MEYSO]. Let us consider example, a non-separable subsampling by 2 of a double indexed signal dnl, nz) is obtained by retain- ing only samples satisfying:

(B7.1)

The resulting points are located on a so-called quincunx sublattice of Z2. Now, one can construct a perfect reconstruction fllter bank involving such sub- sampling because it resembles its one-dimensional counterpart [KOV921. The subsampling rate is 2 to each other with respect to (equal to the determinant of the matrix in (B7.l)), and the filter bank has 2 channels. Iteration of the filter mponent). The function SAX, y) bank on the lowpass branch (see Fig. 18) leads to a discrete wavelet transform, and if the filter is regular ile the functions &9(xy> are (which now depends on the matrix representing the lattice [KOV92]), one can construct non-separable k 2x-q. i= 1,2,3=dj,k,l wavelet bases for square integrable functions over R2 basis for square integrable with a resolution change by 2 (and not 4 as in the solution corresponds to a separable case). An example scaling function is pic- al filter bank with subsam- tured in Fig. 19.

n 2

- 2 - - H1 . 3 .- - Hh @ X- - - - - H1 Hh . %.- g - - horizontally - -Hh ’ 0

D

U

Fig. 18. Iteration of a non-separablejilterbank basec in Rg. 19. Two-dimensionalnon-separable orthonormal xal- non-separable subsampling. This construction le& to non- ingfunction II

34 IEEE SP MAGAZINE OCTOBER 1991

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Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 29,2010 at 16:23:44 EDT from IEEE Xplore. Restrictions apply. CONCLUSION pleting work in the Ph.D. degree in Signal Processing at Telecom University, specializing in wavelet theory, image coding, and fast signal algorithms. We have seen that the Short-Time Fourier Transform and the Wavelet Transform represent alternative ways Martin Vetterli was born in Switzer- to divide the time-frequency (or time-scale) plane. Two land in 1957. He received the Dip. major advantages of the Wavelet Transform are that it El.-Ing. degree from the can zoom in to time discontinuities and that orthonor- Eidgenossische Technische mal bases, localized in time and frequency, can be Hochschule Zurich, Switzerland, in constructed. In the discrete case, the Wavelet Trans- 1981; the Master of Science degree form is equivalent to a logarithmic filter bank, with the from Stanford University, Stanford CA, added constraint of regularity on the lowpass filter. in 1982: and the Doctorat 2s Science The theory of wavelets can be seen as a common Degree from the Ecole Polytechnique framework for techniques that had been developed Federale de Lausanne, Switzerland, in 1986. independently in various fields. This conceptual In 1982, he was a Research Assistant at Stanford unification furthers the understanding of the University, and from 1983 to 1986 he was a researcher mechanisms involved, quantifies trade-offs, and points at the Ecole Polytechnique. He has worked for Siemens to new potential applications. A number of questions and AT&T Bell Laboratories. In 1986, he joined Colum- remain open, however, and will require further inves- bia University in New York where he is currently as- tigations (e.g., what is the “optimal”wavelet for a par- sociate professor of Electrical Engineering, member of ticular application?). the Center for Telecommunications Research, and While some see wavelets as a very promising brand codirector of the Image and Advanced Television new theory [CIP90], others express some doubt that it Laboratory. represents a major breakthrough. One reason for skep- He is a senior member of the IEEE, a member of SIAM ticism is that the concepts have been around for some and ACM, a member of the MDSP committee of the IEEE time, under different names. For example, wavelet Signal Processing Society, and of the editorial boards of transforms can be seen as constant-Q analysis Signal Processing and Image Communication. He [yOU78], wide-band cross-ambiguity functions [SPE67, received the Best Paper Award of EURASIP in 1984 for AUSSO], Frazier-Jawerth transforms [FRA86], perfect his paper on multidimensional subband coding, and the reconstruction octave-band filter banks IMIN85, Research Prize of the Brown Boveri Corporation (Swit- SMI861, or a variation of Laplacian pyramid decomposi- zerland) in 1986 for his thesis. His research interests tion [BUR83], [BUR89]! include multirate signal processing, wavelets, computa- We think that the interest and merit of wavelet theory tional complexity, signal processing for telecommunica- is to unify all this into a common framework, thereby tions and digital video processing. allowing new ideas and developments. REFERENCES ACKNOWLEDGMENTS [AKA901 A.N. Akansu. R.A. Haddad, and H. Caglar, “The The authors would like to thank C. Herley for many Binomial QMF-Wavelet Transform for Multiresolution Signal useful suggestions and for generating the continuous Decomposition.’‘submitted IEEE Trans. Signal Proc., 1990. STFT and WT plots; and Profs. F. Boudreaux-Bartels, [ALL771 J.B. Allen and L.R. Rabiner, “A Unified Approach to M.J.T. Smith and Dr. P. Duhamel for useful suggestions Short-Time Fourier Analysis and Synthesis,” Roc. IEEE, Vol. on the manuscript. We thank B. Shakib (IBM) for 65. No. 11. pp. 1558-1564. 1977. creating the three-dimensional color rendering of phase [ANI901 M. Antonini. M. Barlaud, P. Mathieu and I. Dau- and magnitude of wavelet transforms (so-called bechies. “Image Coding Using Vector Quantization in the “phasemagrams”) used in the cover picture and else- Wavelet Transform Domain.” in Proc. 1990 IEEE Int. Con$ Acoust.. Speech. Signal Proc.. Albuquerque, NM, Apr.3-6, 1990, where: C.A. Pickover (IBM) is thanked for his 3D display pp. 2297-2300. software and J.L. Mannion for his help on software [AUSSO] L. Auslander and I. Gertner, “Wide-Band Ambiguity tools. The second author would like to acknowledge Function and a.x+b group.” in Signal Processing, Part I: Signal support by NSF under grants ECD-88- 1 1 1 1 1 and MIP- Processing Theory, L. Auslander. T. Kailath, S. Mitter eds., 90- 14189. Institute for Mathematics and its Applications, Vol. 22. Springer Verlag, New York. pp. 1-12. 1990. Olivier Rioul was born in Strasbourg, [BAS891 M. Basseville and A. Benveniste, “Multiscale Statisti- France on July 4, 1964. He received cal Signal Processing.“ in Proc. 1989 IEEE Int. Conf. Acoust.. diplomas in Electrical Engineering Speech, Signal Proc.. Glasgow, Scotland, pp. 2065-2068, May 23-26, 1989. from the Ecole Polytechnique, Palaiseau, France, and from Telecom [BER88] J. Bertrand and P. Bertrand. ‘Time-FrequencyRep- resentations of Broad-Band Signals.” in Proc. 1988 IEEE Int. University, Paris, in 1987 and 1989, Cor-$ Acoust., Speech. Signal Proc., New York, NY,Apr. 11-14, respectively. 1988, pp.2 196-2199. Since 1989, he has been with the [BEY891 G. Beylkin. R. Coifman and V. Rokhlin, “Fast Wavelet Centre National d’Etudes des Telecommunications Transforms and Numerical Algorithms. I,” submitted, Dec. (CNET),Issy-Les-Moulineaux, France, where he is com- 1989.

OCTOBER 1991 IEEE SP MAGAZINE 35

Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 29,2010 at 16:23:44 EDT from IEEE Xplore. Restrictions apply. [BOU85] G. F. Boudreaux-Bartels, ‘Time-Varying Signal Int. Conf. Acoust., Speech, Signal Proc., Albuquerque, NM, April Processing Using the Wigner Distribution Time-Frequency Sig- 3-6, 1990, pp. 2455-2458. nal Representation,” in Adu. in Geophysical Data Proc., Vol. 2, pp. 33-79, Jai Press Inc., 1985. [FOUSS] J. B. J. Fourier, ‘Theorie Analytique de la Chaleur,” in Oeuures de Fourier, tome premier, G.Darboux, Ed., Paris: [BUR831 P.J. Burt and E.H. Adelson, “The Laplacian Pyramid Gauthiers-Villars, 1888. asaCompactImageCode,”IEEETrans. onCom.,Vol. 31, N0.4, pp. 532-540, April 1983. IFRA281 P. Franklin, “A Set of Continuous Orthogonal Func- tions,” Math. Annal., Vol. 100, pp.522-529, 1928. [BUR891 P.J. Burt, “Multiresolution Techniques for Image Representation, Analysis, and ‘Smart’ Transmission,” Proc. [FRA861 M. Frazier and B. Jawerth, ‘The cp-Transform and SPIE Conf. on Visual Communication and Image Processing., Decomposition of Distributions,” Proc. Con$ Function Spaces pp. 2-15, Philadelphia, PA, Nov. 1989. and Appl., Lund 1986, Lect. Notes Math., Springer. (CAL64j A. Calderon, “Intermediate Spaces and Interpolation, [GAB461 D. Gabor, ’Theory of Communication,”J. of the IEE, V01.93, pp.429-457, 1946. the Complex Method,” Studia Math., Vol. 24, pp. 113-190, 1964. [GACSl] N. Gache, P. Flandrin, and D. Garreau, ‘‘Fractal Dimension Estimators for Fractional Brownian Motions,” in (CH0911 K.C. Chou, S. Golden, and A.S. Willsky, “Modeling and Estimation of Multiscale Stochastic Processes,” in Proc. 1991 IEEEInt. Conf.Acoust., Speech, SignalProc., Toronto, Proc. 1991 IEEEInt. Con$ Acoust., Speech. SignalProc., Toronto, Ontario, Canada, pp. 3557-3560, May 14-17, 1991. Ontario, Canada, pp. 1709-1712, May 14-17, 1991. lGOU841 P. Goupillaud, A. Grossmann and J. Morlet, “Cycle- [CIPSO] B.A. Cipra, “A New Wave in Applied Mathematics,” Octave and Related Transforms in Seismic Signal Analysis,” Science, Research News, Vol. 249, August 24, 1990. Geoexploration, Vo1.23, pp.85-102. Elsevier Science Pub., 1984/85. [CLABO] T.A.C.M. Classen and W.F.G. Mecklenbrauer, “The [GR084] A. Grossmann and J. Morlet, “Decomposition of Wigner Distribution - A Tool for Time-Frequency Signal Analysis, Part I, 11, 111,” Philips J. Res., Vo1.35, pp. 217-389, Hardy Functions into Square Integrable Wavelets of Constant 1980. Shape,” SD?M J.MathAnal., Vol. 15, N0.4, pp.723-736, July 1984. [COH66] L. Cohen. “Generalized Phase-Space Distribution Functions,”J. Math. Phys., Vol. 7, No. 5, pp. 781-786, 1966. [GR089] A. Grossmann, R. Kronland-Martinet, and J. Morlet, “Reading and Understanding Continuous Wavelet Trans- [COH89] L. Cohen, ‘Time-Frequency Distribution - A Review,” forms,” in [wAV89], pp.2-20, 1989. P~oc.IEEE, Vol. 77, NO. 7, pp. 941-981, 1989. [HAAlO] A. Haar, “Zur Theorie der Orthogonalen Funktionen- [COHSOa] A. Cohen, I. Daubechies and J.C. Feauveau, “Bior- systeme,” [in German] Math. Annal., Vol. 69, pp. 331-371. thogonal Bases of Compactly Supported Wavelets,” to appear 1910. in Comm Pure and Applied Math. [HEISO] C.E. Heil, “Wavelets and Frames,” in Signal Process- [COHSOb] A. Cohen, “Construction de Bases d’ondelettes ing, Part I: Signal Processing Theory, L. Auslander, et al. eds., Holderiennes,”Reuista Matematica Iberoamericana, Vo1.6, No. IMA, Vol. 22, Springer, New York, pp.147-160, 1990. 3 y 4, 1990. [HER711 0. Herrmann, “On the Approximation Probem in (CRI761 A. Croisier, D. Esteban, and C. Galand, “Perfect NonRecursive Digital Filter Design,” IEEE Trans. Circuit Theory, Channel Splitting by Use of Interpolation, Decimation, Tree Vol. CT-18, No. 3, pp. 411-413, May 1971. Decomposition Techniques,” Int. Conf. on Information Scien- ces/Systems, Patras, pp. 443-446, Aug. 1976. [IT921 IEEE Transactions on Information Theory, Special issue on Wavelet Transforms and Multiresolution Signal Analysis, to [CRO76] R.E. Crochiere, S.A. Weber, and J.L. Flanagan, “Digi- appear January, 1992. tal Coding of Speech in Subbands,”Bell Syst. Tech. J.. Vo1.55, [JOH80] J.D. Johnston, “A Filter Family Designed for Use in pp. 1069-1085,Oct.1976. Quadrature Mirror Filter Banks,” in Roc. 1980 IEEE Int. Conf. [CR083] R.E. Crochiere, and L.R. Rabiner, Multirate Digital Acoust., Speech, SignalProc., pp. 291-294, Apr. 1980. Signal Processing, Prentice-Hall, Englewood Cliffs, NJ, 1983. [KAD91] S. Kadambe and G.F. Boudreaux-Bartels, “ACompar- [DAUB81 I. Daubechies, “Orthonormal Bases of Compactly ison of the Existence of ‘CrossTerms’ in the Wavelet Transform, Supported Wavelets,” CorninPure and Applied Math., Vo1.41, Wigner Distribution and Short-Time Fourier Transform,” Sub- NO.7, pp.909-996, 1988. mitted IEEE Trans. Signal Proc., Revised Jan. 1991. [DAUSOa] I. Daubechies. ‘The Wavelet Transform, Time-- [KOV92] J. Kovacevic and M. Vetterli, “Non-separable Multi- Frequency Localization and Signal Analysis,” IEEE Trans. on dimensional Perfect Reconstruction Filter Banks and Wavelet Info. Theory,Vol. 36, No.5, pp.961-1005. Sept. 1990. Bases for R”,” IEEE Trans. on Info. Theory, Special Issue on wavelet transforms and multiresolution signal analysis, to [DAUSOb] I. Daubechies, “Orthonormal Bases of Compactly appear Jan. 1992. Supported Wavelets 11. Variations on a Theme,” submitted to SIAM J. Math. Anal. [LIT371 J. Littlewood and R. Paley, ‘Theoremson Fourier Series and Power Series,”Proc. London Math Soc., Vo1.42, pp. 52-89, [DAUSOc] I. Daubechies and J.C. Lagarias, “Two-ScaleDiffer- 1937. ence Equations 11. Local Regularity. Infinite Products of Matrices and Fractals,”submitted to SIAMJ. Math. Anal, 1990. (MAL89al S. Mallat, “A Theory for Multiresolution Signal Decomposition: the Wavelet Representation,” IEEE Trans. on [DUF52] R.J. Duffin and A.C. Schaeffer, “A Class of Non- PatternAnalysis and Machine Intell, Vol. 11, No. 7, pp.674-693. harmonic Fourier Series,” Am. Math. Soc., Vol. 72, pp. Trans. July 1989. 341-366, 1952. [MAL89b] S. Mallat. “Multifrequency Channel Decompositions [EST771 D. Esteban and C. Galand, “Applicationof Quadrature of Images and Wavelet Models,” Acoust., Speech, Mirror Filters to Split-Band Voice Coding Schemes,” Int. Conf. IEEE Trans. Signal Proc., Vol. 37, No. 12, pp.2091-2110, December 1989. Acoust., Speech, Signal Proc., Hartford, Connecticut, pp. 191- 195, May 1977. [MAL89c] S. Mallat, “Multiresolution Approximations and Wavelet Orthonormal Bases of L2(R),”,Trans. Amer. Math. Soc., IFLA891 P. Flandrin, “Some Aspects of Non-Stationary Signal Vo1.315, No. 1, pp.69-87, September 1989. Processing with Emphasis on Time-Frequency and Time-Scale Methods,”in [wAV89], pp.68-98, 1989. [MAL89d] S. Mallat and S. Zhong, “Complete Signal Repre- sentation with Multiscale Edges,” submitted to IEEE Trans. [FLAgO] P. Flandrin and 0. Rioul, “Wavelets and Mine Pattern Analysis and Machine Intell, 1989. Smoothing of the Wigner-Ville Distribution,”in Proc. 1990IEEE

36 IEEE SP MAGAZINE OClOBER 1991

Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 29,2010 at 16:23:44 EDT from IEEE Xplore. Restrictions apply. IMEY891 Y. Meyer, “Orthonormal Wavelets,” in WAV891, pp. Springer, 315 pp., 1989. 21-37, 1989. WE391 M.V. Wickerhauser, “Acoustic Signal Compression [MEYSO] Y. Meyer, Ondelettes et Operateurs, Tome I. Onde- with Wave Packets,” preprint Yale University, 1989. lettes, Herrmann ed., Paris, 1990. WO0531 P.M. Woodward. Probability and Information Theory [MIN85] F. Mintzer, “Filtersfor Distortion-Free Two-Band Mul- with Application to Radar, Pergamon Press, London, 1953. tirate Filter Banks,” IEEE Trans. on Acoust., Speech, Signal Proc., Vo1.33, pp.626-630, June 1985. WOR901 G.W. Wornell, “A Karhunen-Loeve-like Expansion for l/f Processes via Wavelets,” IEEE Trans. Info. Theory, Vol. 36, [POR801 M.R. Portnoff, “Time-Frequency Representation of NO. 4, pp.859-861, July 1990. Digital Signals and Systems Based on Short-Time Fourier Analysis,”IEEE Trans. on Acoust., Speech Signal Proc.,Vo1.28, IYOU781 J.E. Younberg, S.F. Boll, ”Constant-Q Signal Analysis pp.55-69, Feb. 1980. and Synthesis,” in Proc. 1978 IEEE Int. Conf. on Acoust., Speech, and Signal Proc., Tulsa, OK, pp. 375-378, 1978. [RAM881 T.A. Ramstad and T. Saram&i, “Efficient Multirate Realization for Narrow Transition-Band FIR Filters,” in Proc. 1988 IEEE Int. Syrnp. Circuits and System, Helsinki, Finland, pp. 2019-2022, 1988. EXTE NDE D REF ERE NCES [RI090a] 0. Rioul and P. Flandrin, ‘Time-Scale Energy Dis- tributions: A General Class Extending Wavelet Transforms,”to History of Wavelets: see [HAAlO, FRA28, LIT37, CAL64. appear in IEEE Trans. Signal Roc. YOU78, GOU841. [RIOSOb] 0. Rioul, “A Discrete-Time Multiresolution Theory Books on Wavelets: (see also WAV89, MEYSO]) Unifying Octave-Band Filter Banks, Pyramid and Wavelet I. Daubechies, Ten Lectures on Wavelets, CBMS, SIAM publ., Transforms,” submitted to IEEE Signal Trans. Roc. to appear. [RIOSla] 0. Rioul and P. Duhamel, “Fast Algorithms for Dis- Wavelets and their Applications, R.R. Coifman, I. Daubechies, crete and Continuous Wavelet Transforms,” submitted to IEEE S. Mallat, Y. Meyer scientific eds.. L.A. Raphael, M.B. Ruskai Trans. Information Theory, Special Issue on Wavelet Trans- managing eds., Jones and Bertlel pub., to appear, 1991. forms and Multiresolution Signal Analysis. Tutorials Wavelets: (see also [FLA89, GR089, MAL89b, [RI091b] 0. Rioul, “Dyadic Up-Scaling Schemes: Simple on MEY891) Criteria for Regularity,” submitted to SLAM J. Math. Anal. R.R. Coifman, “Wavelet Analysis and Signal Processing,” in [ROS84] A. Rosenfeld ed., Multiresolution Techniques in Com- Signal Processing, Part I: Signal Processing Theory, L. Ausland- puter Vision, Springer-Verlag, New York 1984. er et al. eds., IMA, Vol. 22, Springer, New York, 1990. ISM1861 M.J.T. Smith and T.P. Bamwell, “Exact Reconstruc- C.E. Heil and D.F. Walnut, “Continuous and Discrete Wavelet tion for Tree-Structured Subband Coders,” IEEE Trans. on Transforms,” SIAM Review, Vol. 31, No. 4, pp 628-666, Dec. Acoust., Speech and Signal Proc., Vol. ASSP-34, pp. 434-441, 1989. June 1986. Y. Meyer, S. Jaffard, 0. Rioul, “L‘Analyse par Ondelettes,” [in [SHESO] M.J. Shensa, ‘The Discrete Wavelet Transform: Wed- FrenchIPour Science, No. 119, pp.28-37, Sept 1987. ding the a Trous and Mallat Algorithms,” submitted to IEEE La Trans. on Acoust., Speech, and Signal Roc., 1990. G. Strang, “Waveletsand Dilation Equations: A Brief Introduc- tion.” SIAMReview, Vol. 31, No. 4, pp. 614-627, Dec. 1989. [SPE67] J.M. Speiser, “Wide-BandAmbiguity Functions,”IEEE Trans. on Info. Theory, pp. 122-123, 1967. Mathematics. Mathematical Physics and Quantum Mechanics: (see also [GR084, DAU88, MEYSO]) IVAl871 P.P. Vaidyanathan, “Quadrature Mirror Filter Banks, M-band Extensions and Perfect-Reconstruction Techniques,” G. Battle, “A Block Spin Construction of Ondelettes, 11. The IEEEASSP Magazine, Vol. 4, No. 3, pp.4-20, July 1987. Quantum FieldTheory (QFT)Connection,” Comm. Math. Phys.. Vol. 114, pp. 93-102, 1988. [vAI88] P.P. Vaidyanathan and P:Q. Hoang, “Lattice Struc- tures for Optimal Design and Robust Implementation of Two- W.M. Lawton. “Necessary and Sufficient Conditions for Con- Band Perfect Reconstruction QMF Banks,” IEEE Ilfans. on structing Orthonormal Wavelet Bases,” Aware Tech. Report # Acoust., Speech and Signal Proc., Vol. ASSP-36, No. 1, pp.81- AD900402. 94, Jan. 1988. P.G. Lemarie and Y. Meyer, “Ondelettes et Bases Hil- [vAI89] P.P. Vaidyanathan and 2. Doganata, “The Role of bertiennes,” [in French] Revista Matematica Iberoamericana, Lossless Systems in Modem Digital Signal Processing,” IEEE V01.2, N0.18~2,pp.1-18, 1986. Trans. Education, Special issue on Circuits and Systems, Vol. T. Paul, “Affine Coherent States and the Radial Schrodinger 32, N0.3, Aug. 1989, pp.181-197. Equation I. Radial Harmonic Oscillator and the Hydrogen [VET861 M. Vetterli, “Filter Banks Allowing Perfect Recon- Atom,” to appear in Ann. Inst. H.Poincare . struction,” SignalProcessing, Vol. 10, No.3, April 1986, pp.219- H.L. Resnikoff, “Foundations or Arithmeticum Analysis: Com- 244. pactly Supported Wavelets and the Wavelet Group,” Aware m89]M. Vetterli and D. Le Gall, “Perfect Reconstruction FIR Tech. Report # AD890507. Filter Banks: Some Properties and Factorizations, IEEE Trans. Regular Wavelets: see [DAU88, DAUSOb, DAUSOc, COHSOb, on Acoust., Speech Signal Proc., Vo1.37, No.7, pp.1057-1071, RIO9 1b] July 1989. Computer-Aided Geometric Design using Regular Interpola- [VETSOa] M. Vetterli and C. Her1ey“‘Waveletsand Filter Banks: tors: Relationships and New Results,” in Proc. 1990 IEEE Int. Con$ Acoust., Speech Signal Roc., Albuquerque, NM, pp. 1723- S.Dubuc, “InterpolationThrough an Iterative Scheme,”J. Math. 1726, Apr. 3-6, 1990. Analysis Appl., Vol. 114, pp. 185-204, 1986. [VETSOb] M. Vetterli and C. Herley, “Wavelets and Filter N. Dyn and D. Levin, “Uniform Subdivision Schemes for the Banks: Theory and Design,”to appear in IEEE naris. on Signal Generation of Curves and Surfaces,”Constructive Approxima- Proc., 1992. tion, to appear. WAV891 Wavelets, Time-Frequency Methods and Phase Space, Numerical Analysis: (see also [BEY8911 Proc. Int. Conf. Marseille, France, Dec. 14-18, 1987, J.M. R.R. Coifman, “Multiresolution Analysis in Nonhomogeneous Combes et al. eds., Inverse Problems and Theoretical Imaging, Media,” in WAV891, pp. 259-262, 1989.

OClOBER 1991 IEEE SP MAGAZINE 37

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Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 29,2010 at 16:23:44 EDT from IEEE Xplore. Restrictions apply. V. Perrier, ‘Toward a Method to Solve Partial Differential Equa- MIN85, SMI86, VAI87, VAI88, VAI89, VET86, VET891) tions Using Wavelet Bases,” in [wAV89],pp. 269-283, 1989. J.D. Johnston, “A Filter Family Designed for Use in Quadrature Multiscale Statistical Signal Processing: see [BAS89. Mirror Filter Banks,” Proc. ICASSP-80, pp.291-294, April 1980. CH0911. T.Q. Nguyen and P.P. Vaidyanathan, ‘Two-Channel Perfect-. Fractals, Turbulence: (see also [GACS1, WOR901) Reconstruction FIR QMF Structures Which Yield Linear-Phase A. Arneodo, G. Grasseau. and M. Holschneider, “Wavelet Analysis and Synthesis Filters,” IEEE Trans. Acoust., Speech, Transform of Multifractals,” Phys. Review Letters, Vo1.6 1. SignalRocessing,Vol. ASSP-37, No. 5, pp.676-690, May 1989. N0.20, pp.2281-2284, 1988. M.J.T. Smith and T.P. Barnwell, “A New Filter Bank Theory for F. Argoul, A. Ameodo, G. Grasseau, Y. Gagne, E.J. Hopfinger, Time-Frequency Representation,” IEEE Trans. on Acous t., and U. Frisch, “Wavelet Analysis of Turbulence Reveals the Speech and Signal Proc., Vol. ASSP-35, N0.3, March 1987, pp. Multifractal Nature of the Richardson Cascade,” Nature, 314-327. Vo1.338, pp.51-53, March 1989. P.P. Vaidyanathan, Multirate Filter Banks, Prentice Hall, to One-Dimensional Signal Analysis: (see also [GR089], [WIG- appear. 891) Pyramid Transforms:see [BUR83, BUR89, ROS841 C. D’Alessandro and J.S. Lienard, “Decomposition of the Multidimensional Mlter Banks: (see also IKOV921) Speech Signal into Short-Time Waveforms Using Spectral Seg- mentation,”inProc. 19881EEEInt. Con$ Acoust., Speech, Signal G. Karlsson and M. Vetterli, “Theory of Two-Dimensional Mul- Roc., New York, Apr.11-14, 1988, pp.351-354. tirate Filter Banks,” IEEE Trans. on Acoust., Speech, Signal Proc., Vo1.38, No.6, pp.925-937, June 1990. ~~ S. -dambe and G’F’ Boudreaux-Bade1s9 “A Of M, Vetterli, “Multi-Dimensional Subband Coding: SomeTheory Wavelet Functions for Pitch Detection of Speech Signals ” in Roc. 1991 IEEEInt. Conf.Acomt.,Speech Signa[Roc.,ToroAto, and Algorithms,” 6, No.2, PP. 97-1 12, Ontario, Canada, pp. 449-452, May. 14-17, 1991. Feb. 1984. M. Vetterli, J. Kovacevic and D. Le Gall, “Perfect Reconstruction R, Kronland-Martinet, J. Morlet, and A, Grossmann, .‘Analysis of Sound PatternsThrough Wavelet Transforms,”Int. J.Pattern Banks for HDTV Representation and Coding3”‘mage Recognition and Artifkial Intelligence, Vol. 1, No.2, pp. 273-302, Communicamn vo1.2* No’3, Oct. 1990*pp.349-364. pp.97-126, 1987. E. Viscito and J. Allebach, ‘The Analysis and Design of Multi- J,L. Larsonneur and J. .‘Wavelets and Seismic Interpre- dimensional FIR Perfect Reconstruction Filter Banks for Ar- tation,” in [wAV89], pp.126-131, 1989. bitrary Sampling Lattices,” IEEE Trans. Circuits and Systems, Vo1.38, pp.29-42, Jan. 1991. F. B. Tuteur, “Wavelet Transformations in Signal Detection ” in Roc. 1988IEEEInt. Con$ Acoust., Speech, Signal Roc., New (see MAL89a’ MAL89c’ MAL89d1)’ York, NY,Apr. 11-14, 1988, pp.1435-1438. Also in WAV891, pp. 132-138, 1989. J.C. Feauveau, “Analyse Multiresolution pour les Images avec Radar,Sonar, AmbiguityFunctions:see e,g, [Ausgo, SPE67, un Facteur de Resolution 2,”[in French],Traitement du Signal, woo531 Vol. 7. NO. 2, pp. 117-128, July 1990. K. Grochenig and W.R. Madych, “Multiresolution analysis, Tirne-Scale Representations: see [BER88, FLA89, FLAgO, RI090al. Haar bases and self-similar tilings of R”,” submitted to IEEE Trans. onrnfo. Theory, Special Issue on wavelet transforms and Filter BankTheory: (see also [CRI76,CR076, EST77, CR083, multiresolution signal analysis, Jan. 1992.

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