Operator-like wavelets

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arXiv:1210

.1808v1

[math.CA

]5Oct2012

Operator-Like Wavelet Bases of L2(Rd)

Ildar Khalidov, Michael Unser, and John Paul Ward

Biomedical Imaging Group, Ecole polytechnique federale de Lausanne (EPFL),Station 17, CH-1015, Lausanne, Switzerland

Abstract

The connections between derivative operators and wavelets are wellknown. Here we generalize the concept by constructing multiresolu-tion approximations and wavelet basis functions that act like differ-ential Fourier multiplier operators. This construction follows from astochastic model: signals are tempered distributions such that the ap-plication of a whitening (differential) operator results in a realizationof a sparse white noise. Using wavelets constructed from these oper-ators, the sparsity of the white noise can be inherited by the wavelet

coefficients. In this paper, we specify such wavelets in full generalityand determine their properties in terms of the underlying operator.

1 Introduction

In the past few decades, a variety of wavelets that provide a complete andstable multiscale representation ofL2(R

d) have been developed. The waveletdecomposition is very efficient from a computational point of view, due tothe fast filtering algorithm. Underlying this approximation procedure is adifferential operator, so our purpose in this paper is to construct wavelets

that behave like a given differential Fourier multiplier operator L, which canbe more general that a pure derivative. In our approach, the multiresolution

This research was funded in part by ERC Grant ERC-2010-AdG 267439-FUN-SP andby the Swiss National Science Foundation under Grant 200020-121763.

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spaces are characterized by generalized B-splines associated with the opera-

tor, and we show that, in a certain sense, the wavelet inherits properties ofthe operator. Importantly, the operator-like wavelet can be constructed di-rectly from the operator, bypassing the scaling function space. What makesthe approach even more attractive is that, at each scale, the wavelet spaceis generated by the shifts of a single function. Our work provides a gen-eralization of some known constructions including: cardinal spline wavelets[5], elliptic wavelets [15], polyharmonic spline wavelets [19, 20], Wirtinger-Laplace operator-like wavelets [21], and exponential-spline wavelets [13].

In applications, it has been observed that many signals are well repre-sented by a relatively small number of wavelet coefficients. Interestingly,the model that motivates our wavelet construction explains the origin of this

sparsity. The context is that of sparse stochastic processes, which are definedby a stochastic differential equation driven by a (non-Gaussian) white noise.Explicitly, the model states that Ls = w where the signal s is a tempereddistribution, L is a shift-invariant Fourier multiplier operator, and w is asparse white noise [18]. The wavelets we construct are designed to act likethe operator L so that the wavelet coefficients are determined by a general-ized B-spline analysis of w. In particular, we define an interpolating spline, corresponding to LL, from which we derive the wavelets = L. Thenthe wavelet coefficients are formally computed by the L2 inner product

s, = s, L

= Ls, = w, .Sparsity of w combined with localization of the B-splines results in sparsewavelet coefficients. This model is relevant in medical imaging applica-tions, where good performance has been observed in approximating func-tional magnetic resonance imaging and positron emission tomography datausing operator-like wavelets that are tuned to the hemodynamic or pharma-cokinetic response of the system [12, 22].

Our construction falls under the general setting of pre-wavelets, whichare comprehensively discussed by de Boor, DeVore, and Ron [7]. Two dis-tinguishing properties of our approach are its operator-based nature and thefact that it is non-stationary. Related constructions have been developed forwavelets based on radial basis functions [4, 6, 17]. In fact [6] also takes anoperator approach; however, the authors were focused on wavelets definedon arbitrarily spaced points.

This paper is organized as follows. In Section 2, we formally define the

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class of admissible operators and the lattices on which our wavelets are de-

fined. In Section 3, we construct the non-stationary multiresolution analysis(MRA) that corresponds to a given operator L and derive approximationrates for functions lying in Sobolev-type spaces. Then, in Section 4, we in-troduce the operator-like wavelets and study their properties; in particular,we derive conditions on L that guarantee that our choice of wavelet yields astable basis at each scale. Under an additional constraint on L, we use thisresult to define Riesz bases ofL2(R

d). In Section 5, we prove a decorrelationproperty for families of related wavelets. Finally, we conclude in Section 6with some examples of the wavelets provided by our construction.

2 PreliminariesThe primary objects of study in this paper are differential Fourier multiplieroperators and their derived wavelets. The operators we consider are shift-invariant operators L that act on the class of square integrable functionsf : Rd C, which will be denoted by L2(Rd). The action of such a Fouriermultiplier operator is defined by its symbol L in the Fourier domain, with

Lf =Lf ,

where the symbol L is a measurable function. Here f denotes the Fouriertransform of ff() =

Rdf(x)eixdx,

and g denotes the inverse Fourier transform of g. Pointwise values of Lwill be required for some of our analysis, so we restrict the class of symbolsby requiring continuity almost everywhere. Additionally, we would like tohave a well-defined inverse of the symbol, so L should not be zero on a set ofpositive measure. To be precise, we define the class of admissible operatorsas follows.

Definition 2.1. Let L be a Fourier multiplier operator, then L is admissibleif its symbol L is a ratio of continuous functions f /g satisfying

1. The set of zeros of f g has Lebesgue measure zero;

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2. The zero sets of f and g are disjoint.

Notice that each such operator defines a subspace of L2, consisting of func-tions whose derivatives are also square integrable, and our approximationresults will focus on these spaces.

Definition 2.2. An admissible operator defines a Sobolev-type subspace ofL2(R

d):

WL2 (Rd) =

f L2(Rd) :

Rd

f()2 (1 + L()2)d < .Having defined the class of admissible operators, we must consider the

lattices on which the multiresolution spaces will be defined. It is importantto use lattices which are nested, so we consider those defined by an expansiveinteger matrix. Specifically, an integer matrix A, whose eigenvalues are alllarger than 1 in absolute value, defines a sequence of lattices

AjZd = {Ajk : k Zd}indexed by an integer j. Here, we state several results about these latticesthat will be used in this paper, most of which can be found in [11]. First, weknow that AjZd can be decomposed into a finite union of disjoint copies ofAj+1Zd; there are |det(A)| vectors {el}|det(A)|1l=0 such that

l

Ajel + Aj+1Zd = AjZd,and using this notation, our convention will be to set e0 = 0.

There are also several important properties that arise when using Fouriertechniques on more general lattices. First, a lattice in the spatial domaincorresponds to a lattice in the Fourier domain known as its dual lattice, andthe dual lattice ofAjZd is given by 2(AT)jZd. Also relevant is the notionof a fundamental domain, which for AjZd is a measurable set j satisfying

kZdj(x + A

jk) = 1

for all x. Finally, just as in the dyadic case, we can define a discrete Fouriertransform (DFT) corresponding to A in terms of the functions which areperiodic with respect to the lattice. The matrix for this transform is givenby

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F = eiklk,l ,and the corresponding inverse DFT matrix is

F1 =1

|det((AT)1)|

eikll,k

,

where k and l range over the cosets of 2(AT)1Zd/Zd and Zd/AZd, respec-tively. The inverse DFT matrix will play a role in deriving the Riesz basiscondition on the wavelet spaces. We summarize the required properties inthe following proposition.

Proposition 2.3. A lattice generated by an expansive integer matrix A de-

fines a discrete Fourier transform. This transform has a matrix representa-tion which is invertible and has nonzero entries.

3 Multiresolution Analysis

The multiresolution framework for wavelet construction was presented byMallat in the late 1980s [14]. In the following years, the notion of pre-wavelets was developed, and a more general notion of multiresolution wasadopted. We consider this more general setting in order to allow for a widervariety of admissible operators.

Definition 3.1. A sequence {Vj}jZ of closed linear subspaces of L2(Rd)forms a non-stationary multiresolution analysis if

1. Vj+1 Vj;2.jZ Vj is dense in L2(R

d) andjZ Vj is at most one-dimensional;

3. f Vj if and only if f( Ajk) Vj for all j Z and k Zd, whereA is an expansive integer matrix;

4. For each j Z, there is an element j Vj such that the collection oftranslates {j( Ajk) : k Zd} is a Riesz basis of Vj , i.e., there areconstants 0 < Aj Bj < such that

Aj c22 kZd

c[k]j( Ajk)2

L2(Rd)

Bj c22 .

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In order to produce non-stationary MRAs, we require additional prop-

erties on the admissible operators. Together with a dilation matrix, theoperator should admit generalized B-splines that satisfy decay and stabilityproperties.

As motivation for our definition, let us consider the one-dimensional ex-ample where L is defined by

Lf(t) =df

dt(t) f(t).

In this case, a Greens function for L is (t) = etH(t), where H is theHeaviside function. In order to produce Riesz bases for the scaling matrixA = (2), we introduce the localization operator Ld,j defined by Ld,jf =

f e2j

f( 2j). The generalized exponential-spline j := Ld,j will then bea compactly supported function whose shifts j( 2j) form a Riesz basis.Notice that, in the Fourier domain, j is given by L1()Ld,j() which is

j() = 1 e2j(i)i .

In this form one can verify the equivalent Riesz basis condition

0 < Aj kZd

j( 2j+1k)

2 Bj < .

In fact, based on the symbol of L, we could have worked entirely in theFourier domain to determine appropriate periodic functions Ld,j. With thisexample in mind, we make the following definition.

Definition 3.2. We say that an operator L and an integer matrix D are aspline-admissible pair of order r > d/2 if the following conditions are satisfied:

1. L is an admissible Fourier multiplier operator;

2. D = aR with R an orthogonal matrix and a > 1;

3. For some constant cL > 0 and all positive h, we have

1 +L(/h)21/2

L(Rd\)

cLhr,

where is the domain [, ]d;

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4. For every j Z, there exists a periodic functionLd,j such that

j() :=

Ld,j()L()1 satisfies the Riesz basis condition0 < Aj

kZd

j( + 2(DT)jk)2 Bj < ,for some Aj and Bj in R. Here we require the periodic functions Ld,jto be of the form

kZd pj [k]e

iDjk for some p 1(Zd).It can be verified that the last condition implies that the function j is

in L2(Rd). We call its inverse Fourier transform j := (

j) a generalized B-

spline for L. Also note that, in this definition, we have restricted the scaling

matrices to be multiples of orthogonal matrices. The lattices generated bythese matrices have some additional nice properties. For example, the latticesgenerated by D and DT are the same, and the lattices DjZd scale uniformlyin j. Also, for such matrices, there are only finitely many possible latticesgenerated by powers of D; i.e., there always exists a positive integer n forwhich Dn = anI.

Proposition 3.3. Given a spline-admissible pair L and D, the spaces

Vj =

kZd

c[k]j( Djk) : c 2(Zd)

form a non-stationary MRA.

Proof. The fact that Vj+1 Vj follows from the definition of D and theRiesz basis conditions on the B-splines. The density property is a result ofthe admissibility of L, the Riesz basis condition, and the inclusion relationVj+1 Vj , cf. [7, Theorem 4.3]. In addition, the conditions imposed on Limply that the dimension of the intersection of the spaces in our MRA is atmost one, cf. [7, Theorem 4.9]. Lastly, Properties 3 and 4 follow directlyfrom the definitions.

Condition 4 in the definition of spline-admissibility is the most challeng-

ing to verify; however, there are many operators for which this condition issatisfied. For instance, it holds for any one-dimensional, constant-coefficientdifferential operator. In addition, it holds in higher dimensions for the Maternoperators characterized by L() = (1 + ||2)/2, as they require no localiza-tion. In Section 6, we shall give a less trivial example and show how this

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Riesz basis property can be verified. As a final point, note that if one is only

interested in analyzing fine-scale spaces, Condition 4 need only be satisfiedfor j smaller than a fixed integer j0, but in this case, it is necessary to includethe space Vj0 in the wavelet decomposition.

We shall close this section by determining approximation rates for thespaces {Vj}jZ, in terms of L and the density of the lattice generated by Dj .In order to state this result, we must define the spline interpolants for the

operator LL, whose symbol isL2. The spline admissibility of this operator

is the subject of the next proposition.

Proposition 3.4. If L is spline admissible of order r > d/2, then LL isspline admissible of order 2r > d.

Proof. Let L be a spline admissible operator of order r > d/2. First noticethat LL is an admissible Fourier multiplier operator. Also, we can see thatLL satisfies Condition 3 of Definition 3.2 with r replaced by 2r. Thereforespline admissibility will follow if we can exhibit B-splines for LL that satisfythe Riesz basis condition, where the integer dilation matrix is the same as

for L. To this end, we claim that j := |j|2 defines the Fourier transformof a B-spline for LL.

An upper Riesz bound for

j can be found by using the norm inequality

between 1 and 2:

kZd

j( + 2(DT)jk)4 kZd

j( + 2(DT)jk)22 B2j .To verify the lower Riesz bound for j, we note that the decay condition

of spline admissibility implies that for R > 0 sufficiently large, we haveL ||R

CRr.

Now for any M > 0 sufficiently large, we can boundkZd

j( + 2(DT)jk)2above by using the decay estimate on L1:

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|k|M

j( + 2(DT)jk)2 + CLd,j()Md2r |det(D)|2jr/d .Therefore

|k|M

j( + 2(DT)jk)2 Aj CLd,j()Md2r |det(D)|2jr/d ,and

|k|M

j( + 2(DT)jk)4 CMd

|k|M

j( + 2(DT)jk)22

CMd

Aj CLd,j()Md2r |det(D)|2jr/d2

Due to the fact that 2r > d, we can always choose M to make the right handside positive, and this establishes a lower Riesz bound for j.

The Riesz basis property of the B-splines for LL imply that the LL-

spline interpolants j(x), given byj() = |det(D)|j |j()|2kZd |j( + 2(DT)jk)|2 (1)

are well-defined and also generate Riesz bases. Importantly, j WL2 doesnot depend on the specific choice of the localization operator, as we can seefrom

j() = |det(D)|j Ld,j()

2

L()

2

Ld,j()2kZd L( + 2(DT)jk)2= |det(D)|j 1

1 +L()2kZd\{0} L( + 2(DT)jk)2 .

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These LL-spline interpolants play a key role in our wavelet construction,

which we describe in the next section; however, for our approximation result,we are more interested in the related functions

mj() =

L()2kZd

L( + 2(DT)jk)2 , (2)which will also be needed for the decorrelation result in Theorem 5.4.

In order to bound the error of approximation from the spaces Vj, weapply the techniques developed in [8]. In that paper, the authors derivea characterization of certain potential spaces in terms of approximation by

closed, shift-invariant subspaces of L2(Rd

). The same techniques can beapplied in our situation, with a few modifications to account for smoothnessbeing determined by different operator norms.

The error in approximating a function f L2(Rd) by a closed functionspace X will be denoted by

E(f, X) = minsX

f sL2(Rd) ,and the approximation rate will be given in terms of the density of the latticein Rd. The lattice determined by Dj has density proportional to |det(D)|j/d ,so we say that Vj provides approximation order r if for every f WL2 (Rd)

E(f, Vj) c |det(D)|jr/d fWL2(Rd) .

The rate of approximation is then determined by the following theorem.

Theorem 3.5. For a spline-admissible pair L and D of order r > d/2, themultiresolution spaces Vj provide approximation order r r ifdet(DT)2jr/d 1 mj()

1 +L()

2

is bounded, independently of j, in L((DT)j), where = [

, ]d.

Proof. This result is a consequence of [8, Theorem 4.3]. To show this let

us introduce the the notation fj() = f(Dj), which implies that fj =|det(D)|jf (DT)j . Notice that the spaces Vj are scaled copies of theinteger shift-invariant spaces

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Vjj := {s(Dj) : s Vj}

=

kZd

c[k]j(Dj( k)) : c 2(Zd)

.

We then can write the error of approximating a function f WL2 (Rd) fromVj in terms of approximation by Vjj as

E(f, Vj) = |det(D)|j/2 E(fj , Vjj )

= (2)d/2 |det(D)|j/2

E(fj,Vjj ),where Vjj is composed of the Fourier transforms of functions in Vjj . Separat-ing this last term, we have

E(f, Vj) (2)d/2 |det(D)|j/2

E(fj,Vjj ) + (1 )fj2

, (3)

where is the characteristic function of the set . We are now left withbounding both terms on the right-hand side of (3). First, we have

(1 )fj22

=Rd\

fj()2 d= |det(D)|2j

Rd\

f((DT)j)2 1 +L((DT)j)2

1 +L((DT)j)2d

and since L is spline-admissible of order r

(1 )fj22 |det(D)|2jr/dj f2WL

2,

so that

|det(D)|j/2(1 )fj

2 |det(D)|jr/d fWL

2. (4)

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In order to bound the remaining term, we will need a formula for the projec-

tion of fj onto Vjj . Notice that1 mj((DT)j) = 1

j (DT)j2kZd |j (DT)j( 2k)|2

= 1 j Dj2

kZd

j Dj( 2k)2 ,so we can apply [8, Theorem 2.20] to get

E(fj,Vjj )2 =

fj2 (1 mj((DT)j))= |det(D)|2j

f (DT)j2 (1 mj((DT)j)).Now, changing variables gives

E(

fj,

Vjj )

2 = |det(D)|j

(DT)j f

2

(1 +

L

2

)1 mj

1 + L2

|det(D)|j f2WL2

1 mj

1 +L2

L((DT)j)

.

Applying our assumption on (1 mj), we have

|det(D)|j/2 E(fj,Vjj ) c |det(D)|jr/d fWL2

. (5)

Substituting the estimates (4) and (5) into (3) yields the result.

4 Operator-Like Wavelets and Riesz Bases

Using the non-stationary MRA defined in the previous section, we define thescale of wavelet spaces Wj by the relationship

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Vj = Vj+1 Wj+1;i.e., Wj+1 is the orthogonal complement ofVj+1 in Vj. Our goal in this sectionis to define Riesz bases for these spaces and for L2(R

d). To begin, let us definethe functions

j+1 = Lj,

which we claim generate Riesz bases for the wavelet spaces, under mild con-ditions on the operator L. First, note that j+1 is indeed in Vj , because its

Fourier transform

j+1 is a periodic multiple of

j, and thus

j+1() = |det(D)|j Ld,j()kZd |j( + 2(DT)jk)|2j().

A direct implication of our wavelet construction is the following property.

Property 4.1. The wavelet function j+1 behaves like a multiscale versionof the underlying operator L in the sense that, for any f WL2 , we havef j+1 = L{f j}. Hence, in the case where j is a lowpass filter,{L{f j}}jZ corresponds to a multiscale representation of L{f}.

The next few results focus on showing that the DjZd/Dj+1Zd shifts of

j+1 are orthogonal to Vj+1 and generate a Riesz basis of Wj+1.Proposition 4.2. The wavelets {j+1( Djk)}kZd\DZd are orthogonal tothe space Vj+1.

Proof. Note that it suffices to show j+1, j+1( Djk) = 0 for everyk Zd\DZd. By definition

j+1, j+1( Djk) =Rdj+1()eiDjkL()j()d

= RdLd,j+1()eiDjkj()d.Now let be a fundamental domain for the lattice generated by 2(DT)j

so that we can write

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j+1, j+1( Djk) =

Ld,j+1()eiDjk nZd

j( 2(DT)jn)d=

Ld,j+1()eiDjkd= 0.

The last equality holds because Ld,j+1 is periodic on a finer lattice.In order to prove that shifts ofj form a Riesz basis of the wavelet space,

we need to introduce some notation, which will help formulate the problem as

a shift-invariant one. Since we view the wavelet space as being shift-invarianton Dj+1Zd, a representative wavelet is chosen for each of the |det(D)| 1cosets. Specifically, recall that there are vectors {el}|det(D)|1l=0 in Zd such that

l

Djel + D

j+1Zd

= DjZd,

so for 1 l |det(D)| 1, we define the wavelets

(l)j+1(x) := j+1(x Djel).

In the following, necessary and sufficient conditions on the operator L will

be given which guarantee that = {(l)j+1}|det(D)|1l=1 generates a Riesz basisof Wj+1. The technique used is called fiberization and it can be applied tocharacterize finitely generated shift-invariant spaces [16]. In this setting, acollection of functions defines a Gramian matrix, and the property of beinga Riesz basis is equivalent to the Gramian having bounded eigenvalues. Inour situation, the Gramian for is

G() =

2(DT)j1Zd

eiDj (ekel)(+)

j+1( + )

2

k,l

.

Let us denote the largest and smallest eigenvalues of G by () and(), respectively. Then the collection is a Riesz basis if and only if and 1/ are essentially bounded (cf. [16] Theorem 2.3.6). To simplify this

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matrix without changing the eigenvalues, we apply the similarity transfor-

mation T()1

G()T(), where T is the diagonal matrix with diagonalentry eiDjel in row l. This transformation multiplies column l of G by

eiDjel and row k of G by e

iDjek. Since the eigenvalues are unchanged,let us call this new matrix G as well. After applying the transformation,we have

G() =

2(DT)j1Zd

eiDj(ekel)

j+1( + )2

k,l

.

Using the fact that

m

em + D

TZd

= Zd and the notation

c(m;) := 2(DT)jZd

j+1( + 2(DT)j1em + )2 ,we can write

G() =

|det(D)|1m=0

c(m;)e2i(ekel)(DT)1em

k,l

.

Since G is Hermitian, its maximum and minimum eigenvalues can be de-termined by analyzing the quadratic form

G =

|det(D)|1k,l=1

kl

|det(D)|1m=0

c(m;)e2i(ekel)(DT)1em

=

|det(D)|1m=0

c(m;)

|det(D)|1k,l=1

(ke2iek (D

T)1em)le2iel(D

T)1em

=

|det(D)|1m=0

c(m;)

|det(D)|1

k=1

ke2iek(D

T)1em

2

over all unit vectors C|det(D)|1. Let us additionally define H to be theinverse DFT matrix for the lattice generated by D, i.e.,

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H = e2iek(DT)1emk,mwith k and m ranging from 0 to |det(D)| 1, and let H0 be the submatrixobtained by removing row 0 from H. Note that each sum in the quadraticform above is given by the inner product of with a column of H0. Thisidentification allows us to prove the following.

Lemma 4.3. For a fixed, the quadratic formG is non-zero if andonly if at most one of the coefficients c(m;) is zero.

Proof. First, suppose that two distinct coefficients, c(m0;) and c(m1;),are zero. Then we can clearly find an that is orthogonal to the remaining

|det(D)| 2 columns of H0. Hence the quadratic form will be 0 for such an.

Next, suppose that all coefficients except possibly c(m0;) are non-zero,and denote the matrix obtained from H0 by removing column m0 by H0,m0.Then, to prove positivity of the quadratic form, we show that the columns ofH0,m0 span C

|det(D)|1. This is equivalent to showing that the determinant ofH0,m0 is non-zero, which follows from the fact that H

1 has non-zero entries(since H0,m0 is a first minor of H).

Lemma 4.4. The collection is a Riesz basis if and only if no two of thecoefficients c(m;) are zero for the same.

Proof. This follows from the definition of j+1 and Lemma 4.3.

Suppose that the conditions of Theorem 4.5 are satisfied so that isa Riesz basis. Then the orthogonality between the shifts of and Vj+1,together with the Riesz basis condition on j+1, implies :={j+1} generates a Riesz basis. The fact that spans Vj follows from a comparisonwith the Riesz basis generated by the Dj+1Zd translates of j( Djel), cf.[1] and [7, Theorem 2.26]. As a consequence of these results, would be astable basis of Wj+1.

In order to interpret this result in terms of the operator L, we note thatj+1() vanishes at p + (DT)j2k for every p Rd such that L(p) = 0.In other words, each zero of L generates a periodic sequence of zeros in thespectrum of the generating wavelet. As long as these periodizations of zerosare essentially disjoint, the conditions of Theorem 4.5 will be satisfied. The

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zero set of the symbol is denoted by N= {p Rd|

L(p) = 0}, and for eachscale j, we define

N(l)j = p + (DT)(j+1)2el + (DT)j2Zd, l = 0, . . . , |det(D)| 1,

where p N.Theorem 4.5. Letj Z be an arbitrary scale. Then the family of functions = {(l)j+1}|det(D)|1l=1 generates a Riesz basis of Wj+1 if and only if the sets

N(l)j satisfy

N(0)

j N(l)

j = (6)for each1 l |det(D)| 1.Proof. This follows from the periodicity of the zeros of j+1 and the conditionon the coefficients from Lemma 4.4.

Our wavelet construction is intended to be general so that we may accountfor a large collection of operators. As a consequence of this generality, we cannot conclude that our wavelet construction always produces a Riesz basis ofL2(R

d). Here, we shall impose additional conditions to ensure a Riesz basisis produced. In order to preserve generality, we shall focus on the fine scale

wavelet spaces and include a B-spline space Vj0 in our Riesz basis.

Theorem 4.6. LetL be a spline admissible operator of order r, and supposethat there exist 0 > 0 and constants C1, C2 > 0 such that the symbol Lsatisfies

C1 ||r L() C2 ||r

for || 0. Then there is an integer j0 such that the collection

{j0+1( )}Dj0Z

d jj0{|det(D)|(j+1)/(2d)j j+1( )}DjZd\Dj+1Zd (7)forms a Riesz basis of L2(R

d).

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Proof. We shall verify the result for the dilation matrix D = 2I, as the other

cases are similar. Let j0 be an integer for which 0 < 22j02

.Considering Lemma 4.4, we can see that the Riesz bounds for the waveletspaces depend on estimating the functions c(m;), the main term being thesum

22jZd

L( + 22j1em + )1

kZd

L( + 22j1em + + 22jk)22

. (8)

The value of this sum depends on the position of the lattice

Xj(m,) := + 22j1

em + 22j

Zd

with respect to the origin. To prove bounds of (8), let us introduce thenotation hj for the fill distance and qj for the separation radius of the latticesXj(m,); i.e,

hj := supxRd

infxkXj(0,0)

|x xk| = 22j1

d

and

qj := infk=k

|xk xk| = 22j1.

Since each lattice Xj(m,) is a translation ofXj(0, 0), the quantities hj andqj are independent of m and .

We shall now bound (8) by considering two cases:

1. dist(0, Xj(m,)) qj/2;2. dist(0, Xj(m,)) < qj/2.

In the first case, all points of the lattice Xj(m,) lie outside of the ballof radius 0 centered at the origin. Therefore, (8) can be reduced to

xkXj(m,)

L(xk)21

. (9)

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Applying Proposition A.1 and Proposition A.2, we can find upper and lower

bounds for this sum which are proportional to 2j1

. Importantly, the pro-portionality constants are independent of j.For the second case, we must be more careful as one of the lattice points

will lie close to the origin. For any fixed , there will be at most one mfor which the lattice Xj(m,) satisfies this second condition. Therefore, asufficient lower bound for the lattice sum will be 0; however, the upper boundmust match the one derived above. Let us consider the cases

2a. L takes the value 0 on the lattice Xj(m,)2b.

L1

< on the lattice Xj(m,)

If the first condition is satisfied, then (8) will be 0. However, if the secondcondition holds, then we can again reduce the sum to (9). Clearly, 0 is alower bound for this expression. For the upper bound, we apply PropositionA.1 as in the previous case.

To finish the proof, we note that Lemma 4.3 implies that the wavelets

{j+1( Djk)}kZd\DZdform a Riesz basis at each level j j0. Furthermore, the bounds on c(m,)imply that the Riesz bounds are all equal. Therefore, the collection (7) is aRiesz basis of L

2(Rd).

5 Decorrelation of Coefficients

As was stated in the introduction, the primary reason for our constructionis to promote a sparse wavelet representation. Our model is based on theassumption that the wavelet coefficients of a signal s are computed by theL2 inner product

j( Djk), s

.

Here, we should point out that, unless the wavelets form an orthogonal basis,reconstruction will be defined in terms of a dual basis. However, as our focusin this paper is the sparsity of the coefficients, we shall be content to workwith the analysis component of the approximation and leave the synthesiscomponent for future study.

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Now, considering our stochastic model, it is important to use wavelets

that (nearly) decorrelate the signal within each scale, and one way to ac-complish this goal is by modifying the underlying operator. Hence, given aspline-admissible pair (L, D), we define a new spline-admissible pair (Ln, D)

by Ln := Ln, and we will see that as n increases, the wavelet coefficients be-come decorrelated. This result follows from the fact that the (Ln)Ln-splineinterpolants (appropriately scaled) converge to a sinc-type function, and itis motivated by the work of Aldroubi and Unser, which shows that a broadfamily of spline-like interpolators converge to the ideal sinc interpolator [2].To state this result explicitly, we denote the B-splines for Ln by n,j = nj .Therefore the (Ln)Ln-spline interpolants are given by

n,j() = |det(D)|j |n,j()|2kZd |n,j( + 2(DT)jk)|2

= |det(D)|j |j()|2nkZd |j( + 2(DT)jk)|2n ,

and we analogously define

mn,j() =|

j()|2n

kZd |j( + 2(DT)jk)|2n .

Decorrelation will depend on showing that the functions mn,j converge to 1almost everywhere on some fundamental domain j of 2(D

T)jZd and to0 almost everywhere outside of j. Our proof relies on techniques used inthe analysis of radial basis functions. For example, a similar method wasused by Baxter to prove convergence of the Lagrange functions associatedwith multiquadric functions [3, Chapter 7]. The idea is to define disjoint setscovering Rd, each with a single point in any given fundamental domain, andanalyze the convergence of mn,j on these sets.

Definition 5.1. For each j Z and x j we define the set

Ej,x := {x + 2(DT)jk : k Zd}.By the Riesz basis condition, each Ej,x has a finite number of elements ywith m1,j(y) of maximal size, and we define Fj to be the set ofx j suchthat there is not a unique y Ej,x where m1,j attains a maximum.

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Lemma 5.2. Letx j\Fj, then fory Ej,x we have mn,j(y) 0 if andonly if m1,j(y) is not of maximal size over Ej,x. Furthermore, if m1,j(y) isof maximal size, then mn,j(y) 1.Proof. Fix x j\Fj and y Ej,x. Notice that the periodicity of thedenominator of m1,j implies that m1,j(y) is maximal iff |1,j(y)| is maximal.

Let us first suppose m1,j(y) is not maximal. If|1,j(y)| = 0, the result isobvious. Otherwise, there is some k0 Zd and b < 1 such that

|1,j(y)| b 1,j(y + 2(DT)jk0) .Therefore

|n,j(y)|2 b2n n,j(y + 2(DT)jk0)2 b2n

kZd

n,j(y + 2(DT)jk)2

and the result follows.Next, suppose m1,j(y) is of maximal size. Since 1,j has no periodic zeros,

|

1,j(y)| = 0. Therefore

mn,j(y) =1

Bn,j(y)

with

Bn,j(y) =kZd

1,j(y + 2(DT)jk)1,j(y)2n .

Since |1,j(y)| is of maximal size, all terms of the sum except one are lessthan 1. In particular, Bn,j will converge to 1 as n increases.

Lemma 5.3. Given j Z, if the Lebesgue measure of Fj is 0, thenkZd

mn,j( + 2(DT)jk)2

converges to 1 in L1(j) as n .

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Proof. Notice that the sum is bounded by 1. Therefore Lemma 5.2 implies

that the sum converges to 1 on the complement of Fj . Hence, we can applythe dominated convergence theorem to obtain the result.

With this theorem, we can show how the wavelets corresponding to Lndecorrelate within scale as n becomes large. The way we characterize decor-relation is in terms of the semi-inner products

(f, g)n :=

Rd

fg Ln2 ,which is a true inner product for the wavelets

n,j+1 = L

n{n,j}.Theorem 5.4. Suppose the Lebesgue measure ofjZFj, as provided in Def-inition 5.1, is 0. Then as n increases, the wavelet coefficients decorrelate inthe following sense. For any j Z,k Zd\{0} we have

(n,j+1, n,j+1( Djk))n 0as n .Proof. The inner product can be expressed as an integral using the definition

(n,j+1, n,j+1( Dkk))n = Rd

n,j+1()(n,j+1( Djk))() Ln2 d= |det(D)|2j

Rd

mn,j()2eiD

jkd,

and we can periodize the integrand to get

Rdmn,j()

2eiDjkd =

l2(DT)jZd j+lmn,j()

2eiDjkd

=

j

eiDjk

l2(DT)jZd

mn,j( l)2d.

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The last expression converges to 0 by the Lebesgue dominated convergence

theorem.Let us now show how this result implies decorrelation of the wavelet

coefficients. To begin, we introduce the notation

cn,j+1,k :=

n,j( Djk), s

for the wavelet coefficients of the random signal s. Note that our stochasticmodel, Ls = w, implies that the coefficients are random variables. Hence, fordistinct k and k, the covariance between cn,j+1,k and cn,j+1,k is determinedby the expected value of their product:

E{cn,j+1,kcn,j+1,k} = E

n,j( Djk), w

n,j( Djk), w

.

Now, as long as the white noise w has zero mean and second-order moments,the covariance satisfies

E{cn,j+1,kcn,j+1,k} =

n,j( Djk), n,j( Djk)

= (n,j+1, n,j+1( Dj+1k))n 0,

where the convergence follows from Theorem 5.4. Therefore, when the stan-dard deviations of cn,j+1,k and cn,j+1,k are bounded below, the correlationbetween them will also converge to zero.

6 Examples

The wavelet construction that we have presented in this paper is quite generaland accommodates many operators. In this section, we consider operatorsthat are defined as polynomials in the Laplacian. In particular, we shallshow how spline admissibility may be verified for such operators. Also, notethat each of these operators satisfies the growth condition of Theorem 4.6.Therefore, the corresponding wavelet spaces may be used to construct Rieszbases of L2(R

d).

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6.1 Matern and Laplace Operators

First, consider the two-dimensional Matern operator that is defined by thesymbol L() = (||2+1)/2, with the parameter > 1. As L()1 satisfiesthe Riesz basis condition, no localization operator is needed. Therefore, theoperator L is spline-admissible of order . Also note that the fractionalLaplacian L = ||3 and iterated Laplacian L = ||4 operators are alsospline-admissible.

Each of these wavelets can be considered with any admissible subsamplingmatrix D. In two dimensions, the choice of D = 2I corresponds to therectangular grid, while

D = 1 11 1characterizes the quincunx subsampling scheme.

6.2 Helmholtz Operator

Operators whose symbols have more general zero sets can also be used. Asan example, consider the polynomial p(x) = 1/4 + x. The operator p() is

known as the Helmholtz operator, and its symbol is L() = 1/4 ||2. Thewavelets corresponding to this operator could potentially have applicationsin optics, as the Helmholtz equation,

u + u = f,

is a reduced form of the wave equation [9, Chapter 5]. In what follows, weshow that this is a spline-admissible operator for the scaling matrix D = 2Ion R2. However, since the wavelets j = L

j will not form a Riesz basis forthe coarse-scale wavelet spaces, we only consider j 1.

The primary issue with proving spline-admissibility is defining discreteoperators Ld,j corresponding to L. It is known that sufficiently smooth func-tions have absolutely convergent Fourier series [10, Theorem 3.2.9], so we

must construct such a function with the same zero set as L. Since L is zeroon a circle and radial, we can choose Ld,j to be the periodization of a radialfunction. More precisely, we define fj to be a function that is periodic withrespect to the lattice dual to 2jZ2 and, on [2, 2]2, satisfies:

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1. The function fj() has continuous derivatives up to order 3 and is

constant for || 1;2. There are constants C1 and C2 such that

0 < C1 fj()(1/4 ||2) < C2 < for || 1.

These conditions are met, for example, by defining fj by

fj() = 2,k(||) 2,k(1/2) if || < 1

2,k(1/2) if

|

| 1,

where 2,k(||) is a Wendland function with 2k continuous derivatives [23, Sec-tion 9.4]. We can therefore apply our wavelet construction, which producesa Riesz basis.

A Discrete Sums

Let X = {xk} be a countable collection of points in Rd, and definehX := sup

xRdinfxkX

|x xk|

qX := infk=k

|xk xk| .

Also, let B(x, r) denote the ball of radius r centered at x. Proving Rieszbounds for the wavelet spaces relies on the following propositions concerningsums of function values over discrete sets.

Proposition A.1. If hX < and r > d/2, then there exists a constantC > 0 (depending only on r and d, not hX) such that

|xk|>2hX|xk|2r Ch2rX

Proof. Since xk 2hX, we havexk hX xk|xk| 21 |xk| ,

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which implies

|xk|2r 22rxk hX xk|xk|

2r

.

Then

k

|xk|2r

xk2hX

|xk|2r

22r

xk2hX

xk hX xk|xk|

2r

Vol (B(xk, hX))

Vol (B(xk, hX))

22rVol (B(0, hX))

xk2hX

B(xk ,hX)

|x|2r dx

ChdX3hX

t2r+(d1)dt

Ch2rX

Proposition A.2. If |xk| > qX/4 and r > d/2, then there exists a constantC > 0 (depending only on r and d, not qX) such that

k

|xk|2r Cq2rX

Proof. Using the fact that |xk| > qX/4, we can writexk + qX8 xk|xk| 2 |xk| ,

which implies

|xk|2r

22r xk + qX8 xk|xk|

2r

.

We now have

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k

|xk|2r 22rk

xk + qX8 xk|xk| 2r Vol (B(xk, qX/8))Vol (B(xk, qX/8)) 2

2r

Vol (B(0, qX/8))

k

B(xk ,qX/8)

|x|2r dx

22r

Vol (B(0, qX/8))

|x|>qX/8

|x|2r dx

CqdXqX/8

t2r+(d1)dt

Cq2r

X .

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