Download - Chapter 02 Continuous Wavelet Transform CWT
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Chapter 02Chapter 02Continuous Wavelet Transform CWTContinuous Wavelet Transform CWTChapter 02Chapter 02Continuous Wavelet Transform CWTContinuous Wavelet Transform CWT
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Definition of the CWTDefinition of the CWTDefinition of the CWTDefinition of the CWT
dtttffsfWsW ss )()(),)((),( ,*
,*
0 , )(, 2 sRsRLf
The continuous-time wavelet transform (CWT)of f(t) with respect to a wavelet (t):
)(2 RL
fW
),)(( sfW
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Mother WaveletMother WaveletDilation / TranslationDilation / TranslationMother WaveletMother WaveletDilation / TranslationDilation / Translation
s
tsts
2/1
, || )( )( )(0,1 tt Mother Wavelet
s Dilation Scale Translation
0 , )(, 2 sRsRLf
dts
ttfsdtttffsfWsW ss
*2/1
,*
,* )()()(),)((),(
44
}|)(| | :{)( 22
dttfCRfRL
Definition of a mother Wavelet (or Wavelet)Definition of a mother Wavelet (or Wavelet)Definition of a mother Wavelet (or Wavelet)Definition of a mother Wavelet (or Wavelet)
0)(
dtt
A real or complex-value continuous-time function (t)satisfying the following properties, is called a Wavelet:
dtt2
)(
1.
2.
CdC 0 )(
2
3.
Finite energy
Wavelet
Admissibility condition.Sufficient, but not a necessary conditionto obtain the inverse.
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The Haar Wavelet and The Morlet WaveletThe Haar Wavelet and The Morlet WaveletThe Haar Wavelet and The Morlet WaveletThe Haar Wavelet and The Morlet Wavelet
tet t
2ln
2cos)(
2
otherwise
12
1 1
2
10 1
)( t
t
t
1
-1
1
Haar Morlet
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Forward / Inverse TransformForward / Inverse TransformForward / Inverse TransformForward / Inverse Transform
dtttffsfWsW ss )()(),)((),( ,*
,*
0 , )(, 2 sRsRLf
Forward
dsdtsWtf s )(),()( ,Inverse
s
tsts
2/1
, || )( Mother Wavelet
s Dilation Scale Translation
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Admissibility conditionAdmissibility conditionAdmissibility conditionAdmissibility condition
0)(
dtt
It can be shown that square integrable functions (t)satisfying the admissibility condition can be used to first analyze and then reconstruct a signal without loss of information.
CdC 0 )(
2
Admissibility condition.
Sufficient, but not a necessary conditionto obtain the inverse.
The admissibility condition implies that the Fourier transform of (t)vanishes at the zero frequency.
0)(2
A zero at the zero frequency also meansthat the average value of the waveletin the time domain must be zero.
Wavelets must have a band-pass like spectrum.
(t) must be oscillatory,it must be a wave.
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Regularity conditions - Vanishing momentsRegularity conditions - Vanishing momentsRegularity conditions - Vanishing momentsRegularity conditions - Vanishing moments
The time-bandwidth product of the wavelet transform is the square of the input signal. For most practical applications this is not a desirable property.Therefor one imposes some additional conditions on the wavelet functionsin order to make the wavelet transform decrease quickly with decreasingscale s. These are the regularity conditions and they state that the waveletfunction should have some smoothness and concentration in both time andfrequency domains.Taylor series at t = 0 until order n (let = 0 for simplicity):
)1()0(
)1(!
)0()0,(
0
1)(2/1
0
)(2/1
nOsMfs
nOdts
t
p
tfssW
n
p
pp
p
n
p
pp dtt
p
tM
p
p )(!
pth moment of the wavelet
Moments up to Mn is zero implies that the coefficients of W(s,t) will decay as fast as sn+2 for a smooth signal.
Oscillation + fast decay =Wave + let = Wavelet
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Dilation / Translation: Haar WaveletDilation / Translation: Haar WaveletDilation / Translation: Haar WaveletDilation / Translation: Haar Wavelet
otherwise
12
1 1
2
10 1
)()(0,1 t
t
tt
Haar
)(0,2 t
1
-1
4
)(0,1 t )(1,2 t
1
-1
4
1
-1
41 2
2
2-1/2 2-1/2
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Dilation / Translation: Morlet WaveletDilation / Translation: Morlet WaveletDilation / Translation: Morlet WaveletDilation / Translation: Morlet Wavelet
Morlet
)(0,2 t)(0,1 t )(1,2 t
tet t
2ln
2cos)(
2
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CWT - Correlation 1CWT - Correlation 1CWT - Correlation 1CWT - Correlation 1
)()()()()(),(0,,0,, sfss RttfttfsW
)()()()()( *, tytxdttytxR yx
CWT
Cross-correlation
CWT W(s,) is the cross-correlation at lag (shift) between f(t) and the wavelet dilated to scale factor s.
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CWT - Correlation 2CWT - Correlation 2CWT - Correlation 2CWT - Correlation 2
2,
22 ||)(||||)(|| |),(| ttfbaW ba
)()(
||)(||||)(|| |),(|
,
2,
22
tft
ttfbaW
ba
ba
W(a,b) always exists
The global maximum of |W(a,b)| occurs if there is a pair of values (a,b)for which ab(t) = f(t).
Even if this equality does not exists, the global maximum of the real part of W2(a,b) provides a measure of the fit between f(t) and the corresponding ab(t) (se next page).
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CWT - Correlation 3CWT - Correlation 3CWT - Correlation 3CWT - Correlation 3
)],(Re[2||)(|| ||)(|| ||)()(|| 2,
22, baWttfttf baba
The global maximum of the real part of W2(a,b)provides a measure of the fit between f(t) and the corresponding ab(t)
ab(t) closest to f(t) for that value of pair (a,b)for which Re[W(a,b)] is a maximum.
)],(Re[2||)(|| ||)(|| ||)()(|| 2,
22, baWttfttf baba
-ab(t) closest to f(t) for that value of pair (a,b)for which Re[W(a,b)] is a minimum.
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CWT - Localization both in time and frequencyCWT - Localization both in time and frequencyCWT - Localization both in time and frequencyCWT - Localization both in time and frequency
The CWT offers time and frequency selectivity;that is, it is able to localize events both in time and in frequency.
Time:The segment of f(t) that influences the value of W(a,b) for any (a,b)is that stretch of f(t) that coinsides with the interval over which ab(t)has the bulk of its energy.This windowing effect results in the time selectivity of the CWT.
Frequency:The frequency selectivity of the CWT is explained using its interpretationas a collection of linear, time-invariant filters with impulse responsesthat are dilations of the mother wavelet reflected about the time axis(se next page).
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CWT - Frequency - Filter interpretationCWT - Frequency - Filter interpretationCWT - Frequency - Filter interpretationCWT - Frequency - Filter interpretation
dtxthtxth )()()(*)(Convolution
)(*)(),( *0, bbfbaW a CWT
CWT is the output of a filter with impulse response *ab(-b) and
input f(b).
We have a continuum of filters parameterized by the scale factor a.
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CWT - Time and frequency localization 1CWT - Time and frequency localization 1CWT - Time and frequency localization 1CWT - Time and frequency localization 1
dtt
dttt
t2
2
0
)(
)(
dt
dt
2
2
0
)(
)(
dtt
dtttt
t2
220
)(
)()(
dt
dt
2
220
)(
)()(
TimeCenter of mother wavelet
FrequencyCenter of the Fourier transformof mother wavelet
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CWT - Time and frequency localization 2CWT - Time and frequency localization 2CWT - Time and frequency localization 2CWT - Time and frequency localization 2
taatata
)()(0,
Time
Frequency
ta
aaa
1
)()(0,
2
1 )()(
productbandwidth -timesmallest thegivesfunctionGaussian
2
1)(
22
2
t
etf
ctaat
Time-bandwidth productis a constant
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CWT - Time and frequency localization 3CWT - Time and frequency localization 3CWT - Time and frequency localization 3CWT - Time and frequency localization 3
taatata
)()(0,
Time
Frequency
ta
aaa
1
)()(0,
Small a: CWT resolve events closely spaced in time.Large a: CWT resolve events closely spaced in frequency.
CWT provides better frequency resolution in the lower end of the frequency spectrum.
Wavelet tool a natural tool in the analysis of signals in which rapidlyvarying high-frequency components are superimposed on slowly varyinglow-frequency components (seismic signals, music compositions, …).
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CWT - Time and frequency localization 4CWT - Time and frequency localization 4CWT - Time and frequency localization 4CWT - Time and frequency localization 4
t
Time-frequency cells for a,b(t) shown for varied a and fixed b.
a=1/2
a=1
a=2
2020
}|)(| | :{
)2,0(
2
2
dttfCRf
L
2
0
int )(2
1hvor )( dtetfcectf tin
nn
n
nmggdttgtggg
ectg
nmnmnm
tinnn
0 , )()(2
1 ,
)(2
0
itetw )( tinnn etww )(hvor }{
Fourier-serie
Ortonormalebasis-funksjoner
DilationGenerering
Basis-funksjoner Basis-funksjoner for Hilbert rommet Lfor Hilbert rommet L22(0,2(0,2).).Fourier transformasjon.Fourier transformasjon.
Basis-funksjoner Basis-funksjoner for Hilbert rommet Lfor Hilbert rommet L22(0,2(0,2).).Fourier transformasjon.Fourier transformasjon.
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Basis-funksjoner Basis-funksjoner for Hilbert rommet Lfor Hilbert rommet L22(R).(R).Wavelet transformasjon.Wavelet transformasjon.
Basis-funksjoner Basis-funksjoner for Hilbert rommet Lfor Hilbert rommet L22(R).(R).Wavelet transformasjon.Wavelet transformasjon.
abab
jjkjkj
kj
fdtttfdta
bttfaabfW
kfWcctf
,,2/1
,,
kj,,
, )()()(||),)((
2
1,
2)(hvor (t))(
Wavelet-serie
Ortonormalebasis-funksjoner
Generering
mlmlkj
jjkj
dxtgtfgf
Zkjktt
,kj,,,
2/,
, )()(,
, )2(2)(
)(t )2(2)(hvor }{ 2/,, ktt jjkjkj
Dilation, translation
a
btatab 2/1
, ||)(
}|)(| | :{
)(
2
2
dttfCRf
RL
2222
Fourier transformertFourier transformertav Wavelet funksjon av Wavelet funksjon Fourier transformertFourier transformertav Wavelet funksjon av Wavelet funksjon
dtetftfFff ti )()}({)(ˆˆ
ibabab eaaa )(ˆ||)(ˆˆ 2/1
;;
Bevis:
ib
uiaib
baui
ti
tiabababab
eaaa
dueeuaa
adueua
dtea
bta
dtettF
)(ˆ||
)(||
)(||
||
)()}({)(ˆˆ
2/1
2/1
)(2/1
2/1
;;;;
2323
CC - Teorem - Teorem CC - Teorem - Teorem
dba
daabgWabfWgfC
2),)((),)((
2424
Teorem Teorem Teorem Teorem
)(ˆ||2
1)(ˆ)(ˆ||
2
1)(ˆ)(ˆ||
2
1
)(ˆ||)(ˆ2
1ˆˆ
2
1),)((
2/12/12/1
2/1;;
bFaafeaaafeaaa
eaaafffabfW
aibib
ibabab
Bevis Bevis Bevis Bevis
)(ˆ)(ˆ)(
)(ˆ)(ˆ)(
axxgxG
axxfxF
a
a
Benytter følgende notasjon:
)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)(ˆ)()(ˆ xfeaxdxxfeaxdxeaxxfdxexFF xixixixiaa
)()(||2
)(ˆ)(ˆ2
1
||2
)(ˆ)(ˆ2
1
||2),)((),)((
22
2
bFbGa
abFbG
a
a
dbbFbGa
adbabgWabfW
aaaa
aa
)(ˆ)(ˆ2
1
||2)(ˆ
2
||)(ˆ
2
||),)((),)((
22/12/1
bFbGa
abG
aabF
aaabgWabfW aaaa
2525
Teorem Teorem Teorem Teorem
dbdya
byygdx
a
bxxf
a
dbgfdbgf
dbdsa
bstgdt
a
bttfadbabgWabfW
abababab
ˆ)(ˆˆ)(ˆ)2(
||
ˆˆˆˆ)2(
1
)()(||),)((),)((
2
1
;;2;;
1
Bevis Bevis Bevis Bevis
)(ˆ)(ˆ)(
)(ˆ)(ˆ)(
axxgxG
axxfxF
a
a
Benytter følgende notasjon:
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Binary dilation / Dyadic translationBinary dilation / Dyadic translationBinary dilation / Dyadic translationBinary dilation / Dyadic translation
)(2 RL
)(t
)2()(hvor }{ ,, ktt jkjkj
Binary dilation
Dyadic translationj2
j
k
2
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Filtering / CompressionFiltering / CompressionFiltering / CompressionFiltering / Compression
)(2 RL
)(tf ),)(( abfW
Data compression
Remove low W-values
Lowpass-filtering
Replace W-values by 0for low a-values
Highpass-filtering
Replace W-values by 0for high a-values
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Inverse Wavelet transformation 1Inverse Wavelet transformation 1Inverse Wavelet transformation 1Inverse Wavelet transformation 1
)(2 RL
)(tf ),)(( abfW
WT
IWT = WT-1
MabfW ),)((
Modifisert
~ Dual
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)(2 RL
)(tf ),)(( abfW
WT
a,b a,b R Ra,b a,b R R
0 )(
|)(|
||
|)(ˆ| 2
dtt
dtt
dC
22/1
2
||)],)([(1
)(a
dadb
a
btaabfW
Ctf
R
~
Dual
Basic Wavelet
Inverse Wavelet transformation 2Inverse Wavelet transformation 2Inverse Wavelet transformation 2Inverse Wavelet transformation 2
Condition
Inverse
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)(2 RL
)(tf ),)(( abfW
WT
a,b a,b R Ra > 0a > 0a,b a,b R Ra > 0a > 0
0 )(
|)(|
||
|)(ˆ|
||
|)(ˆ|
2
1
0
2
0
2
dtt
dtt
ddC
22/1
2
||)],)([(2
)(a
dadb
a
btaabfW
Ctf
R
~
Dual
Inverse Wavelet transformation 3Inverse Wavelet transformation 3Inverse Wavelet transformation 3Inverse Wavelet transformation 3
Condition
Inverse
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)(2 RL
)(tf ),)(( abfW
WT
Condition
Inverse
0 )(
|)(|
||
|)(ˆ| 2
dtt
dtt
dC
22/1
2
||)],)([(1
)(a
dadb
a
btaabfW
Ctf
R
~
Dual
a,b a,b R Ra = 1/2a = 1/2jj
a,b a,b R Ra = 1/2a = 1/2jj
Dyadic Wavelet
Inverse Wavelet transformation 4Inverse Wavelet transformation 4Inverse Wavelet transformation 4Inverse Wavelet transformation 4
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EndEnd