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    Wavelet basicsHennie ter Morsche

    1. Introduction

    2. The continuous/discrete wavelet transform

    3. Multi-resolution analysis

    4. Scaling functions

    5. The Fast Wavelet Transform

    6. Examples

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    1. Introduction

    For a given univariate function f , the Fourier transform of f and the inverse are given by

    f () = f ( t )ei t dt . f ( t ) =

    12 f () e i t d .

    Parseval: ( f , g ) = ( f , g)/ 2 , ( f , g ) =

    f ( t ) g (t ) dt .

    e ( t )

    =ei t ,

    0()

    =(

    0)

    f ( 0) = ( f , e0) = ( f , 0 )

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    Figure 1: The frequency break and its amplitude-spectrum

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    The short time Fourier transform

    Given a Window function g

    g

    L2(IR) , g =1 g is real-valued.

    The short time Fourier transform F (u , ) of a function f isde ned by

    F (u , ) = f ( t )eiut g ( t ) dt , f ( t ) =

    12

    F (u , ) e iut g ( t ) d du ,

    gu, ( t ) :=e iut g ( t ) , F (u , ) = ( f , gu , )

    ( f , gu, ) =1

    2( f , gu, ) ( Parseval) .

    gu, () =ei (u) g ( u ).Fixed window width in time and frequency.

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    2. The continous/discrete Wavelet transform

    The continuous Wavelet transform

    Given in L2

    (IR) .Introduce a family of functions a ,b (a > 0, bIR ) as follows

    a ,b( t ) =1

    a (( t b)/ a ) ( t IR), a ,b = .

    The continuous wavelet transform F (a , b) of a function f isde ned by

    F (a , b) = ( f , a ,b) =1

    a f ( t )(( t b)/ a ) dt .( f , a ,b) =

    12

    ( f , a ,b) Parseval .where

    a ,b() = a e i b

    ( a ),

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    The inverse wavelet transform

    f ( t )

    =C 1

    0

    1

    a2

    F (a , b) a ,b(t ) da db .

    C = 0 |() |2

    d .

    Needed ( 0) =0, i.e.,

    ( t ) dt

    =0.

    This is the reason why the functions a ,b are called wavelets.

    is called the Motherwavelet.

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    Example : The Mexican hat (Morlet wavelet)

    ( t )

    =

    2

    3 14 (1

    t 2)et 2 / 2.

    5 4 3 2 1 0 1 2 3 4 50.4

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    0

    0.2

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    TIJD as2 1.5 1 0.5 0 0.5 1 1.5 2

    0.15

    0.1

    0.05

    0

    0.05

    0.1

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    0.2

    Hertz

    Figure 2: The Mexican hat

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    The wavelet transform of the frequency break using the Mexi-can hat

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    Figure 3: frequency break

    a s

    c h

    a a

    l

    100 200 300 400 500 600 700 800 900 1000

    2

    4

    16

    32

    64

    128

    Figure 4: Grey value picture of the waveletcof cinten

    Horizontal b-axis contains 1000 samples on interval [0, 1].The vertical axis contains the a -values: 2 , 4, . . . , 128.

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    The discrete wavelet transform

    Sampling in the a -b plane.

    a 0 > 1, b0 > 0a =a 0 , b =k a 0 b0, (k , ZZ ).The translation step is adapted to the scale

    k , ( t ) =a/ 2

    0 ( a 0 t k b0).Dyadic wavelets: a0 =2, b0 =1.

    k , ( t ) =2 / 2 ( 2 t k ).( f , k , ) are called waveletcoef cients .

    Discrete Wavelet transform: f ( f , k , )a. Problem of reconstruction:

    f = k , ( f , k , ) k , .b. Problem of decomposition:

    f = k , a k , k ,It would be nice if the functions k , constitute an orthonormalbasis of L2(IR) . (orthogonal wavelets)

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    For orthogonal wavelets the reconstruction formula and the de-composition formula coincide.A biorthogonal wavelets system consists of two sets of waveletsgenerated by a mother wavelet and a dual wavelet

    , for

    which

    ( k , , m,n ) =k ,m ,n ,for all integer values k , , m en n .We assume that ( k , ) constitute a so called Riesz basis (nu-merically stable) of L2(IR) , i.e.

    A ( f , f ) k ,

    k , 2 B ( f , f )

    for positive constants A en B , where f = k , k , k , .The reconstruction formula now reads

    f

    = k ,( f , k , ) k , .

    Examples of biorthogonal wavelets are the bior family imple-mented in the MATLAB Toolbox

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    3. Multi-resolution analysis

    For a given function f , let

    f = k =

    ( f , k , ) k , ,

    Then

    f =

    = f .

    f can be interpreted as that part of f which belongs to thescale .So, f = =f is a decomposition of f to different scalelevels .The function f belongs to the scale space W spanned by( k , ) with xed .

    The space W 0 is spanned by the integer translates of the motherwavelet .

    For integer n the function

    gn( t ) =n1

    = f ( t )

    contains all the information of f up to scale level n 1.So gn

    V n , where

    V n =n1

    =W .

    It follows that V n =V n1W n1 (nZZ ) direct sum.10

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    Properties of the sequence (V n )

    a) V n1V n (n geheel ) ,

    b)n

    ZZ V n =L2(IR) ,

    c)n

    ZZ V n = {0},

    d) f ( t )

    V nf (2t )

    V n+1 ,

    e) f ( t )

    V 0f ( t +1)V 0 .

    If a sequence of subspaces (V n ) satis es the properties a) to e),then it is called a Multi-Resolution-Analysis (MRA) of L2(IR) .

    If there exists a function such that V 0 is spanned by the in-

    teger translates of , then is called a scaling function for theMRA.As a consequence one has that V n is spanned by k ,n , (n xed),

    k ,n =2n/ 2 ( 2n t k )

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    4. Scaling functions

    Suf cient conditions for a compactly supported function tobe a scaling function for an MRA.

    1. There exists a sequence of numbers ( pk ) , from which only a nite number differs from zero, such that

    ( t ) =

    k = pk ( 2t k ) 2-scale relation .

    2. The so-called Riesz function has no zeros on the unit circle.

    Autocorrelation function of : () := ( t + ) ( t ) dt .Riesz function R( z) =

    m=( m) zm .

    3. Partition of the unity

    k

    ( t k ) 1.

    The Laurent polynomial P ( z) = 12 k pk zk is called the twoscale symbol of .

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    Examples

    B-splines of order m:

    P ( z)

    = z

    +1

    2

    m

    The Daubechies scaling function of order 2

    P2( z) =12

    1 + 34 +

    3 + 34

    z +3 3

    4 z2 +

    1 34

    z3 .

    For an orthonormal system one has

    R( z) 1,|P ( z)|2 + |P ( z)|2 1 (| z| =1)

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    Based on a given MRA with scaling function one may con-struct wavelets by rst completing the spaces V to a spaceV +1 by means of a space W , i.e.V +1 = V W in such away that there exists a function such that W is spanned by(( 2 t k )) .To satisfy V 1 =V 0W 0 the following conditions are necessaryand suf cient:

    1. W 0V 1,

    2. W 0 V 0 = {0},3. ( 2t )

    V 0

    W 0 and ( 2t 1)V 0W 0 .It follows that

    ( t ) =

    k =qk ( 2t k ),

    ( 2t ) =

    k

    =

    (a k ( t k ) +bk ( t k )) ( t IR),

    ( 2t 1) =

    k =(ck ( t k ) +d k ( t k )) ( t IR).

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    By introducing the Laurent series A( z) = k a k zk , B( z) =k bk z

    k , C ( z) = k ck zk and D ( z) = k d k zk and the sym-bol Q ( z) = k qk zk for the wavelet , the application of theFourier-transform to the previous equations and the 2-scale re-lation for the scaling function nally lead to the following setof equations, which must hold for complex z with | z| =1.

    A( z2) P ( z) + B( z2) Q ( z) =1/ 2, A( z2) P ( z) + B( z2) Q ( z) =1/ 2,C ( z2) P ( z) + D( z2) Q ( z) = z/ 2,C ( z2) P (

    z)

    + D( z2) Q (

    z)

    = z/ 2,

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    Now let (assuming the inverse exists)

    P ( z) Q ( z)

    P ( z) Q ( z)1

    =H ( z) H ( z)G ( z) G ( z)

    ,

    where

    H ( z) =k

    h k zk ,

    G ( z) =k

    gk zk .

    Then

    A( z2) = ( H ( z) + H ( z))/ 2, B( z2) = (G ( z) +G ( z))/ 2,C ( z2) = z ( H ( z) H ( z))/ 2, D( z2) = z (G ( z) G ( z))/ 2, .

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    We now have

    ( 2t

    k )

    =

    m=h 2m

    k ( t

    m)

    +g2m

    k ( t

    m) ( t

    IR).

    It can be shown that the symbol P ( z) for the dual scaling andthe symbol Q ( z) for the dual wavelet will satisfy

    P ( z) = H ( z1),Q ( z) =Q ( z1).

    For orthogonal wavelets based on an orthogonal scaling func-tion one may choose

    qk = (1)k p1k .

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    5. The Fast Wavelet Transform

    To obtain a wavelet decomposition of a function f in practice,one rst approximates f by a function from a space V n , which

    is close to f . So let us assume that f itself belongs to V n . So

    f =

    k =a k ,nk ,n

    Since V n =n1=W , one has

    f =n1

    =

    k =d k , k ,

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    V n =V n1W n1 implies

    f =

    k

    =

    a k ,n k ,n =

    k

    =

    a k ,n1k ,n1 +

    k

    =

    d k ,n1 k ,n1 .

    Due to

    k ,n =

    m= 2 h 2mk m,n1 + 2 g2mk m,n1 .

    we obtain

    f =

    k

    =

    a k ,n k ,n =

    k

    =

    a k ,n 2 ( m

    =

    (h 2mk m,n1+g2mk m,n1)).

    Our conclusion is

    a m,n1 =

    k = 2 h 2mk a k ,n , d m,n1 =

    k = 2 g2mk a k ,n .

    convolution and subsequently downsampling ( m 2 m) yieldsthe two sequences a (n1) = (a m,n1) en d (n1) = (d m,n1) .

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    A repeated application of the previous operation leads to a de-composition of f to coarser levels, which can be expressed bythe following scheme and ltering proces.

    a (n)-

    @ @

    @ Ra (n1)d (n1)

    - @

    @ @ R

    a (n2)d (n2)

    . . .-

    @ @

    @ Ra (n N )d (n N )

    -

    -

    -

    Lo_d

    Hi_d

    a (n)

    -

    -

    a (n1)

    d (n1)

    ?

    ?

    -

    -

    Figure 5: Decomposition

    Filter coef cients are 2 h k for the low pass lter and 2 gk for the high pass lter.

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    ReconstructionIf a 1 and d 1 are given then we may reconstruct the approx-imation coef cients a .

    f = f 1 +w 1

    f =

    k =a k , k ,

    =

    k =a k , 1k , 1 +

    k =d k , 1 k , 1

    =

    k =

    m=a

    k , 11

    2 p

    m

    2k +m,

    +

    k =

    m=d k , 1

    1 2 qm 2k +m, .

    Hence,

    k =a k , k ,

    = k =

    m=

    1 2 a k , 1 pm2k +d k , 1qm2k m, .

    Conclusion:

    a k , =1

    2

    m=(a m, 1 pk 2m +d m, 1qk 2m ).

    upsampling and subsequently convolution

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    -a( 1)

    -d ( 1)

    6

    6

    -

    -

    Lo_r

    Hi_r

    ?

    6

    -a( )

    Figure 6: Reconstruction

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    6. Examples

    1. Haar wavelet

    General characteristics:OrthogonalSupport width 1Filters length 2Number of vanishing moments for : 1Scaling function yes

    0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.5

    1

    0.5

    0

    0.5

    1

    1.5

    Figure 7: Haar wavelet

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    2. Daubechies family

    General characteristics:

    Order N

    =1, . . .

    OrthogonalSupport width 2 N 1Filters length 2 N Number of vanishing moments for N Scaling function yes

    0 2 4 6 8 0.4

    0.2

    0

    0.2

    0.4

    0.6

    0.8

    1

    1.2db4 : phi

    0 2 4 6 8 1

    0.5

    0

    0.5

    1

    1.5db4 : psi

    Figure 8: Daubechies order 4

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    3. Coi et family

    General characteristics:

    Order N

    =1, . . . , 5

    OrthogonalSupport width 6 N 1Filters length 6 N Symmetry near fromNumber of vanishing moments for 2 N

    0 5 10 15 20 25 0.2

    0

    0.2

    0.4

    0.6

    0.8

    1

    1.2coif4 : phi

    0 5 10 15 20 25 1

    0.5

    0

    0.5

    1

    1.5coif4 : psi

    Figure 9: Coi et order 4

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    Meyer wavelet

    General characteristics:

    OrthogonalCompact support noEffective support [-8, 8]Symmetry yesScaling function yes

    10 5 0 5 10 0.4

    0.2

    0

    0.2

    0.4

    0.6

    0.8

    1

    1.2Meyer scaling function

    10 5 0 5 10 1

    0.5

    0

    0.5

    1

    1.5Meyer wavelet function

    Figure 10: Meyer

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