Dubuc-Deslauriers Interpolation Wavelets Tese

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    DUBUC-DESLAURIERS

    INTERPOLATION WAVELETS

    ISAAC OLUKUNLE ABIODUN

    African Institute for Mathematical Sciences,

    e-mail: [email protected]; [email protected]

    Supervisor : Prof. Johan de Villiers

    Department of Mathematical and Physical Sciences,

    University of Stellenbosch,

    e-mail: [email protected]

    MAY 2006

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    Contents

    Abstract ii

    Dedication iii

    Acknowledgements iv

    Notation v

    Introduction 1

    1 The refinement mask 2

    2 The refinable function 10

    3 The interpolation wavelet 29

    Conclusion 36

    References 38

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    Abstract

    Wavelet decomposition techniques have grown over the last two decades into an important math-ematical tool in signal analysis. The main building block in wavelet construction is a so called

    refinable function (scaling function), with the intrinsic property of being expressible as a linear

    combination of the integer shifts of its own dilation with a factor 2. An important class of refinable

    functions are the Dubuc-Deslauriers refinable functions, which are also fundamental interpolants,

    and can therefore be used to construct corresponding interpolation wavelets by means of the asso-

    ciated interpolation operator.

    In this essay, after proving results on the existence, construction and properties of Dubuc-Deslauriers

    refinable functions, as was done in [1], [7], [8], the method adopted in [4] is shown to yield an

    explicit construction procedure for the Dubuc-Deslauriers interpolation wavelets, and the accom-

    panying decomposition algorithm. A practical application in signal analysis is then investigated.

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    Dedication

    This essay is dedicated to my siblings, Cecilia, Paul and Timothy, you guys mean a lot to me. I setthe record for you to break.

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    Acknowledgements

    I give all glory to God, the alpha and omega, for the grace and strength He gave me in the courseof this essay and the whole of AIMS programme. Indeed, I appreciate the fatherly guidance of my

    supervisor, Prof. Johan de Villiers. You are more than helpful and I will always remember your

    contribution to my life. My gratitude also goes to the administration of AIMS: the founder, Prof.

    Neil Turok, the director, Prof. Fritz Hahne, and others, for the opportunity given me to study here.

    Thanks for all your support. Special regards to Dr Mike Pickles, Nneoma Ogbonna and the rest of

    the tutors at AIMS. I must not fail to appreciate Akwum Onwunta who proved to be a brother to

    me. You are indeed a star.

    My profound gratitude to my parents, Pastor and Mrs Abiodun, and my siblings, Cecilia, Paul

    and Timothy. Your love and care has always been of tremendous help, you are the best. I cannot

    end this acknowledgement without appreciating the Nigerian team at AIMS for the 2005/2006 set,

    Bolaji, Femi, Henry, Gideon, Okeke, Naziga and Doom-null. Our unity is our strength, the G8 is

    going places. In general, a lot of thanks also goes to all the students whose company made AIMS

    an interesting place to be.

    I also appreciate the Redeemed Christian Church of God, Victory Center, Cape Town, for their

    spiritual and moral assistance. Thanks also to my brothers and friends in Nigeria whose love and

    care are well felt while I was thousands of miles away. The likes of Seun Oluwalade, Lekan

    Adesanya, Hafiz Oladejo, Goke Ogun and all others. My regards also to my brothers in South

    Africa here in person of Richard Akinola, Abel Ajibesin and all my senior friends. May God bless

    you all.

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    Notation

    Symbol Definition

    N the set of natural numbers {1, 2, 3, . . .}.

    Z the set of integers {. . . , 2, 1, 0, 1, 2, . . .}.

    Z+ the set of nonnegative integers.

    Zm the set of nonnegative integers m.

    C the set of complex numbers.

    R the real line (, ).

    M(Z) the set of bi-infinite real-valued sequences, i.e. c M(Z)c {cj : j Z} R.

    M0(Z) {c M(Z) :c is finitely supported}.A sequencec M(Z) isfinitely supportedif

    there exist integers Jand Ksuch thatcj =0, j {J, ..., K}.

    M(R) {f : R R}.

    M0(R) {f M(R) : fis finitely supported}. A function f M(R) isfinitely supportedif

    there exists a bounded interval [, ] such that f(x)= 0, x [, ].

    C(R) {f M(R) : fis continuous on R}.

    Ck(R) {f C(R) : f(k) C(R), j = 1, 2, . . . , k}.

    C0(R) C(R) M0(R).

    L1(R) the space of Lebesque integrable functions.

    n the linear space of polynomials of degree n., wheren Z+.

    j,k =

    1, j = k,

    0, j k,j, k Z.The Kronecker delta .

    j =j,0, j Z.

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    CONTENTS CONTENTS

    Symbol Definition

    ( jk

    ) =

    j!

    k!(jk)!, k {0, 1, . . . , j},

    0, k {0, 1, . . . , j},

    are the binomial coefficients {( jk

    ) : j, k Z+}, with the convention that 0! =1.

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    Introduction

    Wavelets have developed into an important mathematical tool in signal and image processing ap-plications such as noise reduction and restoration, texture analysis and segmentation, feature ex-

    traction and detection, approximation theory, just to mention a few. The word wavelet means

    small wave and can be defined as a finitely supported funtion whose dilations and translations

    {(2rx j) : r, j Z} provide localised information on the details of a given signal at di fferent

    resolution levelsr.

    Of fundamental importance in the construction of wavelets is the concept of a refinable (scaling)

    function, that is, a function which satisfies the fundamental equation =

    j

    aj(2 j), where

    the bi-infinite sequencea = {aj : j Z} is called the corresponding refinement mask. The refinable

    function is often not known analytically, and the analysis of its properties is therefore based on

    the explicitly known mask with finitely many non-zero coefficients.

    In this essay, we first study a special class of refinement masks called the Dubuc-Deslauriers masks

    from which we prove the existence of the associated refinable function m. The function m, in turn,

    yields the DD interpolation wavelet

    m. First, in Chapter 1, we give necessary conditions for theconstruction of the refinement mask. In Chapter 2, we discuss in detail the existence and properties

    ofm, and its construction through a subdivision operator. The function m is the main building

    block in the construction of wavelets through multi resolution analysis (MRA) because it forms

    the basis for the construction of the refinement spaces.

    The focus in Chapter 3 is on the Dubuc-Deslauriers interpolation wavelets. Here, the fundamental

    decomposition result is proved, which leads to the decomposition algorithm. Finally we use a

    given signal to demonstrate the efficiency of our decomposition algorithm.

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    1The refinement mask

    In this essay, our ultimate goal is to construct an interpolation wavelet which is suitable for the

    analysis of signals. However, we need to lay a solid foundation by first showing the existence of a

    suitable refinement pair which is the fundamental building block for this wavelet.

    Definition: If a sequencea M0(Z) and a function M0(R) are such that

    =

    j

    aj(2 j), (1.1)

    then (a, ) forms what is known as arefinement pair, the sequencea = {aj} is called therefinement

    mask, and the function is said to be a refinable functioncorresponding to the maska. Equation

    (1.1) is called a refinement equation. A refinable function has the intrinsic characteristic of being

    expressible as a finite linear combination of the integer shifts of its own dilation by factor 2.

    With the eventual view of constructing interpolation wavelets, we shall seek a refinement pair (a, )

    such that the refinable function is interpolatory in the sense that

    (j)=j, j Z. (1.2)

    If both (1.1) and (1.2) hold, we shall call (a, ) aninterpolatory refinement pair. Associated with

    each maska M0(Z) is a Laurent polynomial A, known as themask symbol, which we define by

    A(z)= j

    ajzj, z C\{0}. (1.3)

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    CHAPTER 1. THE REFINEMENT MASK

    We proceed to establish necessary conditions on a mask for the corresponding refinement pair to

    be interpolatory.

    Proposition 1.1 Suppose(a, )is an interpolatory refinement pair. Then

    a2j =j, j Z. (1.4)

    Moreover, the condition (1.4) holds if and only if the corresponding mask symbol A, as defined by

    (1.3), satisfies the identity

    A(z

    )+ A

    (z

    )=

    2, z C\{

    0}.

    (1.5)

    Proof.

    Using (1.1) and (1.2), we obtain, for k Z,

    k = (k)=

    j

    aj(2k j)=

    j

    a2kj(j)=

    j

    a2kjj =a2k,

    thereby proving (1.4).

    To prove the equivalence of (1.4) and (1.5),we use (1.3) to rewrite the left-hand side of (1.5), for

    z C\{0}, as

    A(z) + A(z) =

    j

    ajzj +

    j

    aj(z)j

    =

    j

    a2jz2j +

    j

    a2j+1z2j+1

    +

    j

    a2jz2j

    j

    a2j+1z2j+1

    and thus

    A(z) + A(z)= 2

    j

    a2jz2j, z C\{0}. (1.6)

    Now suppose (1.4) holds. Then (1.6) gives

    A(z) + A(z)= 2

    j

    jz2j =2, z C\{0},

    so that (1.5) holds. If (1.5) holds, then (1.6) implies

    2 = A(z) + A(z) = 2

    ja2jz2j, z C\{0},

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    CHAPTER 1. THE REFINEMENT MASK

    and thus

    j

    a2

    jz2j =1, z C\{0},

    giving (1.4).

    For a given refinement pair (a, ), we define the linear space sequence {V(r) = V(r)

    : r Z} by

    means of

    V(r) =span{(2r j) : j Z}:=

    j

    cj(2r j) :c M(Z)

    , r Z. (1.7)

    The sequence {V(r

    ) :r Z} is nested in the following sense.

    Proposition 1.2 Suppose(a, )is a refinement pair. Then the linear space sequence

    {V(r) :r Z} defined by (1.7) is nested in the sense that

    V(r) V(r+1), r Z.

    Proof. Suppose f V(r), that is, there exists a sequence c M(Z) such that f = j

    cj(2r j).

    Using (1.1), we deduce that

    f =

    j

    cj

    k

    ak(2r+1 2j k)

    =

    j

    cj

    k

    ak2j(2r+1 k) =

    k

    j

    ak2jcj

    (2r+1 k),

    that is, f =

    k

    dk(2r+1 k), withd M(Z) defined bydk=

    j

    ak2jcj, k Z, so that f V(r+1).

    In our eventual interpolation wavelet decomposition technique, we shall require, for a given inter-

    polatory refinement pair, and for every integer N Z, the approximation operator

    QN : M(R) V(N), as defined by

    QNf =

    j

    f

    j

    2N

    (2N j), f M(R). (1.8)

    Our following result shows thatQNis then an interpolation operator.

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    CHAPTER 1. THE REFINEMENT MASK

    Proposition 1.3 Suppose (a, ) is an interpolatory refinement pair, and let QN denote the ap-

    proximation operator defined by (1.8). Then QNis an interpolation operator in the sense that

    (QNf)

    j

    2N

    = f

    j

    2N

    , j Z, f M(R). (1.9)

    Proof.

    For f M(R) and j Z, it follows from (1.8) and (1.2) that

    (QNf)

    j

    2N

    =

    k

    f

    k

    2N

    (j k)=

    k

    f

    k

    2N

    jk= f

    j

    2N

    .

    In our wavelet decomposition application, we shall need the operatorQNto be exact on polynomi-

    als up to some given odd degree, that is, for some integerm N, we shall require that

    QNf = f, f 2m1. (1.10)

    Our next result gives a sufficient condition on an interpolatory refinement pair for (1.10) to hold.

    Proposition 1.4 Suppose (a, ) is an interpolatory refinement pair, with C0(R). If, moreover,

    the mask a satisfies the condition

    k

    a2j+12kkl =

    j + 1

    2

    l, j Z, l Z2m1, (1.11)

    then (1.10) holds.

    Proof.

    Using (1.8), we see that the condition (1.10) has the equivalence formulation

    j

    f

    j

    2N

    (2N j)= f, f 2m1,

    which, in turn, holds if and only if

    j

    j

    2N

    l(2Nx j)= xl, x R, l Z2m1,

    or, equivalently,

    j

    jl(x j)= xl, x R, l Z2m1. (1.12)

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    CHAPTER 1. THE REFINEMENT MASK

    Since the dyadic set { k2r

    :k Z, r Z+} is dense in R, and since is continuous on R, we conclude

    that (1.12) holds if and only if

    j

    jl

    k

    2r j

    =

    k

    2r

    l, k Z, r Z+, l Z2m1. (1.13)

    To prove (1.13), we first show that

    k

    aj2kkl =

    j

    2

    l, j Z, l Z2m1. (1.14)

    To this end, we first note that, since (1.4) holds, we have

    k

    a2j2kkl = jl =

    2j

    2

    l, j Z. (1.15)

    Next, we use (1.11) to obtain

    k

    a2j+12kkl =

    j + 1

    2

    l=

    2j + 1

    2

    l, j Z. (1.16)

    Together, (1.15) and (1.16) then imply (1.14). Using (1.1) and (1.14), we get, for k Z, r Z+,

    andl Z2m1, j

    jl

    k

    2r j

    =

    j

    jl

    n

    an2j

    k

    2r1 n

    (1.17)

    =

    n

    j

    an2jjl

    k

    2r1 n

    = 1

    2l

    j

    jl

    k

    2r1 j

    =

    1

    22l

    jj

    l

    k

    2r2 j

    .

    .

    .

    = 1

    2rl

    j

    jl(k j)= 1

    2rl

    j

    jlkj =

    k

    2r

    l, (1.18)

    thereby proving the desired result (1.13).

    Observe also the following immediate consequence of Proposition 1.4:

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    CHAPTER 1. THE REFINEMENT MASK

    Corollary 1.5 Under the conditions of Proposition 1.4, we have2m1 V(r), r Z .

    Observe that the condition (1.11) has the equivalent formulation

    k

    a2j+12kp(k)= pj + 1

    2

    , j Z, p 2m1. (1.19)

    Now, for a given positive integer m, we consider the problem of finding a minimally supported

    maska such that the conditions (1.4) and (1.19) are satisfied.

    To achieve this, we introduce, form N, the Lagrange fundamental polynomials Lm,k 2m1,

    k= m + 1, ..., m, as defined by

    Lm,k =

    mkj=m+1

    j

    k j, k=m + 1, ..., m, (1.20)

    for which

    Lm,k(j)= kj, k, j =m + 1, ..., m, (1.21)

    andm

    k=m+1

    p(k)Lm,k= p, p 2m1. (1.22)

    Since {Lm,k :k=m + 1, ..., m} is a basis for the polynomial space 2m1, we see that the condition

    (1.19) has the equivalent formulation

    k

    a2j+12kLm,l(k)= Lm,l(j + 12

    ), j Z, l = m + 1, ..., m. (1.23)

    Setting j =0 in (1.23), and using (1.21), we deduce that a necessary condition for (1.19) to hold is

    given by

    a12l+

    k{m+1,...,m}

    a12kLm,l(k)= Lm,l

    12

    , l = m + 1, ..., m,

    or, equivalently,

    a2j+1+

    k{m,...,m1}

    a2k+1Lm,j(k)= Lm,j

    1

    2

    , j = m, ..., m 1. (1.24)

    A minimally supported sequence {a2j+1 : j Z} satisfying (1.24) is given by

    a2j+1 = Lm,j 1

    2 , j =m, ..., m 1, (1.25)

    a2j+1 = 0, j {m, ..., m 1}. (1.26)

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    CHAPTER 1. THE REFINEMENT MASK

    Since, for j m + 1, ..., m, we have

    mjl=m+1

    12

    l

    = 122m+1

    11 2j

    ml=m1

    (2l + 1)= (1)m

    24m+11

    2j 1

    (2m + 1)!

    m!

    2

    andm

    jl=m+1

    (j l)= (1)m+1+j(m + j)!(m + 1 j)!,

    we have from (1.20) that

    Lm,j

    1

    2

    =

    m + 1

    24m+1

    2m + 1

    m

    (1)j+1

    2j 1

    2m + 1

    m + j

    , j = m + 1, ..., m. (1.27)

    Observe from (1.25) and (1.27) that, for j {m, ..., m 1}, we have

    a2j1 =a2(j1)+1 = Lm,j+1

    12

    =

    m + 1

    24m+1

    2m + 1

    m

    (1)j

    2j + 1

    2m + 1

    m + j + 1

    = m + 1

    24m+1

    2m + 1

    m

    (1)j+1

    2(j) 1

    2m + 1

    m j

    = Lm,j

    12

    = a2j+1.

    Hence the maska M0(Z), as given by (1.4), (1.25) and (1.26), satisfies the symmetry condition

    aj =aj, j Z. (1.28)

    We have therefore shown that ifa is a minimally supported symmetric mask satisfying the condi-

    tions (1.4) and (1.19), thenais necessarily given by (1.4), (1.25) and (1.26). Our next result proves

    that this mask does indeed satisfy the condition (1.19).

    Theorem 1.6 [4] The mask a M0(Z)defined by

    a2j+1 = Lm,j

    12

    = m+1

    24m+1

    2m+1

    m

    (1)j

    2j+1

    2m+1

    m+j+1

    , j =m, ..., m 1,

    a2j = j, j Z,

    aj = 0, j {2m + 1, ..., 2m 1},

    (1.29)

    is the minimally supported mask satisfying the conditions (1.4), (1.19) and (1.28).

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    CHAPTER 1. THE REFINEMENT MASK

    Proof. It remain to prove that (1.29) implies (1.19). To this end, let j Z and p 2m1, and note

    that if the polynomialq is defined byq(x)= p(j +x), x R, thenqalso belongs to2

    m1

    . Now we

    use (1.29) and (1.22) to obtain

    k

    a2j+12kp(k) =

    k

    a2k+1p(j k) =

    k

    a2k+1p(j + k)

    =

    mk=m+1

    p(j + k)a2k+1

    =

    mk=m+1

    p(j + k)Lm,k

    12

    =m

    k=m+1

    q(k)Lm,k

    12

    = q( 1

    2)= p(j + 1

    2).

    For a given integer m N and j Z, the maska = am M0(Z) defined by (1.29) was first

    introduced in [3], and will henceforth be referred to as the Dubuc-Deslauriers (DD) maskof order

    m. The corresponding mask symbolA = Am, as defined, in (1.3), by

    Am(z)=

    j

    amjzj, z C\{0}, (1.30)

    will be called theDD mask symbolof orderm.

    Using (1.29) and (1.30), we compute the casesm =1, 2, 3 to be given by

    (i) A1(z)= 1

    2

    1

    z+ 2 +z

    ,

    (ii) A2(z)= 1

    16

    1

    z3 +

    9

    z+ 16 + 9z z3

    , (1.31)

    (iii) A3(z)=

    1

    2563z5

    25

    z3 +

    150

    z + 256 + 150z 25z

    3

    + 3z

    5.

    In a similar manner, we can derive the DD mask symbol up to any desired order.

    In this chapter, we have shown that if (a, ) is an interpolatory refinement pair, if, moreover, the

    interpolation operatorQN : M(R) V(N) possesses the polynomial reproduction property (1.10),

    thena is necessarily the DD mask sequence as defined by (1.29). In the next chapter, we proceed

    to prove that there does indeed exist a function C0(R) such that (amj, ) is an interpolatory

    refinement pair.

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    2The refinable function

    In this chapter, we prove the existence and properties of the refinable function which together with

    the Dubuc-Deslauriers refinement mask obtained in the previous chapter, forms an interpolatory

    refinement pair. We follow the approach employed in [1], [4] and [7].

    Existence and properties of the refinable function

    First, observe from (1.29) and (1.30) that

    Am(z) = 1 +

    m1j=m

    am2j+1z2j+1, z C\{0},

    and thus

    Am(1)= 1 +

    mj=m+1

    Lm,j

    12

    = 1 +

    mj=m+1

    Lm,j

    12

    =1 + 1= 2,

    as obtained by choosing p 2m1 in (1.22) as the constant polynomial p(x) = 1, x R. Hence,

    using also the equivalence of (1.4) and (1.5), we see that Am(1)+Am(1)= 2, and thus Am(1)= 0.

    Recalling also (1.28), we have therefore established the following result.

    Theorem 2.1 For m N, the DD mask symbol Am is a Laurent polynomial of the form (1.30),

    with

    amj = 0, j {2m + 1, ..., 2m 1}, (2.1)

    amj = amj, j Z,

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    CHAPTER 2. THE REFINABLE FUNCTION

    and where

    Am(1)= 2, Am(1)= 0. (2.2)

    The following fundamental existence result will be instrumental in proving the existence of an

    interpolatory functionm C0(R) such that (amj, m) is a refinement pair.

    Theorem 2.2 For a positive integer m, suppose a M0(Z)is a sequence such that aj =0,

    j {2m + 1, ..., 2m 1}, and aj = aj, j Z, and such that the corresponding mask symbol A, as

    defined by (1.3), satisfies the following properties:

    (i) A(z) + A(z)=2, z C\{0}, (2.3)

    (ii) A(1)= 0, (2.4)

    (iii) A(eix )> 0, < x< . (2.5)

    Then there exists a continuous function : R C, such that the equation (1.1) is satisfied.

    Remark: Observe that, since a2

    j = j, j Z, and aj = aj, j Z, we have from (1.3) that, for

    x R,

    A(eix ) = 1 +

    m1j=m

    a2j+1e(2j+1)ix = 1 +

    m1j=0

    a2j+1[e(2j+1)ix + e(2j+1)ix]

    = 1 + 2

    m1j=0

    a2j+1cos(2j + 1)x,

    that is,A(z) R, forz on the unit circle in C.

    Proof of Theorem 2.2.

    We see from (2.4) that there exists an integer N such that

    A(z)= 1

    21(1 +z)B(z), z C\{0}, (2.6)

    with

    B(1)= 1, (2.7)

    and

    B(eix) 0, < x< , (2.8)

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    CHAPTER 2. THE REFINABLE FUNCTION

    having noted also from (2.3) and (2.4) that

    A(1)= 2. (2.9)

    We proceed to define the sequence {r :r N} by

    r(x)= 1

    2r

    rj=1

    Ae

    i x2j

    , x R, r N, (2.10)

    and continue to examine the convergence of the infinite product

    limr

    r(x)=

    j=1

    1

    2

    A ei x

    2j , x R. (2.11)Observe in particular from the remark after the statement of Theorem 2.1 that r M(R).

    Suppose first that x {2l(2k+ 1): l N, k Z}. It follows from (2.4) and (2.10) that

    limr

    r(x)= 0. (2.12)

    Since

    {x R: x = 2k, k Z\{0}}= {x R: x = 2l(2k+ 1), l N, k Z}, (2.13)

    we deduce from (2.12) and (2.13) that

    limr

    r(2k)= 0, k Z\{0}. (2.14)

    We also deduce from (2.9) and (2.10) that

    r(0)= 1, r N, (2.15)

    which, together with (2.14), gives

    limr

    r(2k)= k, k Z. (2.16)

    Now, define the set by

    ={x R : x 2k, k Z\{0}}, (2.17)

    and suppose x . Then, from (2.5), (2.10) and (2.13), we have that Ae

    i x2j

    > 0 for j N, so

    that we can write

    r(x)= expln

    rj=1

    A ei x

    2j 2

    , x , r N,

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    CHAPTER 2. THE REFINABLE FUNCTION

    and thus

    r(x)=exp

    r

    j=1

    lnA e

    i x2j

    2 , x , r N. (2.18)

    Since a series is convergent if its absolute series converges, we deduce from (2.18) that if we can

    show that the series

    j=1

    fj(x), (2.19)

    where

    fj(x)=

    ln

    Ae

    i x2j

    2

    , x , j Z, (2.20)

    converges, then the convergence of the infinite product (2.11) will follow.

    From (2.9) and (2.20) we have, for j N andx ,

    fj(x) =

    A(eix/2j )

    A(1)

    1

    tdt

    1

    min{2,A(eix/2j)}

    |A(ei x

    2j ) A(1)|. (2.21)

    The continuity of the Laurent polynomial A at z = 1 and (2.9) imply that there exists a >0 such

    that

    |z 1|< |A(z) 2|< 1. (2.22)

    Also, from the fact that

    |ei 1| =2

    sin

    2

    ||, R,we get

    |ei x

    2j 1| |x|

    2j, x R, j N. (2.23)

    If we now choose a positive integerx to satisfy

    x max

    1,

    ln |x| ln

    ln 2

    , (2.24)

    we then see from (2.21), (2.22) and (2.23) that

    fj(x)A(ei x2j ) A(1)

    , x , j x. (2.25)Sinceak=0, k {2m + 1, ..., 2m 1}, we have

    A(z)=

    2m1k=2m+1

    akzk, z C\{0}, (2.26)

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    CHAPTER 2. THE REFINABLE FUNCTION

    and consequently, for a fixed j N, by (2.23), we get

    A(ei x2j ) A(1) = 2m1

    k=2m+1

    ak

    ei k

    2j 1

    2m1k=2m+1

    |ak|ei k2j 1

    2m1k=2m+1

    |kak| |x|2j . (2.27)

    But then (2.25) and (2.27) give

    fj(x) M|x|

    2j, x , j x, (2.28)

    with

    M:=

    2m1

    k=2m+1

    |kak|. (2.29)

    From the fact that

    k=11

    2kis a convergent series, we deduce from (2.28), (2.20), (2.18), (2.17) and

    (2.16), that, for each fixed x R, the series (2.19), and therefore also the infinite product (2.11),

    does indeed converge.

    The convergence is uniform on compact sets in R, from (2.24), (2.28), and (2.14), hence the limit

    functiong defined by

    g(x) := limr

    r(x), x R, (2.30)

    is inC(R), with, according to (2.30) and (2.16),

    g(2k) = k, k Z. (2.31)

    We claim thatg L1(R). To prove this, we see from (2.30) and (2.10) that, for x R,

    g(x) = 1

    2A(ei

    x2 ) lim

    k

    k1j=1

    1

    2A(e

    i x2.2j )

    , (2.32)

    and thus

    g(x)= 1

    2A(ei

    x2 )g

    x

    2

    , x R. (2.33)

    It follows from (2.32) and (2.10) that

    g(x)= r(x)g

    x

    2r

    , x R, r N. (2.34)

    Note also from (2.30), (2.10), (2.4) and (2.9) that

    g(x) 0, x R. (2.35)

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    CHAPTER 2. THE REFINABLE FUNCTION

    Now use (2.34), and (2.35), (2.10) to get, for r N,

    2r2r

    |g(x)|dx = 2r

    2r

    g(x)dx = 2r

    2r

    r(x)g

    x2r

    dx =

    Gr(x)g(x)dx, (2.36)

    where

    Gr(x)=

    rj=1

    A

    eix2j1

    , x R, r N. (2.37)

    Note that, similarly to (2.35), we have

    Gr(x) 0, x R, r N. (2.38)

    Sinceg C(R), there exists a positive numberR such that

    |g(x)| R, |x| , (2.39)

    so that (2.36), (2.38) and (2.39) imply

    2r2r

    |g(x)|dx R

    Gr(x)dx, r N. (2.40)

    Next, we define the sequence of Laurent polynomials

    Ar(z)=

    rj=1

    A(z2j1

    ), z C\{0}, (2.41)

    and we define the sequence {r,j : j Z, r N} by

    Ar(z)=

    j

    r,jzj, z C\{0}, r N. (2.42)

    We claim that

    r,2rj =j, j Z, r N. (2.43)

    We prove (2.43) by induction as follows. Forr=1, we see from (2.41) thatA1(z)= A(z),z C\{0},

    and thus, using (1.3) and (2.42), we obtain

    1,j =aj, j Z. (2.44)

    But, according to (2.3) and Proposition 1.1, we have that (1.4) holds, which together with (2.44),

    yields (2.43) forr = 1. The inductive step from rto r+ 1 follows by noting from (2.41) that, for

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    CHAPTER 2. THE REFINABLE FUNCTION

    z C\{0},

    k

    r+1,kzk = Ar(z)A(z2r)

    =

    k

    r,kzk

    j

    ajzj2r

    =

    j

    aj

    k

    r,kzk+j2r

    =

    j

    aj

    k

    r,kj2rzk

    =

    k

    j

    ajr,kj2r

    zk,

    and thus, fork Z,

    r+1,2r+1k=

    j

    ajr,2r+1kj2r =

    j

    ajr,2r(2kj) =

    j

    aj2kj =a2k,

    from the inductive hypothesis, before using again (1.4), and thereby completing our proof of (2.43).

    Observe from (2.37), (2.41) and (2.42) that

    Gr(x)= Ar(eix )=

    j

    r,jei jx , x R, (2.45)

    and thus

    Gr(x)dx =

    j

    r,j

    ei jxdx = 2r,0. (2.46)

    It follows from (2.46) and (2.43) that

    Gr(x)dx = 2, r N. (2.47)

    Now substitute (2.47) into (2.40) to deduce that 2r2r

    |g(x)|dx 2R, r N,

    and thusg L1(R), with

    |g(x)|dx 2R. (2.48)

    Now define the function : R C by the formula

    (t) := 1

    2

    eitx g(x)dx, t R, (2.49)

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    CHAPTER 2. THE REFINABLE FUNCTION

    for which it follows, by virtue of the fact that g L1(R), that is a continuous function on R.

    Moreover, using (2.49), (2.33) and (1.3), we have that, for t R, that

    (t) = 1

    2

    eitx

    12

    k

    akeikx/2

    gx

    2

    dx

    = 1

    2

    e2itx

    k

    akeikx

    g(x)dx=

    k

    ak1

    2

    e(2tk)ix g(x)dx

    = k

    ak(2t k),

    thereby proving that satisfies the equation (1.1).

    We proceed to prove the following properties of the function of Theorem 2.2.

    Theorem 2.3 The function of Theorem 2.2 satisfies the interpolatory property (1.2), and

    C0(R), with

    (j)=0, x (2m + 1, 2m 1). (2.50)

    Proof.

    We use the same notation as in the proof of Theorem 2.2. To prove the interpolatory property (1.2),

    we first show that there exists a real number >0 such that

    g(x), |x| . (2.51)

    From (2.30), (2.35), (2.11) and (2.6), it holds for x R that

    g(x) = |g(x)| =

    j=1

    1 + ei x

    2j

    2

    j=1

    B(ei x2j ) . (2.52)

    Now note that

    j=1

    1 + eix

    2j

    2

    =1, x = 0.Suppose next that x 0. Then, for M N, we have

    2M

    (1 ei x

    2M

    )

    Mj=1

    1 + ei x

    2j

    2 =1 eix . (2.53)

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    CHAPTER 2. THE REFINABLE FUNCTION

    Also, using LHospital rule, we have

    limM

    2M(1 ei x2M )= limM

    1 eix2M

    2M = lim

    Me

    ix2M

    (ix)2M

    ( ln2)2M( ln2)

    = limM

    ixeix2M =i x,

    which, together with (2.53), gives, for x R\{0},

    j=1

    1 + ei x

    2j

    2

    = limM

    1 eix

    2M(1 ei x

    2M )=

    1 eix

    ix= ei

    x2

    sin

    x2

    x

    2

    , (2.54)and thus

    j=1

    1 + ei x

    2j

    2

    =

    j=1

    1 + ei x

    2j

    2

    =

    ei x

    2

    sin( x2 )x2

    , x R\{0},

    1, x = 0.

    . (2.55)SinceB is a Laurent polynomial for which (2.8) holds on the unit circle in C, there exists a positive

    number such that

    |B(eix )| , x . (2.56)

    Also, using (2.7), it can be shown, as in the argument that yielded (2.27), that there exists a real

    numberq > 0 such that

    |1 B(ei x2j )| q|x|2j

    , x R, j N,

    and thus

    |1 B(ei x

    2j )| q

    2j, |x| , j N. (2.57)

    If the positive number j0is chosen to satisfy

    2j0 qe, (2.58)

    then, using (2.57), we obtain

    |1 B(ei x

    2j )| e1, |x| , j j0. (2.59)

    From (2.56),

    j=1

    B(ei x

    2j )

    =j01j=1

    |B(ei x

    2j )|

    j=j0

    |B(ei x

    2j )| j01

    j=j0

    1 |1 B(e

    i x2j )|

    . (2.60)

    We now make use of the inequality

    1 teet, 0te1,

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    CHAPTER 2. THE REFINABLE FUNCTION

    in (2.60) to obtain, for |x| , and keeping in mind also (2.59),

    j=j0

    1 |1 B(e

    i

    x

    2j )|

    j=j0

    exp

    e|1 B(e

    i

    x

    2j )|

    = exp

    e

    j=j0

    |1 B(ei x

    2j )|

    exp

    eq

    j=j0

    1

    2j

    , (2.61)by virtue of (2.57). But

    j=j0

    1

    2j

    = 1

    2j0

    j=0

    1

    2j

    =21j0 ,

    which, together with (2.61) and (2.58), gives

    j=j0

    1 |1 B(e

    i x2j )|

    e2

    1j0 qe e2. (2.62)

    Now combine (2.52), (2.55), (2.60), (2.62), and the inequality

    | sin

    x

    2

    |

    |x2

    |

    2

    , |x| ,

    to deduce that (2.51) does indeed hold, with

    =

    2

    j01e2. (2.63)

    Recall the functionGras given by (2.37), for which, according to (2.45), we have

    1

    2

    Gr(x)ei2rjxdx =

    1

    2

    k

    r,k

    eix(2rjk)dx = r,2rj, j Z, r N. (2.64)

    Combining (2.43) and (2.64) then gives

    1

    2

    Gr(x)ei2rjx dx = j, j Z, r N. (2.65)

    But, according to (2.37), and (2.10), we have for j Z,r N, that

    1

    2

    Gr(x)ei2rjxdx =

    1

    2

    rl=1

    A(eix2l1

    )eix2rjxdx

    = 1

    2

    2r2r

    1

    2r

    rl=1

    A(ei x

    2rl+1 )eix jdx

    = 1

    2

    2r2r

    1

    2r

    rl=1

    A(ei x

    2l )eix jdx

    = 12

    2

    r

    2r

    r(x)eix jdx. (2.66)

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    CHAPTER 2. THE REFINABLE FUNCTION

    We now introduce the notation

    [a,b](x)= 1, x [a, b],0, x [a, b].

    It follows from (2.66) and (2.65) that

    1

    2

    r(x)[2r,2r](x)eix jdx = j, j Z, r N. (2.67)

    We now use (2.34), (2.35) to deduce that, for |x| 2rand j Z,r N, we have

    r(x)[2r,2r](x)eix j = |r(x)| = |g(x)|

    g x2r=

    g(x)

    g

    x2r 1g(x), (2.68)

    as a result of (2.51). It is also clear from (2.30) that

    limr

    r(x)[2r,2r](x)e

    ix j

    = g(x)ei jx, x R, j Z. (2.69)

    Thus, sinceg L1(R), the equations (2.68) and (2.69) imply that the Lebesgue dominated conver-

    gence theorem [9, pp 592] can be applied to the integral in (2.67) to deduce that, for j Z,

    1

    2

    ei jxg(x)dx = 1

    2

    limr

    r(x)

    [2r

    ,2r

    ](x)eix j dx

    = limr

    1

    2

    r(x)[2r,2r](x)eix jdx

    = j. (2.70)

    The result (1.2) then follows as an immediate consequence of (2.70) and (2.49).

    We proceed to prove that (x) R. To this end, first note from (1.1) and (1.2) that

    k

    2

    =

    j

    aj(k j)=

    j

    ajkj =ak, k Z,

    i.e.

    k2

    R, k Z.Similarly, the refinement equation (1.1) then yields

    k

    4

    =

    j

    aj

    k

    2 j

    R, k Z.

    Repeated use of this procedure shows that (x) R, x { k2r

    : k Z, r Z+}. Since the dyadic set

    is dense in R, and is continuous on R, we deduce that indeed M(R) and thus also C(R).

    We proceed to prove the finite support property (2.50) of. To this end, we first show that

    j

    2k

    = 0, j Z, |j| > 2k(2m 1), k=0, 1, 2, ... (2.71)

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    CHAPTER 2. THE REFINABLE FUNCTION

    From (1.2), we have that (2.71) holds for k=0, sincem is a positive integer. For the inductive step

    fromktok+ 1, suppose that (2.71) holds, and let |j|> 2k+1(2m 1). Then, using (1.1), and the fact

    thataj =0, j {2m + 1, ..., 2m 1}, we get

    j

    2k+1

    =

    l

    al

    j

    2k l

    =

    |l|2m+1

    al

    j 2kl

    2k

    = 0,

    since

    |j 2kl| |j| 2k|l| 2k+1(2m 1) 2k(2m 1)= 2k(2m 1),

    hence proving (2.71) for allk {0, 1, ...}.

    Suppose now thatx [2m+1, 2m1], that is, |x| 2m1. Since the dyadic set { j

    2k : j Z, k N}

    is dense in R, we know that there exist a sequence {jk : k N} such that jk2k

    x, k . Since

    |x| > 2m 1, there exist a positive integer Ksuch that jk

    2k

    > 2m 1, k K. Using also the factthat C(R), so that is continuous at x, we deduce from (2.71) that

    |(x)|

    (x)

    jk

    2k

    =

    (x)

    jk

    2k

    0, k ,

    and thus(x)= 0, x [2m + 1, 2m 1]. By continuity, we then also have

    (2m + 1)= (2m 1)= 0,

    thereby completing our proof of (2.50).

    We have shown in Theorem 2.2 and Theorem 2.3 that, if a M0(Z) is a mask satisfying the

    conditions of Theorem 2.2, then there exists a function C0(R) such that (a, ) is an interpolatory

    refinement pair.

    Definition: For a given sequencea M0(Z) we define the operatorSa : M(Z) M(Z) by

    (Sac)j :=

    k

    aj2kck, j Z, c M(Z). (2.72)

    The operatorSa is known as thesubdivision operator[8].

    We proceed to prove the following additional properties of the refinable function.

    Theorem 2.4 For m N, let (a, ) denote the interpolatory refinement pair of Theorems 2.2

    and 2.3, and suppose that the mask a satisfies the condition (1.19). Then the following relations

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    CHAPTER 2. THE REFINABLE FUNCTION

    hold:

    j

    p(j)(x j)= p(x), x R, p 2

    m1

    ; (2.73)

    (x)= (x), x R; (2.74)

    j

    2

    = aj, j Z. (2.75)

    Proof.

    (a) To prove (2.73), we supposel Z2m+1, k Z, andr Z+, and prove that

    j

    jl k

    2r

    j = k

    2r

    l

    , (2.76)

    which then implies (2.73), since the set { k2r

    : k Z, r = 0, 1, ...} is dense in R, and since is a

    finitely supported continuous function on R.

    Ifr = 0, then (2.76) is an immediate consequence of (1.2). Ifr 1, we consecutively use (1.1),

    (1.11) and (1.2), to deduce, that (2.76) holds, as in the steps leading from (1.17) to (1.18).

    (b) Property (2.74) will be proved if we can show that, for k Z andr Z+, it holds that

    k2r

    =

    k

    2r

    . (2.77)

    Forr=0, (2.77) follows from (1.2), whereas forr1, it follows from (1.1), (1.2) and (1.28) that,

    fork Z,r Z+,

    k

    2r

    =

    j

    aj

    k

    2r1 j

    =

    j

    aj

    k

    2r1 + j

    =

    j

    aj

    l

    al

    k

    2r2 + 2j l

    =

    l

    j

    al2jaj

    k

    2r2 + l

    =

    l

    (Saa)l

    k

    2r2 + l

    .

    .

    .

    =

    l

    (Sr1a a)l(k+ l) = (Sr1a a)k, (2.78)

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    CHAPTER 2. THE REFINABLE FUNCTION

    from (1.2). In a similar manner, it can be shown that

    k2r

    =(Sr1a a)k, k Z, r Z+, (2.79)

    which, together with (2.78), proves (2.77) for r1.

    (c) The property (2.75) is an immediate consequence of (1.2) and (1.1).

    We see from Theorems 2.1 to 2.4 that, in order to prove that, for m N, there exists a function

    m C0(R) such that, for the DD mask a = am given by (1.29), we have that (am, m) is an

    interpolatory refinement pair, we shall first have to prove that the corresponding mask symbol

    A= Am, as given by (1.30), satisfies the positivity condition (2.5) of Theorem 2.2.

    We shall rely on the following result.

    Proposition 2.5 [1] For m N, suppose a M0(Z)is a mask that satisfies (1.4), (1.19), and the

    bottom line of (1.29). Then the corresponding mask symbol A defined in (1.3) satisfies

    A(l)(1) = 0, l =0, 1, ..., 2m 1; (2.80)

    A(l)(1) = 0, l =1, 2, ..., 2m 1. (2.81)

    Proof.

    Since (1.3) and (1.4) give

    A(z) = 1 +

    k

    a2k+1z2k+1, z C\{0}, (2.82)

    we see thatA(l)(1)= l+ (1)

    l+1

    k

    hl(2k+ 1)a2k+1, l Z2m1,

    where, for x R,

    h0(x)= 1, hl(x)=

    jZl1

    (x j), l = 1, 2, ..., 2m 1. (2.83)

    Observe thathl l, l Z2m1. Hence, if we define

    pl =hl(2 +1 + 2l), l Z2m1,

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    CHAPTER 2. THE REFINABLE FUNCTION

    that is,

    hl = pl 1 2

    + l , l Z2

    m1

    ,

    then also pl l, l Z2m1. Moreover,

    A(l)(1)= l+ (1)l+1

    k

    pl(l k)a2k+1, l Z2m1, (2.84)

    and

    pl(l + 12

    ) = hl(0)= l, l Z2m1, (2.85)

    from (2.83).

    Sincepl l 2m1, l Z2m1, so that also the polynomialq defined byql = pl(l + ) also belongs

    to2m1for l Z2m1, we have from (2.84), (2.85) and (2.16) that,

    A(l)(1) = l+ (1)l+1

    k

    pla12kql(k)

    = l+ (1)l+1ql(

    12

    )

    = l+ (1)l+1pl(l +

    12

    ) = l[1 + (1)l+1] = 0,

    thereby proving that (2.80) holds.

    To prove (2.81), we follow the same procedure as was used in the steps leading from (2.82) to

    (2.85), to obtain, forl {1, 2, ..., 2m 1},

    A(l)(1)= l+ pl(l + 12

    )=2l =0.

    Proposition (2.5) now enables us to prove the following result.

    Theorem 2.6 For m N, the Dubuc-Deslauriers mask symbol A = Am satisfies the positivity

    condition (2.5) of Theorem 2.2.

    Proof.

    Since the DD maska = am M0(Z) satisfies (1.4) and (1.19), we know that, writing Em = Am,

    withAmas defined by (1.30) and (1.29), we have

    E(l)

    m(1)= E(l)

    m(1)= 0, l Z2m2. (2.86)

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    CHAPTER 2. THE REFINABLE FUNCTION

    Also, since (1.30) and (2.1) yield

    Am(z)=

    2m1j=2m+1

    amjzj, z C\{0},

    we find that the Laurent polynomialEm has the form

    Em(z) = 1

    z2m[e2m+ e2m+1z + ... + e2z

    2m2 + e0z2m + ... + e2m2z

    4m2], z C\{0}. (2.87)

    Combining (2.86) and (2.87), we conclude that

    Em(z)= c

    z2m

    (1 z)2m1(1 +z)2m1, z C\{0},

    for some constantc, and thus

    ieixAm(eix)=

    cieix

    e2mix(1 eix)2m1(1 + eix )2m1, x R. (2.88)

    It follows from (2.88) that

    Am(eix ) Am(e

    i)= ci

    x

    e(12m)i(1 ei)2m1(1 + ei)2m1d. (2.89)

    But, for R, we have

    (1 ei)2m1(1 + ei)2m1 = (1 e2i)2m1

    = [ei(ei ei)]2m1

    = (2i)2m1e(2m1)i

    ei ei

    2i

    2m1

    = (1)m22m1ie(2m1)i(sin )2m1. (2.90)

    We see from (2.89) and (2.90) that, since (2.82) gives Am(1)= 0, we have, for x R, that

    Am(eix ) = ci

    x

    (1)m22m1i(sin )2m1d = c(1)m122m1 x

    (sin )2m1d. (2.91)

    We also have from (2.82) that Am(1)=2, so that by setting x =2in (2.91), we obtain

    2=c(1)m122m1 2

    (sin )2m1d. (2.92)

    Combining (2.91) and (2.92) we then get the formula

    Am(eix)=2 x

    (sin )2m1d 2

    (sin )2m1d, x R. (2.93)

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    CHAPTER 2. THE REFINABLE FUNCTION

    We see that, for x [0, ), we have (sin )2m1 >0, x < , so that

    x

    (sin )2m1d=

    x

    (sin )2m1d 0, x < , and then

    x

    (sin )2m1d =

    x

    (sin )2m1d =

    xx

    (sin )2m1d

    x

    (sin )2m1d

    =

    x

    (sin )2m1d

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    CHAPTER 2. THE REFINABLE FUNCTION

    with Sa =Sam denoting the subdivision operator defined in (2.72), with a = am and where

    ={j : j Z}.

    Proof.

    In this proof, we write for m anda for am. Using (1.1), (2.72) and (1.2), we obtain, for j Z,

    r Z+,

    j

    2r

    =

    k

    ak

    j

    2r1 k

    =

    k

    (Sa)k

    j

    2r1 k

    =

    k(Sa)k

    l

    al j

    2r2 2k l

    =

    k

    (Sa)k

    l

    al2k

    j

    2r2 l

    =

    l

    k

    al2k(Sa)k

    j

    2r2 l

    =

    l

    (S2a)l

    j

    2r2 l

    .

    .

    .

    =

    l

    (Sra)l(j l) = (Sra)j.

    The graphs of the Dubuc-Deslauriers refinable function of orders 1, 2 and 3 are shown in figures

    2.1, 2.2 and 2.3 respectively.

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    CHAPTER 2. THE REFINABLE FUNCTION

    1.5 1 0.5 0 0.5 1 1.50

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1

    Figure 2.1: 1, the DD refinable function of order 1.

    3 2 1 0 1 2 30.2

    0

    0.2

    0.4

    0.6

    0.8

    1

    1.2

    Figure 2.2: 2, the DD refinable function of order 2.

    5 4 3 2 1 0 1 2 3 4 50.2

    0

    0.2

    0.4

    0.6

    0.8

    1

    1.2

    Figure 2.3: 3, the DD refinable function of order 3.

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    3The interpolation wavelet

    We now proceed to construct, from the interpolatory DD refinement pair (am, m) of Chapters

    1 and 2, the Dubuc-Deslauriers interpolation waveletm and the corresponding decomposition

    algorithm.

    The fundamental decomposition result

    Definition: Form N we define, according to Proposition 1.2, as in (1.7), the Dubuc-Deslauriers

    refinement spaces as

    V(r)m =span{m(2r j) : j Z}, r Z, (3.1)

    which satisfy the nesting property

    V(r)m V(r+1)m , r Z. (3.2)

    The following result follows from the interpolatory property (1.2) of = m.

    Proposition 3.1 For m N and r Z, we have f V(r)m if and only if

    f =

    j

    f

    j

    2r

    m(2

    r j). (3.3)

    Proof.

    From the definition (3.1), we see that f V(r)m if and only if there exists a sequencec M(Z) such

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    CHAPTER 3. THE INTERPOLATION WAVELET

    that

    f = k

    ckm(2r k).

    Hence, for j, r Z, we have

    f

    j

    2r

    =

    k

    ckm(j k)= cj,

    thereby completing our proof.

    Next we introduce, form N, r Z, the operatorPm,r :V(r+1)m V

    (r)m by the definition

    Pm,rf =

    jc2jm(2

    r j) for f =

    jcjm(2

    r+1 j), (3.4)

    which, according to Proposition 1.2, is equivalent to the definition

    Pm,rf =

    j

    f

    j

    2r

    m(2

    r j), f V(r+1)m . (3.5)

    Since the operatorQN : M(R) V(N)m defined by (1.11), with = m, coincides withPm,Non V

    (N)m ,

    we have from Proposition 1.3, (1.10), and Corollary 1.5, that

    (Pm,rf)

    j2r

    = f

    j2r

    , j, r Z, f V(r+1)m ,

    Pm,rf = f, f 2m1, (3.6)

    and

    2m1 V(r)m , r Z. (3.7)

    Our following result shows that the polynomial reproduction property (3.6) can be strengthened as

    follows.

    Theorem 3.2 For m N, r Z, the interpolation operator Pm,r : V(r+1)m V

    (r)m , as defined by

    (3.5), is a projection on V(r)m in the sense that

    Pm,rf = f, f V(r)m , r Z. (3.8)

    Proof.

    Suppose f V(r)m , so that, from Proposition 3.1, we have that (3.3) holds. But then (3.8) is an

    immediate consequence of the definition (3.5).

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    CHAPTER 3. THE INTERPOLATION WAVELET

    Next, for r Z, we let f V(r+1)

    m , so that, from (3.5), we obtain, in the simplified notation

    Pr= Pm,r, = m and a =am,

    f = Prf + (f Prf)=

    j

    f

    j

    2r

    (2r j) + (f Prf),

    where from Proposition 3.1, definition (3.5), the refinement equation (1.1), and (1.4), we have that

    f Prf =

    k

    f

    k

    2r+1

    (2r+1 k)

    j

    f

    j

    2r

    (2r j)

    =

    k

    f

    k2r+1

    (2r+1 k)

    j

    f

    j2r

    k

    ak(2r+1 2j k)

    =

    k

    f

    k

    2r+1

    (2r+1 k)

    j

    f

    j

    2r

    k

    ak2j(2r+1 k)

    =

    k

    f

    k

    2r+1

    j

    ak2jf

    j

    2r

    (2r+1 k)

    =

    k

    f

    k

    2r

    j

    a2k2jf

    j

    2r

    (2r+1 2k)

    +

    k

    f

    2k+ 1

    2r+1

    j

    a2k+12jf

    j

    2r

    (2r+1 2k 1)

    =

    k

    f

    2k+ 1

    2r+1

    j

    a2k+12jf

    j

    2r

    (2(2r k) 1)

    =

    k

    f

    2k+ 1

    2r+1

    j

    a2k+12jf

    j

    2r

    (2r k),

    where = (2 1).

    We have therefore now proved the following fundamental decomposition result.

    Theorem 3.3 For m N, r Z, we have

    f =

    j

    f

    j

    2r

    m(2

    r j) +

    j

    dm,r

    j m(2

    r j), f V(r+1)m , (3.9)

    where the functionm V(1)m is defined by

    m =m(2 1), (3.10)

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    CHAPTER 3. THE INTERPOLATION WAVELET

    and where the coefficient sequence dm,r M(Z)is given by

    dm,rj

    = f

    2j + 12r+1

    k

    am2j+12kf

    k2r

    , j Z. (3.11)

    Observe in particular that from (3.10) and (2.50) that

    m(x)= 0, x (m + 1, m). (3.12)

    Also, (3.10) and (1.2) yield

    m(j)=0, j Z. (3.13)

    Now define the linear space sequence {W(r)m :r Z} by

    W(r)m =span{m(2r j) : j Z}, r Z. (3.14)

    Note in particular thatW(r)m V

    (r+1)m , r Z. We see from Theorem 3.3, together with the definitions

    (3.1) and (3.14), that, for r Z, every function fr+1 V(r+1)m can be decomposed in the form

    fr+1 = fr + gr, where fr V(r)m and gr W

    (r)m . The function m V

    (1)m , as defined by (3.10),

    which generates the linear space W(r)m by means of the definition (3.14), is called awaveletand in

    particular here theDD interpolation wavelet of order m, for which the two cases m = 2 andm = 3

    are shown in figures 3.1a and 3.1b.

    3 2 1 0 1 2 30.2

    0

    0.2

    0.4

    0.6

    0.8

    1

    1.2

    (a)2

    2 1 0 1 2 30.2

    0

    0.2

    0.4

    0.6

    0.8

    1

    1.2

    (b)3

    Figure 3.1: DD interpolation wavelets of order 2 and 3.

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    CHAPTER 3. THE INTERPOLATION WAVELET

    The spaces W(r)m : r Z are known as the DD wavelet spaces of order m, whereas, for r Z,

    the sequence{dm,r

    j : j Z} M(Z) in (3.9) is known as the mth order DD wavelet decomposition

    coefficient sequence at resolution level r. The following result, in conjunction with (3.7), is of

    fundamental importance.

    Theorem 3.4 In Theorem 3.3, suppose that, for j, r Z, we have

    f[

    j m+12r

    ,j +m

    2r ]

    2m1. (3.15)

    Then dm,r

    j =0.

    Proof.

    Let j, r Z be given. According to (3.15), there then exists a polynomial p 2m1such that

    f(x)= p(x), j m + 1

    2r x

    j + m

    2r .

    But then, from (3.11), together with bottom line of (1.29) we have

    dm,r

    j = f

    j + 12

    2r

    j+mk=jm+1

    am2j+12kf

    k

    2r

    = p

    j + 12

    2r

    j+mk=jm+1

    am2j+12kp

    k

    2r

    . (3.16)

    With the polynomialq defined by

    q(x)= p

    x

    2r

    , x R,

    we have that q also belongs to 2m1. It follows from (3.16), together with the property (1.19) of

    the DD maska =am, that

    dm,r

    j = q

    j + 1

    2

    j+mk=jm+1

    a2j+12kq(k)

    = qj + 12

    k

    a2j+12kq(k)

    = qj + 1

    2

    q

    j + 1

    2

    = 0.

    Now observe also that (3.12) implies the finite support property

    m(2rx j)=0, x

    j m + 1

    2r ,

    j + m

    2r

    . (3.17)

    We deduce from (3.17) and Theorem 3.4 that, if for, j, r Z, a function f V(r+1)m locally equals

    a polynomial in 2m1 in the support interval jm+12

    r , j+m

    2r of the wavelet m(2r j), then the

    corresponding wavelet coefficientdm,rj

    equals zero.

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    CHAPTER 3. THE INTERPOLATION WAVELET

    Decomposition algorithm

    Our DD interpolation wavelet decomposition technique is as follows.

    Suppose f M(R) is a given signal to be analyzed. First, we use the interpolation operator

    QN = Qm,N : M(R) V(N)m , as defined by (1.8), to define fN = QNf V

    (N)m , and with N chosen

    sufficiently large to adequately capture the features of the signal f. Note in particular from Proposi-

    tion 1.3 that fNinterpolates fon the set { j

    2N : j Z}. The polynomial reproduction property (1.10)

    ofQNensures that local polynomial behaviour of fis preserved by fN = QNf. Here, it should also

    be kept in mind that, from Taylors theorem, if f is locallyCk-smooth for some positive integerk,

    then f is approximated well by the corresponding Taylor polynomial ink1, and one can therefore

    expect the wavelet coefficientdm,rj

    corresponding to the waveletm(2r j) with support entirely

    within the region of smoothness, to be relatively small.

    Once the approximation fN V(N)m is obtained, we can now proceed to the decomposition phase.

    The DD decomposition algorithm is as follows.

    Let the sequence {fr :r= N,N 1, ...,NM} be defined by

    fr=

    j

    f

    j

    2r

    m(2

    r j), r= N,N 1, ...,NM, (3.18)

    so that every successive approximation frV(r)m to fis interpolatory in the sense that

    fr

    j

    2r

    = f

    j

    2r

    , j Z, (3.19)

    i.e. frinterpolates f at increasingly coarse dyadic levels for r = N,N1, ...,N M. Here, the

    integer Mis chosen large enough so that the final approximation fNMis sufficiently blurred.

    But then, according to Theorem 3.3, we have

    fr+1 = fr+ gr, r= N 1,N 2, ...,N M, (3.20)

    where

    gr= j

    dm,r

    j m(2

    r j), r= N 1,N 2, ...,NM, (3.21)

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    CHAPTER 3. THE INTERPOLATION WAVELET

    with

    dm,rj = f

    2j + 12r+1

    k

    am2j+12kf

    k2r

    , j Z, r= N 1,N 2, ...,N M. (3.22)

    It follows from (3.20) that

    fN = fNM+

    N1r=NM

    gr,

    where the wavelet components {gr :r= N 1,N 2, ...,NM} can be interpreted as representing

    the detail components of fN at successively coarse resolution levels r. In particular, according

    to Theorem 3.3, every function grwill provide localized information on the smoothness (or lack

    thereof) of f, as represented by the approximation fN.

    Application

    In order to illustrate the effectiveness of the DD decomposition algorithm derived earlier in this

    chapter, we choose the signal f M0(R) defined by

    f(x)=

    1

    2x

    2

    , x [0, 1),12

    (2x2 + 6x 3), x [1, 2),

    12

    (3 x)2, x [2, 3),

    0. x [0, 3).

    (3.23)

    Our signal f is the cardinal B-spline N3 of order 3, (see e.g [2], Chapter 4). Note from (3.23)

    that f C1(R)\C2(R). In fact, fconsists of quadratic polynomial pieces, with breakpoints at

    x = 0, 1, 2, 3. In particular, the second derivative f has jump discontinuities at these points.

    Hence, form = 2, if we use the DD wavelet2, as given by (3.10), together with the decomposition

    algorithm (3.20), (3.21), (3.22) and (1.31), and the approximation operator Q2,N : M(R) V(N)

    2 ,

    defined by (1.8) to decompose the signal f, we deduce from Theorem 3.3 that, at each resolution

    levelr, it can be expected that the wavelet component grwill have peaks at x = 0, 1, 2, 3, and be

    zero in regions bounded away from these points, since the wavelet coefficientd2,r

    j = 0 whenever

    the support

    j1

    2r ,

    j+2

    2r

    of2(2

    r j) does not overlap with such a breakpoint. Our expectations are

    indeed fulfilled by the graphs in figures 3.3, 3.4 and 3.5, where we chose N =8 and M=3. Note also

    that fNlooks the same to the human eye as our signal fin figure 3.2.

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    CHAPTER 3. THE INTERPOLATION WAVELET

    1 0.5 0 0.5 1 1.5 2 2.5 3 3.5 40.2

    0

    0.2

    0.4

    0.6

    0.8

    f

    Figure 3.2: Signal f.

    1 0.5 0 0.5 1 1.5 2 2.5 3 3.5 40.2

    0

    0.2

    0.4

    0.6

    0.8

    (a) fN1

    1 0.5 0 0.5 1 1.5 2 2.5 3 3.5 41.5

    1

    0.5

    0

    0.5

    1

    1.5x 10

    5

    (b)gN1

    Figure 3.3: First resolution level.

    Observe that the wavelets component gN1 gives the best localization of the discontinuity in the

    second derivative f. This is due to (3.17) and (3.21), together with the comment immediately

    after (3.17). Observe from figure 3.2 that the discontinuity in the second derivative of our signal f

    is not visible to the human eye. However, our wavelet 2sees it.

    Conclusion

    In this essay, we provided a constructive existence proof of the interpolatory Dubuc-Deslauriers

    refinement pair (am, m) of order m, from which we proceeded to construct the corresponding

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    CHAPTER 3. THE INTERPOLATION WAVELET

    1 0.5 0 0.5 1 1.5 2 2.5 3 3.5 40.2

    0

    0.2

    0.4

    0.6

    0.8

    (a) fN2

    1 0.5 0 0.5 1 1.5 2 2.5 3 3.5 41.5

    1

    0.5

    0

    0.5

    1

    1.5x 10

    4

    (b)gN2

    Figure 3.4: Second resolution level.

    1 0.5 0 0.5 1 1.5 2 2.5 3 3.5 40.2

    0

    0.2

    0.4

    0.6

    0.8

    (a) fN3

    1 0.5 0 0.5 1 1.5 2 2.5 3 3.5 45

    4

    3

    2

    1

    0

    1

    2

    3

    4

    5x 10

    4

    (b)gN3

    Figure 3.5: Third resolution level.

    DD interpolation wavelet and decomposition algorithm. Apart from the cardinal B-spline function

    which was used to illustrate the effectiveness of our algorithm, this decomposition algorithm can be

    used in signal analysis to analyse other functions of interest. This essay altogether can adequately

    form a basis for a more substantial study in further research.

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    References

    [1] C.A. Micchelli, Interpolatory Subdivision Schemes and Wavelets,Journal of Approximation,86 (1996), 41-71.

    [2] C.K. Chui, An Introduction to Wavelet Analysis,Academy Press, Boston, 1992.

    [3] G. Deslauriers and S. Dubuc, Symmetric Iterative Interpolation Processes, Constructive Ap-

    proximations, 5 (1989), 49-68.

    [4] J.M. De Villiers, K.M. Goosen & B.M. Herbst, Dubuc-Deslauriers Subdivision for Finite

    Sequence and Interpolation Wavelets on an Interval,SIAM Journal on Mathematical Analysis

    and Applications, 35 (2003), 423-452.

    [5] J.M. De Villiers, Subdivision, Wavelets and Splines, Lecture notes, Department of Mathe-

    matics, University of Stellenbosch, 2005.

    [6] J.M. De Villiers, Introduction to Wavelet Analysis,AIMS review course lecture notes, 2006.

    [7] K.M. Goosen, Subdivision, Interpolation and Splines, Masters thesis, University of Stellen-bosch, 2000.

    [8] K.M. Hunter, Interpolatory Refinable Functions, Subdivisions and Wavelets, Ph.D thesis,

    University of Stellenbosch, 2005.

    [9] S. Mallat, A Wavelet Tour of Signal Processing, 2nd edition,Academy Press, London, 1999.