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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 fand the inverse are given by
f (ω) =∫ ∞
−∞f (t)e−iωt dt .
f (t) = 1
2π
∫ ∞
−∞f (ω)eiωt dω.
Parseval: ( f, g) = ( f , g)/2π , ( f, g) = ∫∫f (t) g(t)dt .
eω(t) = e−iωt , δω0(ω) = δ(ω − ω0)
f (ω0) = ( f, eω0) = ( f , δω0)
0 0.5 1 1.5 2 2.5 3 3.5 4−1
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TIJD0 5 10 15
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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, τ ) of a function f isdefined by
F(u, τ ) =∫ ∞
−∞f (t)e−iut g(t − τ) dt,
f (t) = 1
2π
∫ ∞
−∞
∫ ∞
−∞F(u, τ )eiutg(t − τ) dτ du,
gu,τ (t) := eiut g(t − τ), F(u, τ ) = ( f, gu,τ )
( f, gu,τ ) = 1
2π( f , gu,τ ) ( Parseval).
gu,τ (ω) = e−i(ω−u)τ g(ω − u).
Fixed ” window width” in time and frequency.
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2. The continous/discrete Wavelet transform
The continuous Wavelet transform
Given ψ in L2(IR).Introduce a family of functionsψa,b (a > 0, b ∈ IR) as follows
ψa,b(t) = 1√aψ((t − b)/a) (t ∈ IR),
‖ψa,b‖ = ‖ψ‖.
The continuous wavelet transform F(a, b) of a function f isdefined by
F(a, b) = ( f, ψa,b) = 1√a
∫ ∞
−∞f (t) ψ((t − b)/a) dt .
( f, ψa,b) = 1
2π( f , ψa,b) Parseval.
where
ψa,b(ω) = √a e−iωbψ(aω),
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The inverse wavelet transform
f (t) = C−1ψ
∫ ∞
−∞
∫ ∞
0
1
a2F(a, b) ψa,b(t) da db.
Cψ =∫ ∞
0
|ψ(ω)|2ω
dω.
Needed ψ(0) = 0, i.e.,
∫ ∞
−∞ψ(t) dt = 0.
This is the reason why the functions ψa,b are called wavelets.
ψ is called the Motherwavelet.
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Example: The Mexican hat (Morlet wavelet)
ψ(t) = 2√3π− 1
4 (1 − t2)e−t2/2.
−5 −4 −3 −2 −1 0 1 2 3 4 5−0.4
−0.2
0
0.2
0.4
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TIJD−as−2 −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
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The wavelet transform of the frequency break using the Mexi-can hat
0 0.5 1 1.5 2 2.5 3 3.5 4−1
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Figure 3: frequency break
a s
chaal
100 200 300 400 500 600 700 800 900 1000
2
4
16
32
64
128
Figure 4: Grey value picture of the waveletcoëfficiënten
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.
a0 > 1, b0 > 0a = a−�
0 , b = k a−�0 b0, (k, � ∈ ZZ ).
The translation step is adapted to the scale
ψk,�(t) = a�/20 ψ(a�0t − k b0).
Dyadic wavelets: a0 = 2, b0 = 1.
ψk,�(t) = 2�/2ψ(2�t − k).
( f, ψk,�) are called waveletcoefficients.
Discrete Wavelet transform: f → ( f, ψk,�)
a. Problem of reconstruction:
f = ∑k,�( f, ψk,�)ψk,�.
b. Problem of decomposition:
f = ∑k,� ak,�ψk,�
It would be nice if the functions ψk,� 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 ψ and a dual wavelet ψ , forwhich
(ψk,�, ψm,n) = δk,mδ�,n,
for all integer values k, �,m en n.We assume that (ψk,�) constitute a so called Riesz basis (nu-merically stable) of L2(IR), i.e.
A ( f, f ) ≤ ‖∑k,�
ξk,�‖2 ≤ B ( f, f )
for positive constants A en B, where f = ∑k,� ξk,�ψk,�.
The reconstruction formula now reads
f =∑k,�
( f, ψk,�)ψk,�.
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, ψk,�)ψk,�,
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(ψk,�) with fixed �.
The space W0 is spanned by the integer translates of the motherwavelet ψ .
For integer n the function
gn(t) =n−1∑�=−∞
f�(t)
contains all the information of f up to scale level n − 1.So gn ∈ Vn , where
Vn =n−1∑�=−∞
W�.
It follows that Vn = Vn−1 ⊕ Wn−1 (n ∈ ZZ) direct sum.
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Properties of the sequence (Vn)
a) Vn−1 ⊂ Vn (n geheel),
b)⋃
n∈ZZVn = L2(IR),
c)⋂
n∈ZZVn = {0},
d) f (t) ∈ Vn ⇔ f (2t) ∈ Vn+1,
e) f (t) ∈ V0 ⇒ f (t + 1) ∈ V0.
If a sequence of subspaces (Vn) satisfies the properties a) to e),then it is called a Multi-Resolution-Analysis (MRA) of L2(IR).
If there exists a function φ such that V0 is spanned by the in-teger translates of φ, then φ is called a scaling function for theMRA.As a consequence one has that Vn is spanned by φk,n, (n fixed),
φk,n = 2n/2 φ(2n t − k)
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4. Scaling functions
Sufficient conditions for a compactly supported function φ tobe a scaling function for an MRA.
1. There exists a sequence of numbers (pk), from which only afinite number differs from zero, such that
φ(t) =∞∑
k=−∞pkφ(2t − k) 2-scale relation.
2. The so-called Riesz function has no zeros on the unit circle.
Autocorrelation function of φ: ρ(τ) := ∫ ∞−∞ φ(t + τ) φ(t) dt .
Riesz function
R(z) =∞∑
m=−∞ρ(m) zm.
3. Partition of the unity
∑k
φ(t − k) ≡ 1.
The Laurent polynomial P(z) = 12
∑k pk zk is called the two
scale symbol of φ.
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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) = 1
2
{1 + √
3
4+ 3 + √
3
4z + 3 − √
3
4z2 + 1 − √
3
4z3
}.
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 φ one may con-struct wavelets by first 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 ψ such that W� is spanned by(ψ(2� t − k)).To satisfy V1 = V0⊕W0 the following conditions are necessaryand sufficient:
1. W0 ⊂ V1,
2. W0⋂
V0 = {0},3. φ(2t) ∈ V0 ⊕ W0 and φ(2t − 1) ∈ V0 ⊕ W0.
It follows that
ψ(t) =∞∑
k=−∞qkφ(2t − k),
φ(2t) =∞∑
k=−∞(akφ(t − k)+ bkψ(t − k)) (t ∈ IR),
φ(2t − 1) =∞∑
k=−∞(ckφ(t − k)+ dkψ(t − k)) (t ∈ IR).
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By introducing the Laurent series A(z) = ∑k ak zk, B(z) =∑
k bk zk,C(z) = ∑k ck zk and D(z) = ∑
k dk zk and the sym-bol Q(z) = ∑
k qk zk for the wavelet ψ , the application of theFourier-transform to the previous equations and the 2-scale re-lation for the scaling function φ finally 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
hk 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
φ(2t−k) =∞∑
m=−∞
(h2m−kφ(t−m)+g2m−kψ(t−m)
)(t ∈ IR).
It can be shown that the symbol P(z) for the dual scaling φ andthe symbol Q(z) for the dual wavelet ψ will satisfy
P(z) = H (z−1),
Q(z) = Q(z−1).
For orthogonal wavelets based on an orthogonal scaling func-tion one may choose
qk = (−1)k p1−k.
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5. The Fast Wavelet Transform
To obtain a wavelet decomposition of a function f in practice,one first approximates f by a function from a space Vn, whichis close to f . So let us assume that f itself belongs to Vn. So
f =∞∑
k=−∞ak,nφk,n
Since Vn = ∑n−1�=−∞ W�, one has
f =n−1∑�=−∞
∞∑k=−∞
dk,�ψk,�
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Vn = Vn−1 ⊕ Wn−1 implies
f =∞∑
k=−∞ak,nφk,n =
∞∑k=−∞
ak,n−1φk,n−1+∞∑
k=−∞dk,n−1ψk,n−1.
Due to
φk,n =∞∑
m=−∞
√2 h2m−kφm,n−1 + √
2 g2m−kψm,n−1.
we obtain
f =∞∑
k=−∞ak,nφk,n =
∞∑k=−∞
ak,n
√2 (
∞∑m=−∞
(h2m−kφm,n−1+g2m−kψm,n−1)).
Our conclusion is
am,n−1 =∞∑
k=−∞
√2 h2m−kak,n, dm,n−1 =
∞∑k=−∞
√2 g2m−kak,n.
convolution and subsequently downsampling (m → 2 m) yieldsthe two sequences a(n−1) = (am,n−1) en d (n−1) = (dm,n−1).
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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 filtering proces.
a(n) ��
���
a(n−1)
d(n−1)
��
���
a(n−2)
d(n−2)
. . .�
��
��
a(n−N)
d(n−N)
�
�
�
Lo_d
Hi_d
a(n)
�
�
a(n−1)
d(n−1)
�
�
�
�
Figure 5: Decomposition
Filter coefficients are√
2 hk for the low pass filter and√
2 gk
for the high pass filter.
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ReconstructionIf a�−1 and d�−1 are given then we may reconstruct the approx-imation coefficients a�.
f� = f�−1 + w�−1
f� =∞∑
k=−∞ak,�φk,�
=∞∑
k=−∞ak,�−1φk,�−1 +
∞∑k=−∞
dk,�−1ψk,�−1
=∞∑
k=−∞
∞∑m=−∞
ak,�−11√2
pmφ2k+m,�
+∞∑
k=−∞
∞∑m=−∞
dk,�−11√2
qmφ2k+m,�.
Hence,∞∑
k=−∞ak,�φk,�
=∞∑
k=−∞
∞∑m=−∞
1√2
(ak,�−1 pm−2k + dk,�−1qm−2k
)φm,�.
Conclusion:
ak,� = 1√2
∞∑m=−∞
(am,�−1 pk−2m + dm,�−1qk−2m).
upsampling and subsequently convolution
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6. Examples
1. Haar wavelet
General characteristics:
OrthogonalSupport width 1Filters length 2Number of vanishing moments for ψ : 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 NNumber of vanishing moments for ψ NScaling 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. Coiflet family
General characteristics:
Order N = 1, . . . , 5OrthogonalSupport width 6 N − 1Filters length 6 NSymmetry near fromNumber of vanishing moments for ψ 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: Coiflet order 4
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