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WaveletsDemetrio Labate, Guido Weiss, and Edward Wilson

The subject called “wavelets” is made upof several areas of pure and appliedmathematics. It has contributed to theunderstanding of many problems invarious sciences, engineering and other

disciplines, and it includes, among its notablesuccesses, the wavelet-based digital fingerprintimage compression standard adopted by the FBIin 1993 and JPEG2000, the current standard forimage compression.

We will begin by describing what wavelets arein one dimension and then pass to more generalsettings, trying to keep the presentation at a non-technical level as much as possible. We assumethat the reader knows a bit of harmonic analysis.In particular, we assume knowledge of the basicproperties of Fourier series and Fourier transforms.We start by establishing the basic definitions andnotation which will be used in the following.

The space L2(R) is the Hilbert space of allsquare (Lebesgue) integrable functions endowedwith the inner product 〈f , g〉 =

∫R f g. The Fourier

transform F is the unitary operator that mapsf ∈ L2(R) into the function Ff = f defined by

(Ff )(ξ) = f (ξ) =∫Rf (x) e−2πiξx dx

when f ∈ L1(R)∩ L2(R) and by the “appropriate”limit for the general f ∈ L2(R). We refer to thevariable x as the time variable and to ξ as thefrequency variable. Notice that the function f is

Demetrio Labate is professor of mathematics at the Univer-sity of Houston. His email address is [email protected].

Guido Weiss is professor of mathematics at Washington Uni-versity. His email address is [email protected].

Edward Wilson is professor of mathematics at Washing-ton University. His email address is [email protected].

DOI: http://dx.doi.org/10.1090/noti927

also square integrable. Indeed, F maps L2(R)one-to-one onto itself. The inverse F−1 of F isdefined by

(F−1g)(x) = g(x) =∫Rg(ξ) e2πixξ dξ.

The functions ek(x) = e2πikx : k ∈ Z are 1-periodic and form an orthonormal basis of L2(T),where T is the 1-torus and can be identified withany of the sets (0,1] or [− 1

2 ,12) or [−1,− 1

2) ∪[ 1

2 ,1), …, (all having measure one). We denote theFourier series of f , 1-periodic and in Lp(T), by∑

k∈Z〈f , ek〉T ek ∼ f ,

where 〈f , ek〉T =∫T f ek, and k ∈ Z.

The paper is organized as follows. In the nextsection, “Wavelets in L2(R)”, we introduce one-dimensional wavelets; in the section “Wavelets inHigher Dimensions”, we discuss wavelets in higher-dimensional Euclidean spaces; in “ContinuousWavelets” we introduce continuous wavelets andsome applications; finally, in the last section“Various Other Wavelet Topics, Applications andConclusions”, we discuss other applications andmake some concluding remarks.

Wavelets in L2(R)We consider two sets of unitary operators onL2(R): the translations Tk, k ∈ Z, defined by(Tkf )(x) = f (x − k) and the (dyadic) dilationsDj , j ∈ Z, defined by (Djf )(x) = 2j/2f (2jx). Awavelet (more precisely, a dyadic wavelet) is afunction ψ ∈ L2(R) having the property thatthe system Wψ = ψj,k = (Dj Tk)ψ : j, k ∈ Z isan orthonormal (ON) basis of L2(R). Notice thatthe order of applying first translations and thendilations is important: Dj Tk = T2−jkDj .

In this section we will explain why there aremany wavelets enjoying a large number of useful

66 Notices of the AMS Volume 60, Number 1

properties which makes it plausible that variousdifferent types of functions (or signals) can beexpressed efficiently by appropriate wavelet bases.

It is often stated that in 1910 Haar [19] exhibiteda wavelet ψ = ψH but that it took about seventyyears before a large number of different waveletsappeared in the world of mathematics. This isreally not the case. The Haar wavelet is definedby ψH = χ[0, 12 ) − χ[ 1

2 ,1), and it is not difficult to

show that it is a wavelet (as will be shown below).Another simple example is the Shannon wavelet;it appeared in the 1940s, and we will explain inwhat sense it appeared. It is defined as ψS = χS ,where S = [−1,− 1

2) ∪ [12 ,1). A straightforward

calculation shows that, if ψ ∈ L2(R), then, forj, k ∈ Z,

(1) ψj,k(ξ) = [2−j/2e−2πik2−jξ] ψ(2−jξ).

Let us observe that the sets 2jS, j ∈ Z, form amutually disjoint covering of R \ 0. Moreover,since the system e−kχS : k ∈ Z is an ON basisof L2(S), the functions within the square bracketin (1), restricted to the set 2jS, form an ON basis ofL2(2jS) for each j ∈ Z. It follows immediately thatthe set ψSj,k : j, k ∈ Z is an ON basis of L2(R).This shows that ψS is a wavelet.

We mentioned that it is not difficult to showthat the Haar function ψH is a wavelet. Wewill do this together with the presentation of ageneral method for constructing wavelets, theMultiresolution Analysis (MRA) method introducedby S. Mallat, with the help of R. Coifman andY. Meyer [23], [26].

An MRA is a sequence Vj : j ∈ Z of closedsubspaces of L2(R) satisfying the following condi-tions:

(i) Vj ⊂ Vj+1 for all j ∈ Z.(ii) Vj+1 = D1Vj for all j ∈ Z; that is, f ∈ Vj iff

f (2·) ∈ Vj+1.(iii)

⋂j∈Z Vj = 0.

(iv)⋃j∈Z Vj = L2(R).

(v) There exists φ ∈ V0 such that Tkφ : k ∈Z is an ON basis of V0.

The function φ described in (v) is called a scalingfunction of this MRA.

If Vj : j ∈ Z is an MRA, let Wj be theorthogonal complement of Vj within Vj+1. Animmediate consequence of the above properties isthat the spaces Wj , j ∈ Z, are mutually orthogonaland their orthogonal direct sum

⊕j∈ZWj satisfies

(2)⊕j∈ZWj = L2(R).

If there exists a function ψ ∈ W0 such thatTkψ : k ∈ Z is an ON basis of W0, using theobservation that DjW0 = Wj for each j ∈ Z (aneasy consequence of the MRA properties), we see

that ψj,k = DjTkψ : k ∈ Z is an ON basis of Wj .It follows from (2) that ψj,k = DjTkψ : j, k ∈ Zis an ON basis of L2(R). Thus, ψ is a wavelet. Inthe case where φ = χ[0,1) and V0 is the span of theON system Tkφ : k ∈ Z, it is easy to check thatVj = DjV0 : j ∈ Z is an MRA. Moreover, it is easyto verify that TkψH : k ∈ Z is an ON basis ofthe space W0 defined by W0 = V⊥0 ⊂ V1. It followsthat DjTkψH : j, k ∈ Z is an ON basis of L2(R).This shows that the Haar function ψH is indeed awavelet.

We leave it to the reader to verify that theShannon wavelet ψS is an MRA wavelet as well.In fact, it corresponds to the scaling functionφ(x) = sinc(x) = sinxπ

xπ (note: sinc(0) = 1). Thisis a consequence of the fact that (sinc)∧(ξ) =χ[− 1

2 ,12 )(ξ) = φ(ξ).

We point out that there is an important resultinvolving the function sinc, namely, the followingelementary theorem.

Theorem 1 (Whittaker-Shannon-Kotelnikov Sam-pling Theorem). Let f ∈ L2(R) and supp f ⊂[− 1

2 ,12]. Then

f (x) =∑k∈Zf (k) sinc(x− k),

where the symmetric partial sums of this seriesconverge in the L2-norm, as well as absolutely anduniformly.

We will explain how this result is related towavelets even though, when it was obtained thenotion of wavelets had not yet appeared. Theword sampling reflects the fact that the functionsinvolved are completely determined if we knowtheir values on the countable set Z. The nameShannon is singled out because he is associatedwith many important aspects and applications ofsampling.

Let φ ∈ L2(R), φ not the zero function, andTφ = φk = Tkφ : k ∈ Z. Then Tφ generates theclosed space Vφ := 〈φ〉 = spanφk : k ∈ Z, thatis, the closure of all finite linear combinationsof the functions φk. This space is shift-invariantand is called the Principal Shift-Invariant Space(PSIS) generated by φ. In case φ = sinc, then Tφis an orthonormal system (recall that (sinc)∧(ξ) =χ[− 1

2 ,12 )(ξ)) and we have that

∑k∈Z |f (k)|2 < ∞,

where f is the function in Theorem 1. If V0 = 〈sinc〉,then the set Vj = DjV0 : j ∈ Z is an MRA andφ = sinc is a scaling function for this MRA.For a general MRA with a scaling function φ,there is a bounded 1-periodic function m0 knownas a low-pass filter and an associated high-pass

filter,m1(ξ) = e2πiξm0(ξ + 12), which produce the

so-called two-scale equations:

(3) φ(2ξ) =m0(ξ)φ(ξ), ψ(2ξ) =m1(ξ)φ(ξ).

January 2013 Notices of the AMS 67

In fact, these equations produce the desired waveletψ generated by the scaling functionφ. In the specialcase we are considering, where φ = sinc, the low-pass filter, when restricted to the interval [− 1

2 ,12], is

the function χ[− 14 ,

14 ]

. One can indeed verify that, in

this case, ψ(ξ) = e−iπξ ψS(ξ) = e−iπξ χS(ξ). Thus,the wavelet we obtain is essentially the Shannonwavelet, since the factor e−iπξ is irrelevant to theorthonormality of the system.

Let us point out that the spaces V0 and W0, ingeneral, are PSIS’s, but they have an importantdifference with respect to the dilation operatorswe are considering: Vj = Dj V0 is an increasingsequence of closed spaces as j → ∞, while thespaces Wj = Dj W0 are disjoint and satisfy (2).

The properties of shift-invariant spaces havemany consequences in the theory of wavelets.If φ is not the zero function in L2(R), letpφ(ξ) =

∑j∈Z |φ(ξ + j)|2 and consider the space

Mφ = L2([0,1], pφ) of all 1-periodic functions msatisfying∫ 1

0|m(ξ)|2 pφ(ξ) dξ := ‖m‖2

Mφ<∞.

It is easy to check that the mapping Jφ : Mφ ,〈φ〉 = Vφ defined by Jφm = (m φ)∨ is an isometryonto Vφ ⊂ L2(R). That is, the two spacesMφ andVφ are “essentially equivalent” via the map Jφ. Itis natural, therefore, to ask how the propertiesof the weight pφ correspond to the propertiesof the generating system Tφ. For example, thefunctions e−k(ξ) = e−2πikξ , k ∈ Z, are mappedby the mapping Jφ onto the functions φ(· − k),k ∈ Z. Since the set e−k(ξ) : k ∈ Z is algebraicallylinearly independent, so is the systemTφ. It followsimmediately that

(i) Tφ is an ON system iff pφ(ξ) = 1 a.e.,(ii) pφ(ξ) > 0 a.e. iff there exists an ON basis

of Vφ of the form Tkψ : k ∈ Z for someψ ∈ Vφ.

In [20], [7] many properties of pφ are shownto be equivalent to properties of Vφ or Tφ. Forexample, one can show the following:

(iii) The systemTφ is a frame for Vφ in the sensethat we have constants 0 < A ≤ B <∞ forwhich

A∑k∈Z|〈f , Tkφ〉|2 ≤ 〈f , f 〉 ≤ B

∑k∈Z|〈f , Tkφ〉|2

for each f ∈ Vφ iff

AχΩφ(ξ) ≤ pφ(ξ) ≤ B χΩφ(ξ), a.e.,

where Ωφ = ξ ∈ [0,1] : pφ(ξ) > 0.Note that, in general, when φ = (

χΩφpφ φ)

∨, then

φ ∈ Vφ = Vφ and pφ = χΩ a.e. Moreover, Tφ is aParseval frame (PF) for Vφ; that is, it is a framewith A = B = 1. Slightly more general than ON

MRA wavelets are PF MRA wavelets ψ where Tψis a PF for W0 and there is a scaling functionφ generating a PF for V0. Examples include thefunction ψ given by ψ = χ

U\ 12U

, for U ⊂ [− 12 ,

12),

where U has positive measure and 12U ⊂ U .

One of the most celebrated contributions tothe construction of MRA wavelets was madeby I. Daubechies [1], [2], who used an inge-nious construction to produce MRA waveletsthat are compactly supported and can have highregularity and many vanishing moments, wherethe kth moment of ψ is defined as the integral∫R xkψ(x)dx. These wavelets are very useful for

applications in numerical analysis and engineer-ing, since the wavelet expansions of a piecewisesmooth function converge very rapidly to thefunction. Specifically, suppose that f ∈ CR(R), thespace of R times differentiable functions suchthat ‖f‖CR =max‖f (s)‖∞ : s = 0, . . . , R <∞, andψ is a compactly supported wavelet having atleast R vanishing moments. Choose a bijectionπ : N , Z×Z such that |〈f ,ψπ(k)〉| ≥ |〈f ,ψπ(k+1)〉|for all k ≥ 1; that is, the wavelet coefficients of fare ordered in nonincreasing order of magnitude.Then one can show [21, Thm. 7.16] that

(4) |〈f ,ψπ(m)〉| ≤ C ‖f‖CR m−(R+1/2),

where C is a constant independent of f andm. Theimplication of this is that relatively few coefficientsare needed to get a good approximation of f .In fact, letting fN be the best N-term nonlinearapproximation of f , the nonlinear approximationerror decays as(5)‖f − fN‖2

L2 ≤ C∑m>N

|〈f ,ψπ(m)〉|2 ≤ C ‖f‖CR N−2R.

Remarkably, this result holds also if f is Rtimes continuously differentiable up to finitelymany jump discontinuities. That is, the waveletapproximation behaves as if the functions hadno discontinuities. This behavior is very differentfrom Fourier approximations, in which case theerror rate is of the order O(N−2). These resultshave extensions to higher dimensions (see furtherdiscussion in the section “Wavelets in HigherDimensions”).

Another important property of an MRA is thatit enables an efficient algorithmic implementationof the wavelet decomposition. To explain this,suppose that ψ is a compactly supported MRAwavelet associated with a compactly supportedscaling function φ. Because φ and ψ are in V1,it follows that D−1φ and D−1ψ are in V0. Let usexamine equations (3), which, in the time domain,can be written as


(6)

(D−1φ)(x) =1√2φ(x2) =

∑k∈Zakφ(x− k),

(D−1ψ)(x) =1√2ψ(x2) =

∑k∈Zbkφ(x− k),

where only finitely many coefficients ak and bk arenot zero. Since D−1 is unitary, from (6) we obtain

ak = 〈D−1φ,Tkφ〉 = 〈Dj−1φ,DjTkφ〉,(7)

bk = 〈D−1ψ,Tkφ〉 = 〈Dj−1ψ,DjTkφ〉.We see, therefore, that there are two ON bases forVj = Wj−1 ⊕ Vj−1; they are DjTφ and Dj−1Tψ ∪Dj−1Tφ. From equalities (7) we can calculate thematrices of the change of bases derived fromthese two ON bases (keeping in mind that theorder of dilations and translations is important:TkD1 = D1T2k). For a given compactly supportedfj ∈ Vj , we apply the appropriate change of basismatrix to compute the orthogonal projection fj−1

of f0 into Vj−1 and obtain the “correction term”ej−1 = fj − fj−1 ∈ Wj−1. Iterating this procedure,we see that arithmetic manipulations with finitesets of coefficients are all that are involved tocompute f0 ∈ V0 and ei ∈ Wi , 0 ≤ i ≤ j − 1, so thatfj = f0 + e1 + · · · + ej−1. Together with the excel-lent approximation properties of wavelets, thisremarkably simple technique is one of the main rea-sons why engineers adopted wavelets (specifically,the so-called compactly supported biorthogonalwavelets [8]) in the design of JPEG2000, the indus-trial standard for image compression replacing theolder Fourier-based JPEG standard.

Another property of MRA wavelets is that,considered as members of a subset of the unitsphere in L2(R), they form an arcwise connectedset. In particular, it is not hard to show that thereare continuous paths of wavelets. Suppose thatψ isan MRA wavelet and ψ is a wavelet in the same MRA.Then one can easily show that (ψ)∧ = s ψ, where sis a 1-periodic unimodular function. In particular,we can choose a 1-periodic function θ for which(ψ)∧ = eiθ ψ and set st = eitθ , for t ∈ [0,1], toestablish a continuous path t , (stψ)∨ in L2(R)connecting ψ to ψ. A more complicated argumentshows howψ is continuously connected to the Haarwavelet [13]. Other related questions arise naturally.For example, are all ON wavelets connected? Areany two frame wavelets connected? The answerto this last question is yes, whereas the previousquestion is still an open problem.

Before moving to the topic of wavelets in higherdimensions, let us state that there are many otherfacts about one-dimensional wavelets we have notdiscussed. In particular, there are wavelets notarising from an MRA. Also, wavelets can be definedby replacing dyadic dilations with dilations byr > 1, where r need not be an integer. In thesituation of nondyadic dilations, the construction

of the orthonormal bases associated with thewavelet may require more than one generator;namely, if r = p

q > 1 and p, q are relatively prime,then p − q generators are needed.

Wavelets in Higher DimensionsMany of the concepts in the previous section extendnaturally to n dimensions (n ∈ N, n > 1), withZ-translations replaced by Zn-translations and thedilation set 2j : j ∈ Z replaced by uj : j ∈ Z,where u is an n × n real matrix, each of whoseeigenvalues has magnitude larger than one. Bothfor theoretical and practical purposes, however, itis convenient to focus our attention on PF waveletsrather than on ON wavelets. Thus, given the matrixu, we seek functions ψ ∈ L2(Rn) for which thewavelet system

ψj,k(x) = (Dju Tkψ)(x)(8)

= |detu|j/2ψ(ujx− k), j ∈ Z, k ∈ Zn,is a PF for L2(Rn). That is,∑

j∈Z

∑k∈Zn

|〈f , (Dju Tkψ〉|2 = ‖f‖2L2(R2),

for all f ∈ L2(Rn). For example, let n = 2 and

u =(

2 00 2

). For U ⊂ [− 1

2 ,12)

2 where U has positive

measure and 12U ⊂ U , the function ψ defined by

ψ = χU\ 1

2Uis a PF MRA wavelet. On the other hand,

to obtain an ON MRA wavelet system, we need touse three wavelet generators.

As in the one-dimensional case, we avoid multi-ple wavelet generators by restricting our attentionto n × n integer matrices u with |detu| = 2. Forexample, let u be chosen to be the quincunx matrix

q =(

1 −11 1

), representing a counterclockwise ro-

tation by π/4 multiplied by√

2. We do encounteran “unexpected” fact if we try to find a Haar-typewavelet. There exists a setD ⊂ R2 such that χD is ascaling function for an MRA (defined as the obvioustwo-dimensional analogue of the one-dimensionalMRA) that produces a Haar-type wavelet as thedifference of two disjoint sets. Yet these sets arerather complicated fractal sets, known as the twindragons (see Figure 1), as was observed in [14].

These observations indicate that the generalconstruction of two-dimensional wavelets is signifi-cantly more complicated than the one-dimensionalcase. In particular, it is not known whether thereexist continuous compactly supported ON waveletanalogues of the one-dimensional Daubechieswavelets associated with the dilation matrix q.However, it turns out that a rather simple changein the definition of the dilations in (8) producesmuch simpler constructions of Haar-type waveletsin dimension two.


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−1

−0.5

0

0.5

1

1.5

2

2.5

3

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−1

−0.5

0

0.5

1

1.5

2

2.5

3

Figure 1. On the left is the fractal set known as the “Twin Dragon”, whose characteristic function isthe scaling function for the two-dimensional Haar-type wavelet associated with the dilation matrix qqq.On the right we see the support of the resulting wavelet ψψψ, whose values are 1 on the darker set, -1

on the lighter set, and 0 elsewhere.

Essentially, the idea consists of adding anadditional set of dilations to the ones producedby the integer powers of the quincunx matrix.Specifically, let B be the group of the eightsymmetries of the square, given by B = bj :

j = 0,1, . . . ,7, where b0 =(

1 00 1

), b1 =

(0 11 0

),

b2 =(

0 −11 0

), b3 =

(−1 00 1

), and bj = −bj−4, for

j = 4, . . . ,7. Let R0 be the triangle with vertices(0,0), ( 1

2 ,0), (12 ,

12) and Ri = bi R0 for i = 0, . . . ,7

(see Figure 2). Letφ = 23/4χR0 and V0 be the closedlinear span of Db Tkφ : b ∈ B, k ∈ Z2. Note that|detb| = 1 for all b ∈ B and V0 is the subspace ofL2(R2) of square integrable functions which areconstant on each Z2-translate of the triangles Ri ,i = 0, . . . ,7. It is not difficult to show that thereis a structure very similar to the classical MRAconsisting of the spaces Vj = Djq V0, j ∈ Z. In fact,let us define the vector-valued function

Φ =Db0φ

...Db7φ

=φ0

...φ7

.Then Vj : j ∈ Z is an MRA with a vector-valuedscaling function Φ. To derive a Haar-like wavelet,we observe that R0 = q−1R1 ∪ q−1(R6 + ( 0

1 )) (seeFigure 2). This equality implies that

φ(x) = φ0(x) = φ1(x)+φ6(qx− ( 01 )).

Applying Dbj , j = 1, . . . ,7, to the expressionabove, we obtain similar equalities for φj , j =1, . . . ,7. Now, let ψ(x) = φ1(qx) −φ6(qx − ( 0

1 )).This function is the difference of appropriatelynormalized characteristic functions of disjointtriangles and leads indeed to the desired Haar-likewavelet. In fact, we can define the vector-valuedfunction

Ψ =Db0ψ

...Db7ψ

=ψ0

...ψ7

and observe that the system Djq Tk Ψ : j ∈ Z, k ∈Z2 is an ON basis of L2(R2).

The construction above is representative of amuch more general situation. For example, let

u = 12

(1 −

√3√

3 1

), the matrix of counterclockwise

rotation by π/3, normalized to produce detu =2. Also in this case, there is a fractal Haarwavelet associated with this dilation matrix, butthe introduction of an appropriate additional finitegroup of dilations allows one to derive simplerHaar-like wavelets similar to what we did above.See [18] for details.

In [18] we have shown that all this can beformalized by introducing the notion of waveletsystems with composite dilations, which are thesystems of the form

(9) DaDb Tkψ : j ∈ Z, a ∈ A,b ∈ B, k ∈ Z2,where B is a group of matrices with determinantof absolute value 1 (as in the examples above)and A is a group of expanding matrices, inthe sense that all eigenvalues have magnitudelarger than one (as in the example above, whereA is the group of the integer powers of thequincunx matrix). Many different groups of B-dilations have been considered in the literature,such as the crystallographic and shear groups,which are associated with very different properties.As we will see below, one special benefit ofthis framework is the ability to produce wavelet-like systems with geometric properties going farbeyond traditional wavelets. For example, onecan construct waveforms whose supports arehighly anisotropic and that range not only overvarious scales and locations but also over variousorientations, making these functions particularlyuseful in image processing applications.

As discussed at the beginning of the previoussection, one of the most important propertiesof wavelets in L2(R) is their ability to provide


R2 R1

R0R3

R4 R7

x1

R5 R6

q−1R3

q−1R6 q−1R7

q−1R2

q−1R4 q−1R1 x1

q−1R5 q−1R0

x2x2

Figure 2. Example of construction of a wavelet system with composite dilations. On the left isillustrated the triangle R0R0R0 and the triangles RiRiRi , i = 1, . . . ,7i = 1, . . . ,7i = 1, . . . ,7, obtained as biR0biR0biR0 (the matrices bibibi aredescribed in the text). On the right is illustrated the action of the inverse of the quincunx matrix qqq onthe triangles RiRiRi .

rapidly convergent approximations for piecewisesmooth functions. This property implies thatwavelet expansion is useful to compress functionsefficiently since, as described by the nonlinearapproximation error estimate (5), most of theL2-norm of the function can be recovered from arelatively small number of expansion coefficientsand not much information is lost by discardingthe remaining ones. This idea is the basis forthe construction of several wavelet-based datacompression algorithms, such as JPEG2000 [27].

There is another important, perhaps less ob-vious, implication of the wavelet approximationproperties which has to do with the classicalproblem of data denoising. Suppose that we wantto recover a function f which is corrupted by zero-mean white Gaussian noise with variance σ 2. Letfn denote the noisy function. In this case, Donohoand Johnstone [12] have shown that there is a verysimple and very effective procedure for estimatingf . This consists, essentially, of (i) computing thewavelet expansion of fn, (ii) setting to zero thewavelet coefficients of fn whose magnitudes arebelow a fixed value (which depends on σ ), and (iii)computing an estimator of f as a reconstructionfrom the wavelet coefficients of fn which havenot been set to zero. It turns out that the perfor-mance of this procedure depends directly on thedecay rate of the nonlinear approximation errorestimate (5).

The above observations underline the fundamen-tal importance of the approximation properties ofone-dimensional wavelets for applications. Unfor-tunately, as will be discussed in more detail below,the situation is different in higher dimensions,where the standard multidimensional generaliza-tion of dyadic wavelets does not lead to the

same type of approximation properties as in theone-dimensional case.

Let us restrict our attention to dimension n = 2(the cases n > 2 are similar). Recall that, in dimen-sion n = 1, to achieve the desired approximationproperties of the wavelet expansions we requiredwavelets having compact support and sufficientlymany vanishing moments. In dimension n = 2 wecan easily construct a two-dimensional dyadic ONwavelet system starting from a one-dimensionalMRA with scaling function φ1 and wavelet ψ1, asfollows. We define three wavelets: ψ(1)(x1, x2) =φ1(x1)ψ1(x2), ψ(2)(x1, x2) = ψ1(x1)φ1(x2),ψ(3)(x1, x2) = ψ1(x1)ψ1(x2). Then the system

ψ(`)j,k1,k2(x1, x2)

= 2jψ(`)(2j(x1 − k1),2j(x2 − k2)) :

j, k1, k2 ∈ Z, ` = 1,2,3is an ON basis for L2(R2). This is called a separableMRA wavelet basis. Clearly, we can choose ψ1 andφ1 having compact support and R vanishing mo-ments. In this case, similar to the one-dimensionalresult, if f ∈ CR(R2) and ‖f‖CR <∞, then the mthlargest wavelet coefficient in magnitude satisfiesthe estimate

(10) |〈f ,ψπ(m)〉| ≤ C ‖f‖CR m−(R+1)/2,

where C is a constant independent of f andm. Thisimplies that, letting fN be the bestN-term nonlinearapproximation of f , the nonlinear approximationerror decays as

‖f − fN‖2L2 ≤ C

∑m>N

|〈f ,ψπ(m)〉|2 ≤ C ‖f‖2CR N

−R.

However, while in the one-dimensional case theapproximation properties of the wavelet expansionare not affected if one allows the function f to


have a finite number of isolated singularities, thesituation now is very different. Let us examine,for example, the wavelet approximation of thefunction f = χD, where D ⊂ R2 is a compactset whose boundary has finite length L. Let usconsider in particular the wavelet coefficientsassociated with the boundary of the region D.Since f is bounded, these wavelet coefficients have

size |〈f ,ψ(`)j,k1,k2〉| ∼ 2−j . In addition, since the

boundary of D has finite length and the waveletshave compact support, at each scale j there areabout L2j wavelets with support overlapping theboundary of D. It follows from these observationsthat theNth largest wavelet coefficient in this classis bounded by C(L)N−1, and this implies that thenonlinear approximation error rate is only of theorder O(N−1). Hence, there are a large number ofsignificant wavelet coefficients associated with theedge discontinuity of f , and this is limiting thewavelet approximation rate.

The reason for this limitation of two-dimensionalseparable MRA wavelet bases is the fact that theirsupports are isotropic (they are supported on abox of size ∼ 2−j × 2−j ) so that there are “many”wavelets overlapping the edge singularity andproducing significant expansion coefficients. To ad-dress this issue and produce better approximationsof piecewise smooth multidimensional functions,one has to consider alternative multiscale systemswhich are more flexible at representing anisotropicfeatures. Several constructions have been intro-duced, starting with the wedgelets [3] and ridgelets[5]. Among the most successful constructionsproposed in the literature, the curvelets [6] andshearlets [15] achieve this additional flexibilityby defining a collection of analyzing functionsranging not only over various scales and locations,like traditional wavelets, but also over variousorientations and with highly anisotropic supports.As a result, these systems are able to produce(nonlinear) approximations of two-dimensionalpiecewise C2 functions for which the nonlinearapproximation error decays essentially like N−2,that is, as if the functions had no discontinuities.To give a better insight into this approach andshow how the wavelet machinery can be modifiedto obtain these types of systems, we will brieflydescribe the shearlet construction in dimensionn = 2, which is closely related to the frameworkof wavelets with composite dilations, which wementioned above. To keep the presentation self-contained, we will sketch only the main ideas andrefer the reader to [15] [17] for more details.

For an appropriate function ψ ∈ L2(R2), asystem of shearlets is a collection of functions ofthe form (9), where

(11) a =(

4 00 2

), b =

(1 10 1

).

Note that a is a dilation matrix whose integerpowers produce anisotropic dilations and, moreprecisely, parabolic scaling dilations, since the dila-tion factors grow quadratically in one coordinatewith respect to the other one; the shear matrix b isnonexpanding and, as we will see below, its integerpowers control the orientations of the elementsof the shearlet system. The generator ψ of thesystem is defined in the frequency domain as

ψ(ξ) = ψ(ξ1, ξ2) = w(ξ1) v(ξ2

ξ1

),

where w , v ∈ C∞(R), suppw ⊂ [− 12 ,

12] \ [−

116 ,

116]

and suppv ⊂ [−1,1]. Furthermore, it is possible tochoose the functionsw ,v so that the correspondingsystem (9) is a PF (Parseval frame) for L2(R2). Thegeometrical properties of the shearlet system aremore evident in the Fourier domain. In fact, adirect calculation gives that

ψj,`,k(ξ) := (DjaD`b Tkψ)∧(ξ)(12)

= 2−3j/2w(2−2jξ1)v(2jξ2

ξ1−`)e2πiξa−jb−`k,

implying that the functions ψj,`,k are supportedin the trapezoidal regions

(ξ1, ξ2) ∈ R2 :

ξ1 ∈ [−22j−1,22j−1] \ [−22j−4,22j−4],

|ξ2

ξ1− `2−j | ≤ 2−j.

The last expression shows that the frequencysupports of the elements of the shearlet systemare increasingly more elongated at fine scales (asj →∞) and orientable, with orientation controlledby the index ` (this is illustrated in Figure 3).These properties show that shearlets are muchmore flexible than “isotropic” wavelets and explainwhy shearlets can achieve better approximationproperties for functions which are piecewisesmooth. Similar properties hold for the curveletsand can be extended to higher dimensions.

As indicated above, shearlets and curvelets areonly some of the methods introduced during thelast decade to extend or generalize the waveletapproach. We also recall the construction ofbandelets [25]; these systems achieve improvedapproximations for functions which are piecewiseCα (α may be larger than 2) by using an adaptiveconstruction. We refer to the volume [24] for thedescription of several such systems.

Continuous WaveletsIn this section we examine continuous waveletson Rn. The general linear group GL(n,R) ofn × n invertible real matrices acts on R by lineartransformations. The semidirect product G of


Figure 3. The frequency supports of theelements of the shearlet system are pairs oftrapezoidal regions defined at various scalesand orientations, dependent on jjj and `respectively. The figure shows the frequencysupports of two representative shearletelements: the darker region corresponds toj = 0j = 0j = 0, ` = 0` = 0` = 0 and the lighter region to j = 1j = 1j = 1, ` = 1` = 1` = 1.

GL(n,R) and Rn is called the general affine groupon Rn, since each invertible affine map on Rn hasthe form (a, t) ·x = a(x+ t) = ax+at for a unique(a, t) ∈ G. Thus, the group law (a, t) · (b, s) =(ab, b−1t+s) forG corresponds to the compositionof the associated affine maps and (a, t)−1 =(a−1,−a−1t). We have a unitary representation τof G acting on L2(Rn) defined by

(τ(a,t)ψ)(x) = |deta|−12ψ((a, t)−1 · x)

= |deta|−12ψ(a−1x− t) := ψa,t(x).

We then have that

(τ(a,t)ψ)∧(ξ) = |deta|1/2ψ(a∗ξ) e−2πiξ·at .

For ψ ∈ L2(Rn), the continuous wavelet transformWψ associated with ψ and G is defined by

(Wψf )(a, t) = 〈f ,ψ(a,t)〉

(13)

= |deta|−12

∫Rnf (x)ψ(a−1x− t) dx,

and it maps f ∈ L2(Rn) into a space of functionson G. For D a closed subgroup of GL(n,R),H = (a, t) : a ∈ D, t ∈ Rn is a closed subgroupof G and the left Haar measures on H are theproduct measures dλ(a, t) = dµ(a)dt , where µis a left Haar measure on D. In the special casewhere D is a discrete subgroup of GL(n,R) suchas uj : j ∈ Z, for some u ∈ GL(n,R), we cantake µ to be the counting measure on D.

We then seek conditions on D and µ for whichrestricting Wψf to H gives an isometry from

L2(Rn) into L2(H, dλ) or, equivalently, we have acontinuous reproducing formula

f =∫H〈f , τ(a,t)ψ〉τ(a,t)ψdλ(a, t),

for each f ∈ L2(Rn). As shown in [4], this holds ifand only if

(14)∫D|ψ(a∗ξ)|2 dµ(a) = 1, for a.e. ξ ∈ Rn,

in which caseψ is a continuous wavelet (with respectto D). In the special case D = aIn : a > 0, whereIn is the n×n identity matrix and dµ(aIn) = da

a , theexpression (14) reduces to the classical Calderóncondition

(15)∫∞

0|ψ(aξ)|2 da

a= 1, for a.e. ξ ∈ Rn,

which is easy to satisfy. Indeed, when n = 1, theset

g ∈ L2(R) : c g satisfies (15) for some c > 0is dense in L2(R) (a similar observation holds forn > 1). When D = uj : j ∈ Z, for u > 1, one canshow that if ψ(uj ,k) : j ∈ Z, k ∈ Z is a Parsevalframe, then ψ is also a continuous wavelet withrespect to D. Conversely, (15) is only necessarybut not sufficient for ψ(uj ,k) : j ∈ Z, k ∈ Z to bea Parseval frame.

Forψ ∈ L2(Rn), the modified continuous wavelettransform Wψ associated with ψ and G is definedby

(Wψf )(a, t) = (Wψf )(a, a−1t)(16)

= |deta|−12

∫Rnf (x)ψ(a−1(x− t)) dx,

and it also maps f ∈ L2(Rn) into functions onG = D × Rn. The modification arises from using

the analyzing function |deta|−12ψ(a−1(x − t))

rather than |deta|−12ψ(a−1x− t), and it simplifies

the problem of estimating the asymptotic decayproperties of the continuous wavelet transform.

Indeed, forG = D×Rn, whereD = aIn : a > 0,let us consider the modified continuous wavelettransform

(Wψf )(a, t) := (Wψf )(aIn, t)

(17)

= a−n/2∫Rnf (x)ψ(a−1(x− t)) dy.

A fundamental property of this transform isits ability to characterize the local regularity offunctions. For example, let f be a bounded functionon R which is Hölder continuous at x0, withexponent α ∈ (0,1]; that is, there is C > 0 forwhich

|f (x0 + h)− f (x0)| ≤ C|h|α.


Suppose that∫R(1 + |x|)|ψ(x)|dx < ∞ and that

ψ(0) = 0. Since the last condition implies that∫Rψ(x)dx = 0, then

(Wψf )(a, t)

= a−1/2∫R(f (x)− f (x0))ψ(a−1(x− t))dx.

Thus, using the Hölder continuity and a change ofvariables, we have

|(Wψf )(a, t)|

(18)

≤ a−1/2∫R|f (x)− f (x0)| |ψ(a−1(x− t))|dx

≤ C aα+1/2∫R|y + a−1(t − x0)|α |ψ(y)|dy.

(19)

This shows that, at t = x0, the continuous wavelettransform of f decays (at least) like aα+1/2, as a → 0.Under slightly stronger conditions on ψ, one canshow that the converse also holds, hence providinga characterization result. It is also possible toextend this analysis to discontinuous functionsand even distributions. For example, if f has ajump discontinuity at x0, then one can show thatthe continuous wavelet transform of f decays likea1/2, as a → 0, and similar properties hold inhigher dimensions (cf. [22]).

While the continuous wavelet transform (17) isable to describe the local regularity of functions anddistribution and detect the location of singularitypoints through its decay at fine scales, it does notprovide additional information about the geometryof the set of singularities. In order to achieve thisadditional capability, one has to consider wavelettransforms associated with more general dilationgroups.

For example, in dimension n = 2, let M be thesubgroup of GL(2,R) of the matricesma,s =

a −a1/2 s

0 a1/2

: a > 0, s ∈ R

,and let us consider the corresponding generalizedcontinuous wavelet transform

(Wψf )(a, s, t) := (Wψf )(ma,s , t)

(20)

= a−3/4∫R2f (x)ψ(m−1

a,s(x− t)) dx,

where a > 0, s ∈ R, and t ∈ R2. It is easy to verify

that we have the factorizationma,s =( 1 −s

0 1

) ( a 0

0 a1/2

);

that is, ma,s is the product of an anisotropicdilation matrix and a shear matrix. As a result, theanalyzing function ψa,s,t = a−3/4ψ(m−1

a,s(x − t))associated with this transform ranges over variousscales, orientations, and locations, controlled by

the variables a, s, t respectively. This is similar tothe discrete shearlets in the section “Wavelets inHigher Dimensions”. The transform (Wψf )(a, s, t)is called the continuous shearlet transform of f .

Thanks to the properties associated with dilationgroup M , the continuous shearlet transform isable to detect not only the location of singularitypoints through its decay at fine scales but also thegeometric information of the singularity set. Inparticular, there is a general characterization ofstep discontinuities along 2D piecewise smoothcurves, which can be summarized as follows [16].Let B = χS , where S ⊂ R2 and its boundary ∂S is apiecewise smooth curve.

• If t ∉ ∂S, then WψB(a, s, t)has rapid asymptoticdecay, as a → 0, for each s ∈ R. That is,

lima→0a−NWψB(a, s, t) = 0, for all N > 0.

• If t ∈ ∂S and ∂S is smooth near t , thenWψB(a, s, t) has rapid asymptotic decay, asa → 0, for each s ∈ R unless s = tanθ0 and(cosθ0, sinθ0) is the normal orientation to ∂Sat t . In this last case, WψB(a, s0, t) ∼ a

34 , as

a → 0. That is,

lima→0a−

34 WψB(a, s, t) = C 6= 0.

• If t is a corner point of ∂S and s = tanθ0 where(cosθ0, sinθ0) is one of the normal orientationsto ∂S at t , then WψB(a, s0, t) ∼ a

34 , as a → 0.

For all other orientations, the asymptotic decayof WψB(a, s, t) is faster and depends in acomplicated way on the curvature of theboundary ∂S near t [16].

Similar results hold in higher dimensions and forother types of singularity sets.

Note that, in the definition of WψB(a, s, t), we aretaking the inner product of f with the continuousshearlets TtD−1

mas . On the other hand, the discreteshearlets in “Wavelets in Higher Dimensions”involve the reverse order of operators. Despitethis fact, the decay properties of the continuousshearlet transform are related to the approximationproperties of discrete shearlets.

Various Other Wavelet Topics, Applications,and ConclusionsThe number of researchers who have worked andare working on wavelets is very large. This fieldand its applications are enormous. We could notcover all the topics that are most important andinteresting in such a short article. We make noclaim that we have chosen to cover all “the mostimportant” topics on wavelets. In this article wehave defined wavelets to be elements of the Hilbertspace L2(Rn). In fact, we can extend the wavelettechniques we described to the Banach spaces


Lp(Rn), p 6= 2. As usual, the flavor for 1 < p < 2 isvery different from p ≥ 2. The roles played by Rand the 1-torus T were shown to extend to higherdimensions. It is well known that the harmonicanalysis involving (Z,T) extends to the setting(G, G) where G is a locally compact Abelian groupand G its dual. Wavelet theory extends to (G, G)and other abstract settings.

Wavelets continue to stimulate and inspire activeresearch going beyond the area of harmonic analy-sis, where they were originally introduced. Whileduring the 1980s and 1990s most of wavelet theorywas devoted to the construction of “nice” waveletbases and their applications to denoising and com-pression, during the last decade wavelet researchwas focused more on the subject of approximationsand the so-called sparse approximations. As wementioned above, several generalizations and ex-tensions of wavelets were introduced with the goalof providing improved approximation propertiesfor special classes of functions where the moretraditional wavelet approach is not as effective.This research has stimulated the investigation ofredundant function systems (that is, frames whichare not necessarily tight) and their applicationsusing techniques coming not only from harmonicanalysis but also from approximation theory andprobability. The emerging area of compressedsensing, for example, can be seen as a methodfor achieving the same nonlinear approximationproperties of wavelets and their generalization byusing linear measurements defined according to acertain clever strategy.

Some fundamental ideas from wavelet theory,most notably the multiresolution analysis, haveappeared in other forms in very different contexts.For example, the theory of diffusion wavelets [11]provides a method for the multiscale analysis ofmanifolds, graphs, and point clouds in Euclideanspace. Rather than using dilations as in the classicalwavelet theory, this approach uses “diffusionoperators” acting on functions on the space. Forexample, let T be a diffusion operator (e.g., theheat operator) acting on a graph (the graph canbe a discretization of a manifold). The study ofthe eigenfunctions and eigenvalues of T is knownas spectral graph theory and can be viewed as ageneralization of the theory of Fourier series onthe torus. The main idea of diffusion wavelets isto compute dyadic powers of the operator T toestablish a scale for performing multiresolutionanalysis on the graph. This approach has manyuseful applications, since it allows one to apply theadvantages of multiresolution analysis to objectsthat can be modeled as graphs, such as chemicalstructures, social networks, etc.

In order to describe the more recent applicationsinspired by wavelets, we quote Coifman from [9,p. 159]:

Over the last twenty years we have seenthe introduction of adaptive computationalanalytic tools that enable flexible transcrip-tions of the physical world. These toolsenable orchestration of signals into con-stituents (mathematical musical scores) andopened doors to a variety of digital imple-mentations/applications in engineering andscience. Of course I am referring to waveletand various versions of computationalHarmonic Analysis. The main concepts un-derlying these ideas involved the adaptationof Fourier analysis to changing geometriesas well as multiscale structures of naturaldata. As such, these methodologies seem tobe limited to analyze and process physicaldata alone. Remarkably, the last few yearshave seen an explosion of activity in machinelearning, data analysis and search, implyingthat similar ideas and concepts, inspiredby signal processing might carry as muchpower in the context of the orchestrationof massive high dimensional data sets. Thisdigital data, e.g., text documents, medicalrecords, music, sensor data, financial data,etc., can be structured into geometries thatresult in new organizations of language andknowledge building. In these structures, theconventional hierarchical ontology buildingparadigm merges with a blend of HarmonicAnalysis and combinatorial geometry. Con-ceptually these tools enable the integrationof local association models into global struc-tures in much the same way that calculusenables the recovery of a global functionfrom a local linear model for its variation.As it turns out, such extensions of differ-ential calculus into the digital data fieldsare now possible and open the door to theusage of mathematics similar in scope tothe Newtonian revolution in the physicalsciences. Specifically we see these toolsas engendering the field of mathematicallearning in which raw data viewed as cloudsof points in high dimensional parameterspace is organized geometrically much thesame way as in our memory, simultaneouslyorganizing and linking associated events,as well as building a descriptive language.

Let us illustrate three examples that will givethe reader a more concrete idea of what Coifmanasserts (see also [10]).

(a) In the oil exploration and mining industry,one needs to decide where to drill or mine to


greatest advantage for finding oil, gas, copper,or other minerals. This involves an analysis ofthe composition and structure of the soil in acertain region. From such analysis one would findproperties that optimize where these resourcesare most likely to be found.

(b) Suppose that we would like to decompose alarge collection of books into subclasses of “simi-lar” books, e.g., novels, histories, physics books,mathematical books, etc. Possibly we also wantto assign “distances” between these subclasses.It is not unreasonable that the distributions ofmany particular words contained in each book canidentify various kind of books so that they can beassigned in a specific subclass.

(c) In medical diagnostics, important informationcan be gleaned from the analysis of data obtainedfrom radiological, histological, chemical tests,and this is important for arriving at an earlydetection of potentially dangerous tumors andother pathologies.

What is surprising is that the analysis of thesevery different types of data can be performedvery efficiently using the type of mathematicsbased on the ideas presented in this paper. Thepractical impact of these ideas in applicationsis remarkable. For example, several hospitalshave adopted medical diagnostics methods thatwere developed by Coifman and his group usingwavelet-based methods.

In conclusion, we want to stress that waveletsis a huge field. Many have helped to create it.We want to state, however, that Yves Meyer hascontributed and introduced many ideas that weremost important in its creation. The material in thefirst chapter of [21] describes many constructionswhich are due to him and the ideas that paved theway for many of the topics (e.g., the MRA) we havepresented in this paper.

References[1] I. Daubechies, Orthonormal bases of compactly sup-

ported wavelets, Comm. Pure Appl. Math. 41(7) (1988),909–996.

[2] , Ten Lectures on Wavelets, SIAM, Philadelphia,1992.

[3] D. L. Donoho, Wedgelets: Nearly-minimax estimationof edges, Ann. Statist. 27 (1999), 859–897.

[4] A. P. Calderón, Intermediate spaces and interpola-tion, the complex method, Studia Math. 24 (1964),113–190.

[5] E. J. Candès and D. L. Donoho, Ridgelets: The key tohigh dimensional intermittency?, Philos. Trans. Roy.Soc. Lond. Ser. A 357 (1999), 2495–2509.

[6] , New tight frames of curvelets and optimal rep-resentations of objects with C2 singularities, Comm.Pure Appl. Math. 57 (2004), 219–266.

[7] P. G. Casazza, O. Christensen, and N. J. Kalton,Frames of translates, Collect. Math. 52 (2001), 35–54.

[8] A. Cohen, I. Daubechies, and J. C. Feauveau,Biorthogonal bases of compactly supported wavelets,Comm. Pure Appl. Math. 45(5) (1992), 485–560.

[9] J. Cohen and A. I. Zayed (eds.), Wavelets and Multi-scale Analysis: Theory and Applications, Birkhäuser,2011.

[10] R. R. Coifman and M. Gavish, Harmonic analysis ofdigital data bases, in [9], pp. 161–197.

[11] R. R. Coifman and M. Maggioni, Diffusion wavelets,Appl. Comp. Harm. Anal. 21(1) (2006), 53–94.

[12] D. L. Donoho and I. M. Johnstone, Ideal spatial adap-tation by wavelet shrinkage, Biometrika 81(3) (1994),425–455.

[13] G. Garrigos, E. Hernández, H. Sikic, F. Soria,G. Weiss, and E. N. Wilson, Connectivity in the set oftight frame wavelets, Glas. Mat. 38(1) (2003), 75–98.

[14] K. Gröchenig and W. R. Madych, Multiresolutionanalysis, Haar bases, and self-similar tilings of Rn,IEEE Trans. Info. Theory 38(2) (1992), 556–568.

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[19] A. Haar, Zur Theorie der orthogonalen Funktionen-systeme, Math. Ann. 69(3) (1910), 331–371.

[20] E. Hernández, H. Sikic, G. Weiss, and E. Wilson, Onthe properties of the integer translates of a squareintegrable function, Contemp. Math., vol. 505, Amer.Math. Soc., Providence, RI, 2010, pp. 233–249.

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