MultiwaveletsFritz Keinert

Iowa State University, Ames, Iowa, USA

Article Outline

Glossary 1I. Definition of the Subject and Its Importance 2II. Introduction 3III. Refinable Function Vectors 4IV. Multiresolution Approximation and Discrete Multiwavelet Transform 6V. Moments and Approximation Order 8VI. The Discrete Multiwavelet Transform (DMWT) Algorithm 9VII. Pre- and Postprocessing and Balanced Multiwavelets 11VIII. Boundary Handling 13IX. Applications 14X. Polyphase Factorization 15XI. Lifting 15XII. Two-Scale Similarity Transform (TST) 16XIII. Examples and Software 18XIV. Future Directions 20XV. Bibliography 20Books and Reviews 20Primary Literature 20

Glossary

Balanced MultiwaveletA multiwavelet for which the expansion coefficients for polynomials up to a certain degree are

polynomial sequences. Balanced multiwavelets do not require pre- or postprocessing.Discrete Multiwavelet Transform (DMWT)The algorithm which decomposes a signal into a coarse approximation and fine detail at several

levels. A direct generalization of the Discrete Wavelet Transform (DWT) for scalar wavelets.LiftingA method for modifying an existing multiwavelet, or building one from scratch. Lifting can be

used to impose approximation order, balancing, or symmetry.Modulation MatrixThe modulation matrix

M(ξ) =(H(ξ) H(ξ + π)G(ξ) G(ξ + π)

),

can be used to describe the action of the DMWT. It is also used in the construction of multiwaveletsby lifting or TST.

Multiresolution, Multiresolution Approximation (MRA)Multiresolution is the fundamental concept underlying everything related to any kind of wavelet.

A signal is decomposed into a low resolution approximation, plus fine detail at one or more levels ofresolution. An MRA is a nested chain of subspaces of L2 which describes this concept mathemati-cally.

Multiscaling Function, Multiwavelet Function, MultiwaveletA multiwavelet of multiplicity r consists of a multiscaling function and a multiwavelet function.

Both are functions from R to Cr, that is, function vectors. Multiwavelets are not the same asmultivariate wavelets, which are functions from Rn to C.

Polyphase Decomposition, Polyphase Matrix1

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The polyphase decomposition separates a coefficient sequence into two sequences, by even andodd subscripts. It can be used to describe or implement the DMWT.

The polyphase matrix

P (z) =(H(0)(z) H(1)(z)G(0)(z) G(1)(z)

),

is useful in the construction of multiwavelets, especially orthogonal multiwavelets.Pre- and PostprocessingBefore a DMWT can be applied to a signal, the signal needs to be written as a multiscaling

function series. The DMWT works on the coefficients of the series expansion, not on the values ofthe signal.

For scalar wavelets the distinction can usually be ignored, but not for multiwavelets. Having amultiwavelet of approximation order p does not mean that the coefficients of a polynomial up toorder p− 1 form a polynomial sequence.

Preprocessing converts point samples into expansion coefficients. Postprocessing does the oppo-site.

Refinable Function Vector, Refinement EquationA refinable function vector of multiplicity r is a function from R to Cr which satisfies a refinement

equation of the form

φ(x) =√

2k1∑

k=k0

Hk φ(2x− k)

with recursion coefficients Hk which are r × r matrices.SymbolGiven any sequence a = {ak}, the symbol of a is defined as

a(ξ) =∑

k

ake−iξ or a(z) =

∑k

akzk, z = e−iξ,

possibly with a normalizing factor. The sequence may represent point samples of a signal, therecursion coefficients of a two-scale refinement equation, or other quantities.

Both the trigonometric and polynomial notations are useful, depending on the setting; it is trivialto switch back and forth between them.

Two-Scale Similarity Transform (TST)The TST is a way of moving approximation orders in a biorthogonal multiwavelet pair from one

side to the other. For scalar wavelets it corresponds to moving a factor of (1 + e−iξ)/2 back andforth.

The TST can be used to modify an existing multiwavelet, or to build multiwavelets from scratch.It can also be used to impose or characterize symmetry, approximation order and balancing.

I. Definition of the Subject and Its Importance

Classical (scalar) wavelets have been around since the late 1980s and have become an indispen-sible tool in signal processing, with further applications in numerical analysis, operator theory, andother fields. Wavelets have been generalized in many ways: wavelet packets, multivariate wavelets,ridgelets, curvelets, vaguelettes, slantlets, second generation wavelets, frames, and other construc-tions.

One such generalization are multiwavelets, which have been around since the early 1990s. Mul-tiwavelets use several scaling and wavelet functions. Their construction is more complicated thanthat of scalar wavelets, but the underlying multiresolution concepts and the decomposition andreconstruction algorithms are very similar.

Scalar wavelets are functions from R to C. Multiwavelets are functions from R to Cr. Theyinclude scalar wavelets as the special case r = 1. Multiwavelets are used to analyze one-dimensionalsignals, or higher-dimensional signals by using tensor products, just like scalar wavelets. Theyshould not be confused with multivariate wavelets, which are functions from Rn to C, used toanalyze higher-dimensional signals.

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Multiwavelets have several advantages over scalar wavelets: they can have short support coupledwith high smoothness and high approximation order, and they can be both symmetric and orthog-onal. They also have some disadvantages: the algorithms require preprocessing and postprocessingsteps.

The applications are the same as for scalar wavelets: signal compression, signal denoising, fastoperator evaluation in numerical analysis, Galerkin methods for differential and integral equations.Performance of multiwavelets is similar to that of scalar wavelets, but implementation requires a bitmore effort, expecially because of the need for pre- and postprocessing. Multiwavelets are best usedin situations where their advantages (symmetry or short support) outweigh the extra effort.

II. Introduction

The fundamental concept underlying everything related to any kind of wavelet is multiresolution.A function (or signal or image) is decomposed into a low resolution approximation plus fine detailat one or more levels of resolution (see fig. 3). Scalar wavelets use a scaling function for the coarseapproximation, a wavelet function for the fine detail. Multiwavelets use several scaling functionsand wavelet functions, combined into function vectors.

The first occurrence of multiwavelets is in the work of Alpert [9], which uses piecewise poly-nomial multiwavelets of high multiplicity. The Donovan–Geronimo–Hardin–Massopust (DGHM)multiwavelet [25] is commonly considered to be the first nontrivial example.

The DGHM multiscaling function is a vector

φ(x) =(φ1(x)φ2(x)

)which satisfies a recursion relation

φ(x) =120

[(12 16

√2

−√

2 −6

)φ(2x) +

(12 0

9√

2 20

)φ(2x− 1)

+(

0 09√

2 −6

)φ(2x− 2) +

(0 0

−√

2 0

)φ(2x− 3)

].

The first scaling function φ1 is supported on [0, 1] and is symmetric about x = 1/2. The secondscaling function φ2 is supported on [0, 2], symmetric about x = 1. These functions and their integertranslates are orthonormal, that is,∫

φi(x)φj(x− k) dx = δijδ0k.

They have approximation order 2, which means that f(x) = 1 and f(x) = x can be written aslinear combinations of integer shifts of φ1, φ2. The scaling functions as well as the correspond-ing wavelet functions are shown in fig. 1. The wavelet functions have support on [0, 2] and aresymmetric/antisymmetric about x = 1. They are also orthogonal to each other and to the φj .

0 1 2−1

0

1

2

DGHM scaling function φ1

0 1 2−1

0

1

2

DGHM scaling function φ2

0 1 2−2

−1

0

1

2

DGHM wavelet function ψ1

0 1 2−2

−1

0

1

2

DGHM wavelet function ψ2

Figure 1. DGHM multiwavelet

Many other examples have been constructed since then. Much of the theory is due to Strela andPlonka, in their individual and joint papers in the late 1990s.

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The most comprehensive treatment of multiwavelets in the literature is the book [3]. Usefulsurvey articles include [4] and [6].

The theory of multiwavelets parallels the theory of scalar wavelets. It is highly recommendedthat anyone studying multiwavelets become familiar with scalar wavelets first. Good introductionsto scalar wavelets can be found in [2], [3], or [5].

This article assumes that the reader is familiar with scalar wavelets. It is brief on the aspects ofmultiwavelets which are essentially the same as for scalar wavelets, and gives more details on theareas where they differ. Similarities and differences are pointed out in many places.

The main body of this article is divided into three logical parts. The first part (sections IIIthrough V) describes the basic theory and properties of the multiscaling and multiwavelet functions.The second part (sections VI through IX) describes the practical implementation of the DiscreteMultiwavelet Transform (DMWT). The third part (sections X through XI) describes techniques forbuilding multiwavelets, or modifying existing multiwavelets.

For conciseness, we make some simplifying assumptions:• The dilation factor is 2, not a general m ≥ 2.• All functions satisfy recursion relations with finitely many coefficients and have compact

support.• All multiscaling functions satisfy the basic regularity conditions described in section III,

and lie in L1 and L2.

III. Refinable Function Vectors

A refinable function vector is a vector-valued function

φ(x) =

φ1(x)...

φr(x)

, φj : R → C,

which satisfies a two-scale matrix refinement equation of the form

φ(x) =√

2k1∑

k=k0

Hk φ(2x− k). (1)

r is called the multiplicity of φ. The recursion coefficients Hk are r × r matrices. Scalar waveletsare a special case, for r = 1.

A pair φ, φ of refinable function vectors is called biorthogonal if

〈φ(x), φ(x− k)〉 =∫φ(x)φ(x− k)∗ dx = δ0kI.

Here φ∗

is the complex conjugate transpose of φ, so this inner product produces an r× r matrix. Ifφ is biorthogonal to itself, it is called orthogonal.

Note that the refinement equation (1) is similar to an eigenvalue problem: if φ is a solution, sois any multiple of φ.

There has to be a factor of 2 or√

2 in all formulas based on eq. (1), just as there has to be afactor of 2π somewhere in every definition of the Fourier transform, but different authors put it indifferent places. Formulas from other sources may differ slightly from those in this article.

The support of φ is contained in the interval [k0, k1]. It may be strictly shorter if the first or lastrecursion coefficient Hk0 , Hk1 is nilpotent. This happens in the case of the DGHM multiwavelet.

The symbol of a refinable function vector is the trigonometric matrix polynomial

H(ξ) =1√2

k1∑k=k0

Hke−ikξ. (2)

Equivalently we could use the matrix polynomial

H(z) =1√2

k1∑k=k0

Hkzk, z = e−iξ

5

The refinement equation (1) can only have an L2-solution which leads to stable algorithms if itsatisfies the Basic Regularity Conditions:

• H(0) satisfies Condition E. That is, H(0) has a simple eigenvalue of 1, and all other eigen-values are less than 1 in magnitude.

• The coefficients Hk satisfy the sum rules of order 1. That is, y∗0H(π) = 0, where y∗0 is theleft eigenvector of H(0) to eigenvalue 1. (See eq. (6) for general sum rules).

The three main ways to prove existence and uniqueness of φ and obtain smoothness estimatescarry over from scalar wavelets: infinite product, Cascade Algorithm, and eigenvalue problem. Thesecond and third methods are also practical ways of obtaining point values and graphs of φ.

We define the Fourier transform as

f(ξ) =1√2π

∫ ∞

−∞f(x)e−ixξ dx.

The Fourier transform of refinement equation (1) is

φ(ξ) = H(ξ

2)φ(

ξ

2).

This leads to the formal infinite product

φ(ξ) =

[ ∞∏k=1

H(2−kξ)

]φ(0).

The infinite product can only rarely be evaluated in closed form, but its convergence can be studied.Since everything is done on the Fourier transform side, this approach can be used to investigatedistribution solutions.

The Cascade Algorithm is fixed point iteration applied to the recursion relation. Select a suitablestarting function φ(0)(x), and iterate:

φ(n+1)(x) =√

2k1∑

k=k0

Hkφ(n)(2x− k).

The transition operator or transfer operator for the symbol H(ξ) is defined by

TF (ξ) = H(ξ)F (ξ)H(ξ)∗ +H(ξ + π)F (ξ + π)H(ξ + π)∗. (3)

If the transition operator satisfies condition E, the cascade algorithm converges for any startingfunction φ(0) which satisfies

y∗0

k1∑k=k0

φ(0)(k) = c 6= 0.

This condition essentially states that φ(0) must have a component in the direction of φ. Compareeq. (5).

A third approach is the eigenvalue method. Write out the refinement equation at all integer pointsin the support:

φ(j) =√

2k1∑

k=k0

Hk φ(2j − k) =√

2k1∑

k=k0

H2j−k φ(k), j = k0, . . . , k1.

This is an eigenvalue problemΦ = TΦ,

where

Φ =

φ(k0)

φ(k0 + 1)...

φ(k1)

, Tjk =√

2H2j−k, k0 ≤ j, k ≤ k1.

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The basic regularity conditions guarantee that (y∗0,y∗0, . . . ,y

∗0) is a left eigenvector to eigenvalue 1,

so a right eigenvector also exists. The right eigenvector is often unique, but not always, so thismethod can fail.

Unless Hk0 or Hk1 have an eigenvalue of 1/√

2, the values of φ at k0 and k1 are zero, and we canreduce the size of Φ and T .

Once the values of φ at the integers have been determined, we can use the refinement equationto obtain values at points of the form k/2, k ∈ Z, then k/22, and so on to any desired resolution.

A necessary condition for biorthogonality is∑k

HkH∗k−2j = δ0jI,

or equivalentlyH(ξ)H(ξ)∗ +H(ξ + π)H(ξ + π)∗ = I.

These conditions are sufficient if the cascade algorithm for φ, φ converges.

IV. Multiresolution Approximation and Discrete Multiwavelet Transform

The contents of this section parallel corresponding results for scalar wavelets.A Multiresolution Approximation (MRA) of L2 is a doubly infinite nested sequence of subspaces

of L2

· · · ⊂ V−1 ⊂ V0 ⊂ V1 ⊂ V2 ⊂ · · ·with properties

(i)⋃

n Vn is dense in L2.(ii)

⋂n Vn = {0}.

(iii) f(x) ∈ Vn ⇐⇒ f(2x) ∈ Vn+1 for all n ∈ Z.(iv) f(x) ∈ Vn ⇐⇒ f(x− 2−nk) ∈ Vn for all n, k ∈ Z.(v) There exists a function vector φ ∈ L2 so that

{φj(x− k) : j = 1, . . . , r, k ∈ Z}forms a stable basis of V0.

The vector of basis functions φ is called the multiscaling function. The MRA is called orthogonal ifφ is orthogonal.

Condition (v) means that any f ∈ V0 can be written uniquely as

f(x) =∑k∈Z

f∗k φ(x− k)

with convergence in the L2-sense; and there exist constants 0 < A ≤ B, independent of f , so that

A∑

k

‖fk‖22 ≤ ‖f‖22 ≤ B∑

k

‖fk‖22.

Condition (iii) expresses the main property of an MRA: each Vn consists of the functions in V0

compressed by a factor of 2n. Thus, a stable basis of Vn is given by {φn,k : k ∈ Z}, where

φn,k(x) = 2n/2φ(2nx− k).

The factor 2n/2 preserves the L2-norm.Since V0 ⊂ V1, φ can be written in terms of the basis of V1 as

φ(x) =∑

k

Hkφ1k(x) =√

2∑

k

Hkφ(2x− k)

for some coefficient matrices Hk. In other words, φ is refinable.If the MRA is not orthogonal, further development requires the existence of a second MRA based

on a dual multiscaling function φ biorthogonal to φ.The projection of an arbitrary function f ∈ L2 onto Vn is given by

Pnf =∑

k

〈f, φn,k〉φn,k.

7

The basis functions φn,k are shifted in steps of 2−n as k varies, so Pnf cannot represent any detailon a scale smaller than that. We say that the functions in Vn have resolution 2−n or scale 2−n. Pnfis called an approximation to f at resolution 2−n. An MRA provides a sequence of approximationsPnf of increasing accuracy to a given function f . For f ∈ L2, Pnf → f in L2 as n→∞.

The true power of the multiresolution approach arises from considering the differences betweenapproximations at different levels. The difference between the approximations at resolution 2−n and2−n−1 is called the fine detail at resolution 2−n:

Qnf(x) = Pn+1f(x)− Pnf(x).

Qn is also a projection (orthogonal if the MRA is orthogonal). Its range Wn satisfies

Vn ⊕Wn = Vn+1.

V−1

W−1

V0

W0

V1

Figure 2. The spaces Vn and Wn.

The two sequences of spaces {Vn} and {Wn} and their relationships can be graphically representedas in figure 2.

0 2 4 6

−1

0

1

P0f

0 2 4 6

−1

0

1

Q0f

0 2 4 6

−1

0

1

Q1f

0 2 4 6

−1

0

1

P2f

Figure 3. P0f (top left), Q0f (top right), Q1f (bottom left), and P2f = P0f +Q0f +Q1f (bottom right) for f(x) = sinx.

8

The sequence of spaces {Wn} satisfies conditions similar to conditions (i) through (v) of an MRA,except that the Wn are linearly independent (mutually orthogonal if the MRA is orthogonal) insteadof nested.

A basis for W0 is given by the integer translates of a function vector ψ called the multiwaveletfunction. The multiscaling function φ and multiwavelet function ψ together form a multiwavelet.

For multiwavelets there is no simple formula for finding ψ, like in the scalar case. A constructionfor finding ψ is given in section X.

In terms of the multiwavelet functions, the projection Qn is given by

Qnf =∑

k

〈f, ψn,k〉ψn,k.

We now come to the main concept: the Discrete Multiwavelet Transform (DMWT).Given a function f ∈ L2, we can start at any level N and represent f by its the approximation

at resolution 2−N plus all the detail at finer resolution:

f = PNf +∞∑

k=N

Qkf.

For practical applications, we need to reduce this to a finite sum. We replace f by Pnf for somen. Then

Pnf = PNf +n−1∑k=N

Qkf.

This equation describes the DMWT: a high-resolution approximation Pnf to the original functionor signal f gets decomposed into a coarse approximation PNf , and fine detail at several resolutions.See fig. 3 for illustration. The decomposition as well as the reconstruction can be performed veryefficiently on a computer. Implementation details are presented in section VI.

V. Moments and Approximation Order

One of the main properties of interest is the approximation order of a multiscaling function. A highapproximation order is the basis for good performance in data compression and other applications.The results in this section are similar to corresponding results for scalar wavelets.

The kth discrete moment of φ, ψ is defined by

Mk =1√2

∑j

jkHj , Nk =1√2

∑j

jkGj .

Discrete moments are r×r matrices. They are uniquely defined and easy to calculate. In particular,M0 = H(0).

The kth continuous moment of φ, ψ is

µk =∫xkφ(x) dx, νk =

∫xkψ(x) dx.

Continuous moments are r-vectors.Continuous and discrete moments are related by

µk = m−kk∑

t=0

(kt

)Mk−tµt, νk = m−k

k∑t=0

(kt

)Nk−tµt. (4)

In particular,µ0 = M0µ0 = H(0)µ0.

µ0 is only defined up to a constant multiple. Its scaling depends on the scaling of φ. For abiorthogonal pair φ, φ , the correct scaling is given by

µ∗0

(∑k

φ(k)

)= µ∗0

(∫φ(x) dx

)= µ∗0µ0 = 1. (5)

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Unlike the scalar case, the sum of point values at the integers and the integral do not have to bethe same. They just have to have the same inner product with µ0 (which is the same as y0 definedbelow).

For orthogonal φ,

‖φ‖2 =r∑

k=1

‖φk‖2 = r, ‖µ0‖ = 1.

For biorthogonal multiwavelets we cannot in general achieve both ‖µ0‖ = 1 and ‖µ0‖ = 1.Once µ0 has been chosen, all other continuous moments are uniquely defined and can be computed

from eqs. (4).The multiscaling function φ provides approximation order p if

‖f(x)− Pnf(x)‖ = O(2−np), ‖Qnf‖ = O(2−np).

whenever f has p continuous derivatives.φ has accuracy p if all polynomials up to order p− 1 can be represented as

xn =∑

k

c∗n,kφ(x− k), n = 0, . . . , p− 1

for some coefficient vectors cn,k.The recursion coefficients {Hk} satisfy the sum rules of order p if there exist vectors y0, . . . ,yp−1

with y0 6= 0, which satisfyn∑

t=0

(nt

)2t(−i)n−ty∗tD

n−tH(0) = y∗n,

n∑t=0

(nt

)2t(−i)n−ty∗tD

n−tH(π) = 0∗,

(6)

for n = 0, . . . , p−1. D stands for the derivative operator. The vectors yn are called the approximationvectors. Note that y0 = µ0. For scalar wavelets, the sum rules of order p reduce to “H(ξ) has azero of order p at ξ = π.”

As in the scalar case, approximation order p, accuracy p, and the sum rules of order p areequivalent for sufficiently regular φ. They are also equivalent to the fact that the dual multiwaveletfunction has p vanishing moments, except that we need to specify vanishing continuous moments.In the scalar case, vanishing continuous moments are equivalent to vanishing discrete moments, butin the multiwavelet case the discrete moments are matrices. They have to annihilate certain vectors,but they do not have to be zero matrices.

For multiwavelets, accuracy p does not mean that the DMWT preserves polynomial sequencesup to order p− 1. See section VII for details.

Approximation order p is also equivalent to a certain factorization of the symbol, but not as simpleas in the scalar case. This TST factorization requires a lot of machinery, and will be presented insection XII.

VI. The Discrete Multiwavelet Transform (DMWT) Algorithm

In this section we describe the implementation of the DMWT. The idea behind it was alreadyexplained in section IV. We describe it in four different ways. All of them, except for the modulationformulation, can be used as the basis of a computer implementation.

We assume that the original signal is s(x). The algorithm starts at some resolution level n withthe coefficient sequence sn = {sn,k} from

Pns(x) =∑

k

〈s, φn,k〉φn,k(x) =∑

k

s∗n,kφn,k(x).

The decomposed signal consists of sn−1, dn−1. Note that these are sequences of vectors, and therecursion coefficients are matrices. We can interpret the algorithm in terms of convolutions anddown- and upsampling as in the scalar case, but they are block convolutions and block down- andupsampling.

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The operation count for the complete algorithm is O(N), as in the scalar case. All formulationsof the algorithm are straightforward generalizations of the scalar Discrete Wavelet Transform.

Direct FormulationDecomposition:

sn−1,j =∑

k

Hk−2jsn,k,

dn−1,j =∑

k

Gk−2jsn,k.

Reconstruction:sn,k =

∑j

H∗k−2jsn−1,j +

∑j

G∗k−2jdn−1,j .

Matrix FormulationThe decomposition and reconstruction steps can be interpreted as infinite matrix–vector products.

The formulation becomes nicer if we interleave the s- and d-coefficients:

(sd)n−1,j =(

sn−1,j

dn−1,j

), Lk =

(H2k H2k+1

G2k G2k+1

).

Then

...(sd)n−1,−1

(sd)n−1,0

(sd)n−1,1

...

=

· · · · · · · · ·· · · L−1 L0 L1 · · ·

· · · L−1 L0 L1 · · ·· · · L−1 L0 L1 · · ·

· · · · · · · · ·

...sn,−1

sn,0

sn,1

...

, (7)

or simply(sd)n−1 = L sn.

L is an infinite banded block Toeplitz matrix with blocks of size 2r × 2r.The reconstruction step can be similarly written as

sn = L∗(sd)n−1.

The perfect reconstruction condition is expressed as

L∗L = I.

Modulation FormulationThis is a way of thinking about the algorithm and verifying the perfect reconstruction conditions.

It is not a way to actually implement it.We associate with the signal sequence sn = {sn,k} its symbol

sn(ξ) =∑

k

sn,ke−ikξ.

We can write the algorithm in matrix form as(sn−1(2ξ)dn−1(2ξ)

)= M(ξ) · 1√

2

(sn(ξ)

sn(ξ + π)

),

1√2

(sn(ξ)

sn(ξ + π)

)= M(ξ)∗

(sn−1(2ξ)dn−1(2ξ)

).

The matrix

M(ξ) =(H(ξ) H(ξ + π)G(ξ) G(ξ + π)

)is called the modulation matrix. The perfect reconstruction condition becomes

M(ξ)∗M(ξ) = I.

The modulation matrix of an orthogonal multiwavelet is paraunitary, that is,

M(ξ)∗M(ξ) = I.

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Polyphase FormulationThe phases of the signal sn = {sn,k} are defined by

s(0)n,k = sn,2k, s(1)

n,k = sn,2k+1.

The corresponding polyphase symbols are given by

s(0)n (z) =

∑k

sn,2kzk, s(1)

n (z) =∑

k

sn,2k+1zk.

The phases and polyphase symbols of the coefficient sequences Hk, Gk are defined similarly. Notethat the symbols, defined in eq. (2), have a factor of 1/

√2 in front, the polyphase symbols do not.

In matrix notation, the polyphase DMWT algorithm can be written as

(sn−1(z)dn−1(z)

)= P (z)

(s(0)n (z)

s(1)n (z)

),(

s(0)n (z)

s(1)n (z)

)= P (z)∗

(sn−1(z)dn−1(z)

),

where

P (z) =(H(0)(z) H(1)(z)G(0)(z) G(1)(z)

)is the polyphase matrix. The polyphase matrix has many uses in building and modifying multi-wavelets.

The perfect reconstruction condition is

P (z)∗P (z) = I.

The polyphase matrix of an orthogonal multiwavelet is paraunitary.

VII. Pre- and Postprocessing and Balanced Multiwavelets

The DMWT algorithm requires the initial expansion coefficients sn,k. Frequently the availabledata consists of σn,k, which are equally spaced samples of the signal s. Converting σn,k to sn,k

is called preprocessing or prefiltering. After an inverse DMWT, converting sn,k back to functionvalues is called postprocessing or postfiltering. Postprocessing has to be the inverse of preprocessingto achieve perfect reconstruction.

For real-valued scalar wavelets,

2−n/2σn,k = 2−n/2s(2−nk) ≈ sn,k = 〈s, φn,k〉,

and we can usually ignore the distinction. This is not true in general for multiwavelets, with someexceptions discussed below. Preprocessing and postprocessing steps are necessary.

For simplicity we assume in the remainder of this section that everything takes place at leveln = 0, and drop the subscript 0.

For a given signal s(x),

s∗k = 〈s(x), φ(x− k)〉 =∫s(x)φ(x− k)∗ dx,

σ∗k =1√r

(s(k), s(k +

1r), . . . , s(k +

r − 1r

)).

The factor 1/√r in σk insures that for s(x) = 1 in the orthogonal case, ‖sk‖ = ‖σk‖ = 1.

Example: For the DGHM multiwavelet,

µ0 =1√3

(√2

1

).

This means that the function s(x) = 1 is represented by coefficients of the form s∗k = 3−1/2(√

2, 1),while σ∗k = 2−1/2(1, 1).

12

If we use σk as the input for the DMWT, it will not be preserved during decomposition, nor willthe d coefficients be zero. A preprocessing step needs to map (1, 1) into a multiple of (

√2, 1).

A quasi-interpolating prefilter of order p produces correct σk for all polynomials up to degreep − 1. An approximation order preserving prefilter of order p produces σk which correspond to apolynomial of the correct degree with correct leading term, but possibly different lower-order terms.This is sufficient to achieve good results in practice.

Some common ways of constructing prefilters are listed below.Interpolating PrefiltersTry to determine sk so that the multiscaling function series matches the function values at the

points xk,j = k + j/r: ∑n

s∗jφ(2xkj − n) = s(xkj).

This may or may not be possible for a given multiwavelet. This approach preserves the approxi-mation order but not orthogonality. See [64].

Quadrature-Based PrefiltersApproximate the integral defining sk by a quadrature rule. See [35].Quadrature-based prefilters can always be found. They preserve approximation order as long as

the accuracy of the quadrature rule is at least as high as the approximation order. They do notusually preserve orthogonality.

Hardin–Roach PrefiltersThese prefilters are designed to preserve orthogonality and approximation order. Preprocessing

is assumed to be linear filtering

sk =∑

j

Qk−jσj ⇔ s(ξ) = Q(ξ)σ(ξ).

This is an orthogonal transform if Q(ξ) is paraunitary.It is proved in [32] that such prefilters exist for arbitrarily high approximation orders. Approxi-

mation order preserving prefilters can be shorter than quasi-interpolating prefilters.Details are described in [32], [12]. Only approximation order 2 has been worked out so far.Other PrefiltersOther approaches to prefiltering can be found in [48], [56], [62], and [65].Balanced MultiwaveletsBalanced multiwavelets are specifically constructed to not require preprocessing. A multiwavelet

is balanced of order k if the coefficient sequence sk of any polynomial up to order k−1 is a polynomialsequence of the same order.

The following two characterizations of balanced multiwavelets are given in [40].

Theorem 1. A multiwavelet is balanced of order p if and only if one of the following equivalentconditions is satisfied:

(a) There exist constants r0 = 1, r1, . . . , rp−1 so that the symbol satisfies the sum rules of orderp with approximation vectors of the form

y∗k = (ρk(0r), ρk(

1r), . . . , ρk(

r − 1r

)),

where

ρk(x) =k∑

j=0

(kj

)rk−jx

j .

(b) The symbol factors as

H(ξ) =12pC(2ξ)pH0(ξ)C(ξ)−p (8)

with

C(ξ) = I − e0e∗0, e∗0 =1√r(1, 1, . . . , 1).

13

Part (a) are the standard sum rules (eq. (6)) with approximation vectors of a special form. Part(b) is the TST factorization of theorem 3 with all factors Ck equal, and of a special form.

Other conditions are derived in [52].Any multiwavelet with approximation order 1 can be balanced of order 1, by replacing φ with

Qφ(x) for a constant matrix Q which satisfies y∗0Q = e∗0. In the orthogonal case, this is the Hardin–Roach prefilter for approximation order 1.

Balancing of higher order is harder to enforce. In [16] it is shown how balancing conditions canbe imposed via lifting steps. (Lifting is explained in section XI.)

Other examples of balanced multiwavelets are given in [16], [38], [40], [41], [52] to [53], [54], [66],[67], and [69].

Other Multiwavelets Which Do Not Require PreprocessingExamples are totally interpolating (biorthogonal) multiwavelets (see [68]), and full rank multi-

wavelets from [17].

VIII. Boundary Handling

The DMWT operates on infinite sequences of coefficients. In real life we can only work on finitesequences. We need some procedures for handling the boundary.

The finite length DMWT should be linear, so it should be of the form

(sd)n−1 = Lnsn

for some matrix Ln, in analogy with eq. (7). In order to preserve the usual definition of the DMWTas much as possible, we postulate the form

Ln =

Lb

Li,n

Le

,

where• The interior part Li,n is a segment of the infinite block Toeplitz matrix L, and each row

contains a complete set of coefficients. This part will make up most of the matrix. Li,n

approximately doubles in size when n is increased by 1.• The matrices Lb at the beginning and Le at the end are fairly small and remain unchanged

at all levels. These matrices will handle the boundaries.• The entire matrix Ln has the same block structure as L (each block row shifted by one

compared to its neighbors).• Ln is invertible, and its inverse matrix L∗n has an analogous structure.

In the orthogonal case, we also would like to preserve L−1n = L∗n.

There are a number of ways to find suitable boundary coefficients. An excellent overview of thescalar case can be found in [5, section 8.5]. For multiwavelets, all the standard approaches that workfor scalar wavelets appear to carry over in practice, but very little has actually been proved.

Data Extension ApproachThis is easy to implement. We artificially extend the signal across the boundaries so that each

extended coefficient is a linear combination of known coefficients.For example, suppose the left border is at 0. We are given s0,k for k ≥ 0, but not for k < 0, and

we want to computes−1,0 = H−1s0,−1 + H0s0,0 + H1s0,1 + · · · .

If our extension method iss0,−1 = As0,0 +Bs0,1,

thens−1,0 = (H0 +AH−1)s0,0 + (H1 +BH−1)s0,1 + · · · .

The H- and G-coefficients that “stick out over the side” when L is truncated to Ln are wrappedback inside.

This gives us Ln, and its inverse will be L∗n. There is no guarantee that Ln will not be singular,or that Ln = L−∗n will have the correct form, but it often works in practice.

14

Special cases include the following:

• Periodic Extension: This is easy to do, and always works. It preserves orthogonalityand approximation order 1. Periodic extension is not usually a good idea unless the dataare truly periodic. The jump at the boundary leads to spurious large d-coefficients.

• Symmetric Extension: In the scalar case there are four possibilities at each end: evenor odd extension, and whole-sample or half-sample, depending on whether the boundarycoefficient is repeated or not. This is analyzed in detail in [18], where it is shown that if youmatch the type of data extension correctly to the type of symmetry of the scaling function,the DWT will preserve the symmetry across levels.

For multiwavelets there are many more possibilities of symmetry and symmetric ex-tension. Some cases are treated in [63]. An ad hoc symmetric extension for the DGHMmultiwavelet is used in [56].

• Zero, Constant or Linear Extension: These methods appear to work for multiwavelets.They preserve the corresponding approximation order (0, 1, or 2), but not symmetry.

Matrix Completion ApproachThis is a linear algebra approach based on finding suitable end blocks which guarantee LnL

∗n = I.

It is described for scalar wavelets in [43] and [5]. The extension of these results to multiwaveletsis a subject of ongoing research by the author of this article. Preliminary results indicate that thisapproach works in practice, but there are some new phenomena.

For example, for a multiscaling function of multiplicity 2 and support length 3 one would expectto need one boundary multiscaling function at each end, each composed of two individual scalingfunctions. It turns out that sometimes one has to use three individual scaling functions at one end,and one at the other.

Boundary Function ApproachThis approach is the most time-consuming, but it can preserve both orthogonality and approxi-

mation order. The idea is to introduce special boundary functions at each end of the interval, andwork out the resulting decomposition and reconstruction algorithms.

For scalar wavelets, this approach was pioneered in [21] and [11]. For multiwavelets, this has onlybeen worked out in detail for the case of the cubic Hermite multiscaling function with one particulardual. The details are given in [22], and they are quite lengthy.

Other ApproachesThere is one variation on the boundary function approach that is easy to do and has no counterpart

for scalar wavelets. If we have multiwavelets with support on [−1, 1], there is exactly one boundary-crossing multiscaling function at each end, and it is already orthogonal to everything inside. Wecan simply restrict the boundary function vector to the inside of the interval, and orthonormalizethe components among themselves. Examples for this approach are given in [30] and [31].

This would also work for scalar wavelets, of course, except there are no wavelet pairs with supportin [−1, 1] except the Haar wavelet. For multiwavelets, it is possible to achieve arbitrarily highapproximation order, plus symmetry, on [−1, 1] by taking the multiplicity high enough.

IX. Applications

Scalar wavelets and multiwavelets both deal with one-dimensional signals. They have the sameapplications: signal compression, signal denoising, fast operator evaluation in numerical analysis,Galerkin methods for differential and integral equations.

By far the largest part of wavelet applications in the literature deal with scalar wavelets. Thereare relatively few articles that report on implementations and performance for multiwavelets.

Some studies have compared the performance of scalar wavelets and multiwavelets in imagedenoising and compression, including [19], [26], [27], [34], [55], [56], [58], and [63]. The use ofmultiwavelets for video compression is reported in the thesis of Tham [59].

It appears that multiwavelets can do as well or better than scalar wavelets, but careful atten-tion must be paid to preconditioning and handling of boundaries. The authors of [56] report thatmultiwavelet filters with short support produce fewer artifacts in the reconstruction of compressedimages.

15

The main advantage of multiwavelets over scalar wavelets in numerical analysis lies in their shortsupport, which makes boundaries much easier to handle.

For integral equations, multiwavelets with support [0, 1] can be used. At least some of the basisfunctions necessarily must be discontinuous, but for integral equations that is not a problem.

Indeed, the first appearance of such multiwavelets was in the thesis and papers of Alpert (see [9] to[7]), before the concept of multiwavelets was invented. Multiwavelet methods for integral equationsare also discussed in [20], [44], [46], [47], [57], and [61].

For differential equations, multiwavelets with support [−1, 1] can be used. Regularity and ap-proximation order can be raised to arbitrary levels by taking the multiplicity high enough.

There is only one multiscaling function that crosses each boundary. It is already orthogonalto all the interior functions, so constructing the boundary multiscaling function is an easy matter:orthonormalize the truncated boundary-crossing multiscaling function. This automatically preservesapproximation order. Finding the boundary multiwavelet function still takes a little effort.

If symmetric/antisymmetric multiwavelets are used, it is even possible to use only the antisym-metric components of the boundary function vector for problems with zero boundary conditions.Examples of suitable multiwavelets can be found in [22] and [31].

Other papers about adapting multiwavelets to the solution of differential equation include [8],[15], [14], [42], and [45].

For an overview of the use of wavelets (including multiwavelets) in numerical analysis, see thecollection [1].

X. Polyphase Factorization

An orthogonal projection factor of rank k, 2 ≤ k ≤ r − 1, is a linear paraunitary matrix of theform

F (z) = (I − UU∗) + UU∗z,

where U has k orthonormal columns.

Theorem 2. Assume that P (z) is the polyphase matrix of an orthogonal multiwavelet P (z) =P0 + P1z + · · ·+ Pnz

n, with P0 6= 0, Pn 6= 0. Then P (z) can be factored in the form

P (z) = QF1(z) · · ·Fn(z),

where Q is a constant unitary matrix, and each Fj(z) is a projection factor. The number of factorsequals the polynomial degree of P (z).

The proof is constructive and produces the factors one by one. The factors are not necessarilyunique.

The unitary matrix Q could also be put on the right, but placing it on the left has one distinctadvantage: the factorization can be computed even if only the top rows of P are know. Thecompletion problem (finding the multiwavelet function if the multiscaling function is known) canthen be reduced to the problem of completing a constant orthogonal matrix, which is easy.

Other completion methods can be found in [28], [39], and [6].The factorization theorem for biorthogonal multiwavelets is considerably more complex. In addi-

tion to projection factors it also requires so-called atoms. Details can be found in [36], [3]. In [51],atoms are called pseudo-identity matrix pairs.

The polyphase factorization can be used to construct orthogonal or biorthogonal multiwaveletsfrom scratch.

XI. Lifting

The lifting process is for constructing or modifying scalar wavelets or multiwavelets. It does notpreserve orthogonality.

Assume we have a pair of matrix polynomials satisfying P (z)∗P (z) = I. We can interpret theseas the polyphase matrices of a biorthogonal multiwavelet pair. The DMWT algorithm works forpurely algebraic reasons, whether or not the coefficients are actually associated with multiscalingand multiwavelet functions.

16

Given any other pair satisfying L(z)∗L(z) = I, we obtain a new pair

[L(z)P (z)]∗[L(z)P (z)

]= I.

The idea behind lifting is to use L(z), L(z) of a special form.A lifting step is based on

L(z) =(I A(z)0 I

), L(z) =

(I 0

−A(z)∗ I

)for arbitrary A(z). The effect on the symbols is

Hnew(z) = H(z) +A(z2)G(z)

Gnew(z) = G(z),

Hnew(z) = H(z),

Gnew(z) = G(z)−A(z2)∗H(z)

This means that the multiscaling function φ changes, but its dual φ does not.A dual lifting step is based on

L(z) =(

I 0B(z) I

), L(z) =

(I −B(z)∗

0 I

).

It has a similar effect, with the roles of φ, φ reversed. Note that some authors use a reverse definitionof lifting and dual lifting steps.

For scalar wavelets, it is shown in [23] that every polyphase matrix can be factored entirely intolifting steps. An implementation of the DWT based on this factorization is faster than a directimplementation.

There is a corresponding theorem for multiwavelets (see [24]), but it requires some extra factors(unit triangular and diagonal matrices). It is not clear that this factorization is as useful as in thescalar case.

In a lifting step, conditions can be put on A(z) that will create a new φ with higher approximationorder, while keeping φ and its approximation order the same. Likewise, a dual lifting step canincrease the approximation order of φ. For details see [37].

A lifting procedure which imposes symmetry conditions is described in [60]. A lifting procedurewhich imposes balancing conditions is described in [16]. Other papers on multiwavelet lifting include[13], [24], and [29].

XII. Two-Scale Similarity Transform (TST)

The two-scale similarity transform (TST) is a new, nonobvious construction for multiwaveletsthat has no counterpart for scalar wavelets (or rather, the concept is so trivial there that it did notneed a name). Like lifting, it does not preserve orthogonality.

One application is a characterization of approximation order which is useful for both theoreticaland practical purposes. It leads to the counterpart of the statement “the symbol H(ξ) satisfies thesum rules of order p if and only if it contains a factor of (1 + e−iξ)p.” The TST factorization canalso be used to characterize balanced multiwavelets and symmetric multiwavelets.

Assume that φ is a refinable function vector, and let

φnew(x) =∑

k

Ckφ(x− k)

for some coefficient matrices Ck. Then

φnew(ξ) = C(ξ)φ(ξ), C(ξ) =∑

k

Cke−ikξ.

17

If C(ξ) is nonsingular for all ξ, then

φnew(mξ) = C(2ξ)φ(2ξ) = C(2ξ)H(ξ)φ(ξ)

= C(2ξ)H(ξ)C(ξ)−1φnew(ξ).

This means that φnew is again refinable with symbol Hnew(ξ) = C(2ξ)H(ξ)C(ξ)−1. This is a basischange which leaves all the spaces Vn in the MRA invariant.

We can also allow singular C(ξ) of a special type, and this is actually the more interestingapplication of this idea.

A TST matrix is a 2π-periodic, continuously differentiable matrix-valued function C(ξ) whichsatisfies

• C(ξ) is invertible for ξ 6= 2πk, k ∈ Z.• C(0) has a simple eigenvalue 0 with left and right eigenvectors l and r.• This eigenvalue satisfies λ′(0) 6= 0.

The last statement requires a brief explanation: as ξ varies, the eigenvalues of C(ξ) vary continuouslywith ξ. Simple eigenvalues vary in a differentiable manner. λ(ξ) is the eigenvalue for which λ(0) = 0.In some neighborhood of the origin, λ(ξ) is uniquely defined and differentiable. This derivative mustbe nonzero at 0.

If C(ξ) is a TST matrix, then

C0(ξ) = (1− e−iξ)C(ξ)−1

is well-defined for all ξ.The standard example is

C(ξ) = I − rl∗e−iξ = (I − rl∗) + rl∗(1− e−iξ),

C(ξ)−1 = (I − rl∗) +rl∗

1− e−iξ, ξ 6= 2πk,

C0(ξ) = rl∗ + (I − rl∗)(1− e−iξ),

where l and r are normalized to l∗r = 1. Here r(ξ) = r, λ(ξ) = 1− e−iξ, so λ′(0) = i 6= 0.Main DefinitionHnew is a TST of H if

Hnew(ξ) =12C(2ξ)H(ξ)C(ξ)−1

for a TST matrix C(ξ) for which C(0) and H(0) share a common right eigenvector r.Hnew is an inverse TST of H if

Hnew(ξ) = 2C(2ξ)−1H(ξ)C(ξ)

for a TST matrix C(ξ) for which C(0), H(0), H(π) share a common left eigenvector l.If C is a TST matrix for H, it automatically satisfies the conditions to be an inverse TST

matrix for Hnew. The eigenvector conditions ensure that Hnew has an approximation order onehigher than before, and Hnew has an approximation order one lower. This is a way of movingapproximation orders from one side to the other. For scalar wavelets, it corresponds to moving afactor of (1+e−iξ)/2. The TST can be extended to cover the multiwavelet functions as well (see [3]).

Repeated application of TSTs leads to the following result.

Theorem 3. If H(ξ) has approximation order p ≥ 1, it can be factored as

H(ξ) = Hp(ξ) =12Cp(2ξ)Hp−1(ξ)Cp(ξ)−1 = · · ·

= 2−pCp(2ξ) · · ·C1(2ξ)H0(ξ)C1(ξ)−1 · · ·Cp(ξ)−1,

where each Ck(ξ) is a TST matrix.

As mentioned in section VII, additional conditions on the TST matrices Cj characterize balancedmultiwavelets.

18

TSTs were defined in [6]. A special case which required H(0) to have eigenvectors of a particularstructure was developed independently in [50]. The two approaches were reconciled in the jointpaper [49].

XIII. Examples and Software

For the reader who wants to experiment with multiwavelets, this section lists some of the morecommonly used coefficients. There is also a brief list of relevant software at the end.

In all listings, p = approximation order, s = Sobolev exponent, and α = Holder exponent. If αis not listed, s − 1/2 is a lower bound for α. A common factor for all coefficients in its column islisted separately, for easier readability. All examples have multiplicity r = 2.DGHM (Donovan–Geronimo–Hardin–Massopust [25]) (orthogonal)

Support [0, 2], p = 2, α = 1, s = 1.5. φ1 is symmetric about x = 1/2, φ2 is symmetric aboutx = 1. See fig. 1 in the introduction for graphs.

Hk Gk

k = 0(

12 16√

2−√

2 −6

) (−√

2 −62 6

√2

)1

(12 0

9√

2 20

) (9√

2 −20−18 0

)2

(0 0

9√

2 −6

) (9√

2 −618 −6

√2

)3

(0 0

−√

2 0

) (−√

2 0−2 0

)factor 1/(20

√2) 1/(20

√2)

BAT O1 (Lebrun–Vetterli [40]) (orthogonal, balanced)Support [0, 2]. φ2 is reflection of φ1 about x = 1 and vice versa; wavelet functions are symmet-

ric/antisymmetric about x = 1. p = 2, balanced of order 1, s = 0.6406.The shape of ψ1 is most likely responsible for the name BAT wavelet. There are also BAT O2

and BAT O3, which are balanced of order 2 and 3.

Hk Gk

k = 0(

0 2 +√

70 2−

√7

) (0 −20 1

)1

(3 11 3

) (2 2

−√

7√

7

)2(

2−√

7 02 +

√7 0

) (−2 0−1 0

)factor 1/(4

√2) 1/4

0 1 2−1

0

1

2

BAT O1 scaling function φ1

0 1 2−1

0

1

2

BAT O1 scaling function φ2

0 1 2−3

−2

−1

0

1

2

3

BAT O1 wavelet function ψ1

0 1 2−3

−2

−1

0

1

2

3

BAT O1 wavelet function ψ2

Figure 4. BAT O1 multiwavelet

HC (Hermite Cubics) (biorthogonal)

19

Hermite cubics are C1 piecewise cubic polynomials on [−1, 1] which satisfy φ1(0) = 1, φ′1(0) = 0and φ2(0) = 0, φ′2(0) = 1. They are not orthogonal, so there are many biorthogonal completions.Many authors also use a different scaling for φ2, so not even the Hk will match what is listed here.

The completion listed here is the smoothest symmetric completion with support length 4 (see[33].)

Support of φ is [−1, 1], support of φ is [−2, 2]; ψ and ψ have support [−1, 2]; all functionsare symmetric/antisymmetric about the center of their support. p = 4, p = 2, α = 2, s = 2.5,s = 0.8279.

Hk Gk Hk Gk

k = −2(−2190 −154013914 9687

)−1

(4 6−1 −1

) (5427 567−1900 −120

) (9720 3560−60588 −21840

)0

(8 00 4

) (−19440 −605887120 21840

) (23820 0

0 36546

) (−2 −13 1

)1

(4 −61 −1

) (28026 0

0 56160

) (9720 −356060588 −21840

) (4 00 4

)2

(−19440 60588−7120 21840

) (−2190 1540−13914 9687

) (−2 1−3 1

)3

(5427 −5671900 −120

)factor 1/(8

√2) 1/(19440

√2) 1/(19440

√2) 1/(8

√2)

−1 0 1

0

1

HC scaling function φ1

−1 0 1

0

1

HC scaling function φ2

−2 −1 0 1 2

−10

0

10

HC dual φ1

−2 −1 0 1 2

−10

0

10

HC dual φ2

−2 −1 0 1 2

−1

0

1

HC wavelet function ψ1

−2 −1 0 1 2

−1

0

1

HC wavelet function ψ2

−1 0 1 2−8

−4

0

4

8

HC dual ψ1

−1 0 1 2−8

−4

0

4

8

HC dual ψ2

Figure 5. Hermite Cubic multiwavelet

There are many toolboxes available for scalar wavelets. Most of them are for Matlab or are stand-alone programs, but there are tools for Mathematica, MathCAD and other systems as well. An exten-sive list can be found at www.amara.com/current/wavesoft.html. MathWorks (www.mathworks.com)is the maker of Matlab, so theirs is the “official” Matlab wavelet toolbox. These programs cannotusually handle multiwavelets.

There are only two software packages for multiwavelets available, both of them for Matlab.MWMP (the Multiwavelet Matlab Package) was written by Vasily Strela; it can be found at

www.mcs.drexel.edu/~vstrela/MWMP. A set of multiwavelet Matlab routines from the author of

20

this entry is available through http://www.math.iastate.edu/keinert or the CRC Press downloadpage at www.crcpress.com/e_products/downloads/default.asp.

XIV. Future Directions

The basic properties of multiwavelets are quite well understood by now. Many types of multi-wavelets have been constructed, and new construction methods continue to be found. Nevertheless,multiwavelets have not been applied as much as initially expected. The theory is more complicatedthan for scalar wavelets, and the need for pre- and postprocessing is a deterrent in applications.

For future theoretical development, the biggest need is for improved methods of preprocessing, oreven better, for multiwavelets that do not require preprocessing. The study of balanced multiwaveletsis a very active field at the moment.

To a lesser degree, a comprehensive theory of boundary handling is also needed. Most multiwaveletusers sidestep the issue by padding the data.

In applications, the experience has been that multiwavelets can can do as well as scalar waveletsin the usual applications, and possibly better. However, they are not spectacularly better, and theextra effort is not usually warranted. The exception to this are applications that utilize one of theparticular strengths of multiwavelets.

One of these strengths is that multiwavelets can be both orthogonal and symmetric. This is usefulin applications where symmetry is important.

The other, more important strength is high approximation order/high smoothness coupled withshort support. This is important in using multiwavelets as basis functions in the solution of dif-ferential and integral equations. Short support reduces the overlap between segments, resulting insparser matrices, and also resolves much of the boundary problem.

For integral equations, multiwavelets with support [0, 1] can be used (which are necessarily dis-continuous). Differential equations require multiwavelets with support [−1, 1].

XV. Bibliography

Books and Reviews

[1] Bramble JH, Cohen A, and Dahmen W. Multiscale problems and methods in numerical simulations, volume 1825of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2003. Lectures given at the C.I.M.E. Summer School held

in Martina Franca, September 9–15, 2001, Edited by C. Canuto.[2] Burrus CS, Gopinath RA, and Guo H. Introduction to wavelets and wavelet transforms: a primer. Prentice Hall,

New York, 1998.

[3] Keinert F. Wavelets and multiwavelets. Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton,FL, 2004.

[4] Plonka G and Strela V. From wavelets to multiwavelets. In Mathematical methods for curves and surfaces, II

(Lillehammer, 1997), pages 375–399. Vanderbilt Univ. Press, Nashville, TN, 1998.[5] Strang G and Nguyen T. Wavelets and filter banks. Wellesley-Cambridge Press, Wellesley, MA, 1996.

[6] Strela V. Multiwavelets: theory and applications. PhD thesis, Massachusetts Institute of Technology, 1996.

Primary Literature

[7] Alpert BK, Beylkin G, Coifman R, and Rokhlin V. Wavelets for the fast solution of second-kind integral equations.

Technical Report YALEU/DCS/RR-837, Yale University, New Haven, CT, 1990.

[8] Alpert B, Beylkin G, Gines D, and Vozovoi L. Adaptive solution of partial differential equations in multiwaveletbases. J. Comput. Phys., 182(1):149–190, 2002.

[9] Alpert BK. Sparse representation of smooth linear operators. Technical Report YALEU/DCS/RR-814, Yale Uni-

versity, New Haven, CT, 1990.[10] Alpert BK. A class of bases in L2 for the sparse representation of integral operators. SIAM J. Math. Anal.,

24(1):246–262, 1993.[11] Andersson L, Hall N, Jawerth B, and Peters G. Wavelets on closed subsets of the real line. In Recent advancesin wavelet analysis, pages 1–61. Academic Press, Boston, MA, 1994.

[12] Attakitmongcol K, Hardin DP, and Wilkes DM. Multiwavelet prefilters II: optimal orthogonal prefilters. IEEETrans. Image Proc., 10(10):1476–1487, 2001.

[13] Averbuch AZ and Zheludev VA. Lifting scheme for biorthogonal multiwavelets originated from hermite splines.

IEEE Trans. Signal Process., 50(3):487–500, 2002.[14] Averbuch A, Israeli M, and Vozovoi L. Solution of time-dependent diffusion equations with variable coefficients

using multiwavelets. J. Comput. Phys., 150(2):394–424, 1999.

21

[15] Averbuch A, Braverman E, and Israeli M. Parallel adaptive solution of a Poisson equation with multiwavelets.

SIAM J. Sci. Comput., 22(3):1053–1086, 2000.

[16] Bacchelli S, Cotronei M, and Lazzaro D. An algebraic construction of k-balanced multiwavelets via the liftingscheme. Numer. Algorithms, 23(4):329–356, 2000.

[17] Bacchelli S, Cotronei M, and Sauer T. Multifilters with and without prefilters. BIT, 42(2):231–261, 2002.

[18] Brislawn C. Classification of nonexpansive symmetric extension transforms for multirate filter banks. Appl. Com-put. Harmon. Anal., 3(4):337–357, 1996.

[19] Bui TD and Chen G. Translation-invariant denoising using multiwavelets. IEEE Trans. Signal Process.,

46(12):3414–3420, 1998.[20] Chen Z, Micchelli CA, and Xu Y. The Petrov–Galerkin method for second kind integral equations II: multiwavelet

schemes. Adv. Comput. Math., 7(3):199–233, 1997.

[21] Cohen A, Daubechies I, and Vial P. Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon.Anal., 1(1):54–81, 1993.

[22] Dahmen W, Han, B Jia, R.-Q, and Kunoth A. Biorthogonal multiwavelets on the interval: cubic Hermite splines.Constr. Approx., 16(2):221–259, 2000.

[23] Daubechies I and Sweldens W. Factoring wavelet transforms into lifting steps. J. Fourier Anal. Appl., 4(3):247–

269, 1998.[24] Davis GM, Strela V, and Turcajova R. Multiwavelet construction via the lifting scheme. In Wavelet analysis and

multiresolution methods (Urbana-Champaign, IL, 1999), pages 57–79. Dekker, New York, 2000.

[25] Donovan GC, Geronimo JS, Hardin DP, and Massopust PR. Construction of orthogonal wavelets using fractalinterpolation functions. SIAM J. Math. Anal., 27(4):1158–1192, 1996.

[26] Downie TR and Silverman BW. The discrete multiple wavelet transform and thresholding methods. IEEE Trans.

Signal Process., 46(9):2558–2561, 1998.[27] Efromovich S. Multiwavelets and signal denoising. Sankhya Ser. A, 63(3):367–393, 2001.

[28] Goh SS and Yap VB. Matrix extension and biorthogonal multiwavelet construction. Linear Algebra Appl.,

269:139–157, 1998.[29] Goh SS, Jiang Q, and Xia T. Construction of biorthogonal multiwavelets using the lifting scheme. Appl. Comput.

Harmon. Anal., 9(3):336–352, 2000.[30] Han B and Jiang Q. Multiwavelets on the interval. Appl. Comput. Harmon. Anal., 12(1):100–127, 2002.

[31] Hardin DP and Marasovich JA. Biorthogonal multiwavelets on [−1, 1]. Appl. Comput. Harmon. Anal., 7(1):34–53,

1999.[32] Hardin DP and Roach DW. Multiwavelet prefilters I: orthogonal prefilters preserving approximation order p ≤ 2.

IEEE Trans. Circuits Systems II Analog Digital Signal Process., 45(8):1106–1112, 1998.

[33] Heil C, Strang G, and Strela V. Approximation by translates of refinable functions. Numer. Math., 73(1):75–94,1996.

[34] Hsung TC, Lun DPK, and Ho KC. Optimizing the multiwavelet shrinkage denoising. IEEE Trans. Signal Process.,

53(1):240–251, 2005.[35] Johnson BR. Multiwavelet moments and projection prefilters. IEEE Trans. Signal Process., 48(11):3100–3108,

2000.

[36] Kautsky J and Turcajova R. Discrete biorthogonal wavelet transforms as block circulant matrices. Linear AlgebraAppl., 223/224:393–413, 1995.

[37] Keinert F. Raising multiwavelet approximation order through lifting. SIAM J. Math. Anal., 32(5):1032–1049,2001.

[38] Kessler B. Balanced scaling vectors using linear combinations of existing scaling vectors. In Approximation theory

XI: Gatlinburg 2004, Mod. Methods Math., pages 197–210. Nashboro Press, Brentwood, TN, 2005.[39] Lawton W, Lee SL, and Shen Z. An algorithm for matrix extension and wavelet construction. Math. Comp.,

65(214):723–737, 1996.[40] Lebrun J and Vetterli M. High-order balanced multiwavelets: theory, factorization, and design. IEEE Trans.Signal Process., 49(9):1918–1930, 2001.

[41] Lian JA. Armlets and balanced multiwavelets: flipping filter construction. IEEE Trans. Signal Process.,

53(5):1754–1767, 2005.[42] Lin EB and Xiao Z. Multiwavelet solutions for the Dirichlet problem. In Wavelet analysis and multiresolution

methods (Urbana-Champaign, IL, 1999), volume 212 of Lecture Notes in Pure and Appl. Math., pages 241–254.Dekker, New York, 2000.

[43] Madych WR. Finite orthogonal transforms and multiresolution analyses on intervals. J. Fourier Anal. Appl.,3(3):257–294, 1997.

[44] Maleknejad K and Yousefi M. Numerical solution of the integral equation of the second kind by using waveletbases of Hermite cubic splines. Appl. Math. Comput., 183(1):134–141, 2006.

[45] Massopust PR. A multiwavelet based on piecewise C1 fractal functions and related applications to differentialequations. Bol. Soc. Mat. Mexicana (3), 4(2):249–283, 1998.

[46] Micchelli CA and Xu Y. Using the matrix refinement equation for the construction of wavelets II: smooth waveletson [0, 1]. In Approximation and computation (West Lafayette, IN, 1993), pages 435–457. Birkhauser, Boston, MA,1994.

[47] Micchelli CA and Xu Y. Using the matrix refinement equation for the construction of wavelets on invariant sets.Appl. Comput. Harmon. Anal., 1(4):391–401, 1994.

22

[48] Miller J and Li CC. Adaptive multiwavelet initialization. IEEE Trans. Signal Process., 46(12):3282–3291, 1998.

[49] Plonka G and Strela V. Construction of multiscaling functions with approximation and symmetry. SIAM J.

Math. Anal., 29(2):481–510, 1998.[50] Plonka G. Approximation order provided by refinable function vectors. Constr. Approx., 13(2):221–244, 1997.

[51] Resnikoff HL, Tian J, and Wells RO Jr. Biorthogonal wavelet space: parametrization and factorization. SIAM

J. Math. Anal., 33(1):194–215, 2001.[52] Selesnick IW. Multiwavelet bases with extra approximation properties. IEEE Trans. Signal Process., 46(11):2898–

2908, 1998.

[53] Selesnick IW. Balanced multiwavelet bases based on symmetric FIR filters. IEEE Trans. Signal Process.,48(1):184–191, 2000.

[54] Shen L and Tan HH. On a family of orthonormal scalar wavelets and related balanced multiwavelets. IEEE

Trans. Signal Process., 49(7):1447–1453, 2001.[55] Shi H, Cai Y, and Qiu Z. On design of multiwavelet prefilters. Appl. Math. Comput., 172(2):1175–1187, 2006.

[56] Strela V, Heller PN, Strang G, Topiwala P, and Heil C. The application of multiwavelet filter banks to imageprocessing. IEEE Trans. Signal Process., 8(??):548–563, 1999.

[57] Tausch J. Multiwavelets for geometrically complicated domains and their application to boundary element meth-

ods. In Integral methods in science and engineering (Banff, AB, 2000), pages 251–256. Birkhauser, Boston, MA,2002.

[58] Tham JY, Shen L, Lee SL, and Tan HH. A general approach for analysis and application of discrete multiwavelet

transform. IEEE Trans. Signal Process., 48(2):457–464, 2000.[59] Tham JY. Multiwavelets and scalable video compression. PhD thesis, National University of Singapore, 2002.

Available at www.cwaip.nus.edu.sg/thamjy.

[60] Turcajova R. Construction of symmetric biorthogonal multiwavelets by lifting. In Unser MA, Aldroubi A, andLaine AF, editors, Wavelet applications in signal and image processing VII, volume 3813, pages 443–454. SPIE,

1999.

[61] von Petersdorff T, Schwab C, and Schneider R. Multiwavelets for second-kind integral equations. SIAM J. Numer.Anal., 34(6):2212–2227, 1997.

[62] Vrhel MJ and Aldroubi A. Prefiltering for the initialization of multiwavelet transforms. IEEE Trans. SignalProcess., 46(1):3088, 1988.

[63] Xia T and Jiang Q. Optimal multifilter banks: design, related symmetric extension transform, and application

to image compression. IEEE Trans. Signal Process., 47(7):1878–1889, 1999.[64] Xia XG, Geronimo JS, Hardin DP, and Suter BW. Design of prefilters for discrete multiwavelet transforms. IEEE

Trans. Signal Process., 44(1):25–35, 1996.

[65] Xia XG. A new prefilter design for discrete multiwavelet transforms. IEEE Trans. Signal Process., 46(6):1558–1570, 1998.

[66] Yang S and Peng L. Construction of high order balanced multiscaling functions via PTST. Sci. China Ser. F,

49(4):504–515, 2006.[67] Yang S and Wang H. High-order balanced multiwavelets with dilation factor a. Appl. Math. Comput., 181(1):362–

369, 2006.

[68] Zhang JK, Davidson TN, Luo ZQ, and Wong KM. Design of interpolating biorthogonal multiwavelet systemswith compact support. Appl. Comput. Harmon. Anal., 11(3):420–438, 2001.

[69] Zhou S and Gao X. A study of orthonormal multi-wavelets on the interval. Int. J. Comput. Math., 83(11):819–837,2006.