introduction to wavelets · 2009-03-19 · frequency domain • parseval the wavelet coefficients...
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![Page 1: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/1.jpg)
1
Introduction to Wavelets
![Page 2: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/2.jpg)
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Discrete Wavelet Transform
• A wavelet is a function of zero average centered in the neighborhood of t=0 and is normalized
• The translations and dilations of the wavelet generate a family of functions over which the signal is projected
• Wavelet transform of f in L2(R) at position u and scale s is
1
0)(
=
=∫+∞
∞−
ψ
ψ dtt
⎟⎠⎞
⎜⎝⎛ −
=s
uts
tsu ψψ 1)(,
,1( , ) , ( )
22
u s
j
j
t uWf u s f f t dtss
su k
ψ ψ+∞
∗
−∞
−⎛ ⎞= = ⎜ ⎟⎝ ⎠
=
= ⋅
∫
![Page 3: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/3.jpg)
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Wavelet transform
Ψu,s(t)
t
t
Ψ0,s(t)
Wf(0,s) ⇔ correlation for u=0
0
![Page 4: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/4.jpg)
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Wavelet transform
Ψu,s(t)
t
t
Ψn2j,s(t)
u=n 2j
Wf(n 2j,s) ⇔ correlation for u=n 2j
![Page 5: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/5.jpg)
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Wavelet transform
Ψu,s(t)
t
t
Ψ(n+1)2js(t)
u= (n+1) 2j
Wf((n+1)2j,s) ⇔ correlation at u=(n+1)2j
![Page 6: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/6.jpg)
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Changing the scale
Ψu,s(t)
Ψu,s(t)
Ψu,s(t)
finer
coarser
s=2j+1
s=2j
s=2j+2
multiresolution
![Page 7: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/7.jpg)
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Fourier versus Wavelets
![Page 8: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/8.jpg)
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Scaling
![Page 9: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/9.jpg)
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Shifting
t t
![Page 10: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/10.jpg)
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Recipe
![Page 11: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/11.jpg)
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Recipe
![Page 12: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/12.jpg)
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Wavelet Zoom
• WT at position u and scale s measures the local correlation between the signal and the wavelet
(small)
(large)
small scale large scale
![Page 13: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/13.jpg)
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Frequency domain
• Parseval
The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where the energy of the corresponding wavelet function (respectively, its transform) is concentrated
•• time/frequency localizationtime/frequency localization• The position and scaleposition and scale of high amplitude coefficients allow to characterize the
temporal evolutiontemporal evolution of the signal
• Time domain signals (1D) : Temporal evolution• Spatial domain signals (2D) : Localize and characterize spatial singularities
Stratching in time ↔ Shrinking in frequency (and viceversa)
ωωωπ
ψ dFdtttfsuWf susu )()(21)()(),( ,
*,
* ∫∫+∞
∞−
+∞
∞−
Ψ==
sjsusu ess
sut
st ωωωψψ −Ψ=Ψ⇔⎟
⎠⎞
⎜⎝⎛ −
= )()(1)( ,,
![Page 14: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/14.jpg)
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Example
approximation
details
Wavelet representation = approximation + details approximation ↔ scaling functiondetails ↔ wavelets
![Page 15: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/15.jpg)
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A different perspective
detail signald2
1f
Ad2j f = Ad
2j +1f + d2
j +1f
approximation at resolution 21
Ad21f
approximation at resolution 20
Ad20 f
ϕ21
ϕ20
Ψ2j+1
![Page 16: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/16.jpg)
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Haar pyramid [Haar 1910]
sig0
sig1
sig2
sig3
Haar basis function Haar waveletϕ20
signal=approximation at scale n + details at scales 1 to n
details
![Page 17: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/17.jpg)
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What wavelets can do?
![Page 18: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/18.jpg)
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Wavelets and linear filtering
• The WT can be rewritten as a convolution product and thus the transform can be interpreted as a linear filtering operation
,
*
*
1( , ) , ( ) ( )
1( )
ˆ ˆ( ) ( )
ˆ (0) 0
u s s
s
s
t uWf u s f f t dt f uss
ttss
s s
ψ ψ ψ
ψ ψ
ψ ω ψ ω
ψ
+∞∗
−∞
−⎛ ⎞= = = ∗⎜ ⎟⎝ ⎠
−⎛ ⎞= ⎜ ⎟⎝ ⎠
=
=
∫
→ band-pass filter
![Page 19: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/19.jpg)
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Wavelets & filterbanksQuadrature Mirror Filter (QMF)
![Page 20: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/20.jpg)
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Analysis or decomposition
![Page 21: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/21.jpg)
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Analysis or decomposition
![Page 22: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/22.jpg)
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Synthesis or reconstruction
upsampling
![Page 23: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/23.jpg)
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Multi-scale analysis
![Page 24: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/24.jpg)
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Famous waveletsHaar
Mexican hat
![Page 25: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/25.jpg)
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Daubechie’s
![Page 26: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/26.jpg)
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Bi-dimensional wavelets
)()(),()()(),()()(),(
)()(),(
3
2
1
yxyxyxyxyxyx
yxyx
ψψψ
ϕψψ
ψϕψ
ϕϕϕ
=
=
=
=
![Page 27: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/27.jpg)
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Fast wavelet transform algorithm (DWT)
Decomposition step
![Page 28: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/28.jpg)
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Fast wavelet transform algorithm (DWT)
![Page 29: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/29.jpg)
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Filters
![Page 30: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/30.jpg)
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Fast DWT for images
![Page 31: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/31.jpg)
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Fast DWT for images
![Page 32: Introduction to Wavelets · 2009-03-19 · Frequency domain • Parseval The wavelet coefficients Wf(u,s) depend on the values of f(t) (and F(ω)) in the time-frequency region where](https://reader033.vdocuments.us/reader033/viewer/2022042311/5ed917726714ca7f4769222d/html5/thumbnails/32.jpg)
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Subband structure for images
cD1(h)
cD1(v) cD1(d)
cD2(v) cD2(d)
cD2(h)cA2