7/31/2019 Wavelets 1
1/26
Wavelet basicsHennie ter Morsche
1. Introduction
2. The continuous/discrete wavelet transform
3. Multiresolution analysis
4. Scaling functions
5. The Fast Wavelet Transform
6. Examples
1
7/31/2019 Wavelets 1
2/26
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 )
0 0.5 1 1.5 2 2.5 3 3.5 41
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
TIJD0 5 10 15
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
HERTZ
Figure 1: The frequency break and its amplitudespectrum
2
7/31/2019 Wavelets 1
3/26
The short time Fourier transform
Given a Window function g
g
L2(IR) , g =1 g is realvalued.
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.
3
7/31/2019 Wavelets 1
4/26
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 ),
4
7/31/2019 Wavelets 1
5/26
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.
5
7/31/2019 Wavelets 1
6/26
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
0.2
0
0.2
0.4
0.6
0.8
1
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
0.15
0.2
Hertz
Figure 2: The Mexican hat
6
7/31/2019 Wavelets 1
7/26
The wavelet transform of the frequency break using the Mexican hat
0 0.5 1 1.5 2 2.5 3 3.5 41
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
TIJD0 5 10 15
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
HERTZ
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 baxis contains 1000 samples on interval [0, 1].The vertical axis contains the a values: 2 , 4, . . . , 128.
7
7/31/2019 Wavelets 1
8/26
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)
8
7/31/2019 Wavelets 1
9/26
For orthogonal wavelets the reconstruction formula and the decomposition 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 (numerically 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 implemented in the MATLAB Toolbox
9
7/31/2019 Wavelets 1
10/26
3. Multiresolution 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
7/31/2019 Wavelets 1
11/26
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 MultiResolutionAnalysis (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 )
11
7/31/2019 Wavelets 1
12/26
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 ) 2scale relation .
2. The socalled 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 .
12
7/31/2019 Wavelets 1
13/26
Examples
Bsplines 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)
13
7/31/2019 Wavelets 1
14/26
Based on a given MRA with scaling function one may construct 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).
14
7/31/2019 Wavelets 1
15/26
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 symbol Q ( z) = k qk zk for the wavelet , the application of theFouriertransform to the previous equations and the 2scale relation 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,
15
7/31/2019 Wavelets 1
16/26
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, .
16
7/31/2019 Wavelets 1
17/26
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 function one may choose
qk = (1)k p1k .
17
7/31/2019 Wavelets 1
18/26
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 ,
18
7/31/2019 Wavelets 1
19/26
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) .
19
7/31/2019 Wavelets 1
20/26
A repeated application of the previous operation leads to a decomposition 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.
20
7/31/2019 Wavelets 1
21/26
ReconstructionIf a 1 and d 1 are given then we may reconstruct the approximation 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
21
7/31/2019 Wavelets 1
22/26
a( 1)
d ( 1)
6
6


Lo_r
Hi_r
?
6
a( )
Figure 6: Reconstruction
22
7/31/2019 Wavelets 1
23/26
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
23
7/31/2019 Wavelets 1
24/26
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
24
7/31/2019 Wavelets 1
25/26
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
25
7/31/2019 Wavelets 1
26/26
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
26