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11
Wavelet TransformWavelet TransformWavelet TransformWavelet Transform
22
Definition of The Continuous Wavelet Transform Definition of The Continuous Wavelet Transform CWTCWTDefinition of The Continuous Wavelet Transform Definition of The Continuous Wavelet Transform CWTCWT
dxxfxfbafWbaW baba )()(),]([),( ,,
0 , )(, 2 aRbaRLf
The continuous-time wavelet transform (CWT)of f(x) with respect to a wavelet (x):
][ fW
),]([ bafW)(xf
)(xL2(R)
a
bxaxba
2/1, || )(
33
Mother WaveletMother WaveletDilation / TranslationDilation / TranslationMother WaveletMother WaveletDilation / TranslationDilation / Translation
a
bxaxba
2/1, || )(
)( )(0,1 xx Mother Waveleta Dilation Scaleb Translation
dxxfa
bxadxxfxfbafWbaW baba )()()(),]([),(
2/1
,,
dxxdxxba
22
, )()(
44
}|)(| | :{)( 22
dxxfCRfRL
Properties of a Basic WaveletProperties of a Basic WaveletProperties of a Basic WaveletProperties of a Basic Wavelet
0)(
dxx
dxx2
)(
1.
2.
CdC 0
)(2
Finite energy (Let)fast decay
Oscillation (Wave)
Admissibility condition. Necessary condition to obtain the inverse from the CWT by the basic Wavelet .Sufficient, but not a necessary condition to obtain the inverse by general Wavelet.
)( )(ˆ ),( 2 RLxxxx
L2(R) is called a Basic Wavelet if the following admissibility condition is satisfied:
Oscillation + fast decay =Wave + let = Wavelet
55
Haar Wavelet Haar Wavelet Dilation / TranslationDilation / TranslationHaar Wavelet Haar Wavelet Dilation / TranslationDilation / Translation
otherwise
12
1 1
2
10 1
)()( 0,1 x
x
xx
Haar
)(0,2 x
1
-1
4
)(0,1 x )(1,2 x
1
-1
4
1
-1
41 2
2
2-1/2 2-1/2
a
bxaxba 2/1
, || )(
66
Morlet Wavelet Morlet Wavelet Dilation / TranslationDilation / TranslationMorlet Wavelet Morlet Wavelet Dilation / TranslationDilation / Translation
Morlet
)(0,2 x)(0,1 x )(1,2 x
a
bxaxba 2/1
, || )(
xex x
2ln
2cos)(
2
77
Forward / Inverse TransformForward / Inverse Transform [1/5] [1/5]Forward / Inverse TransformForward / Inverse Transform [1/5] [1/5]
0 , )(, 2 aRbaRLf
Forward
Inverse
a
bxaxba 2/1
, || )(
dxxfxfbafWbaW baba )()(),]([),( ,,
dadbxbaWaC
xf ba )(),(11
)( ,2
CdC 0
)(2
Admissibility condition.
88
Forward / Inverse TransformForward / Inverse Transform [2/5] [2/5]Forward / Inverse TransformForward / Inverse Transform [2/5] [2/5]
a
bxaxba 2/1
, || )(
a
bxaxba
2/1
, )(
)(ˆ
)(
)(
)()()(ˆ
22/1
222/1
)(22/1
22/1
2,,,
auaea
dsesaea
adsesa
dxea
bxa
dxexxFu
ubj
ausjubj
basuj
uxj
uxjbababa
)(ˆ)(ˆ 22/1
, auaeau ubjba Theorem
cwt_001
Proof
)(ˆ)(ˆ 22/1
, auaeau ubjba
0 , )(, 2 aRbaRLf
99
Forward / Inverse TransformForward / Inverse Transform [3/5] [3/5]Forward / Inverse TransformForward / Inverse Transform [3/5] [3/5]
a
bxaxba 2/1
, || )(
)(ˆ)(ˆ])(*)([ ugufxgxfF Theoremcwt_002
Proof
)(ˆ)(ˆ
)()(
)()(
)()(
)()(
)()()()()](*)([
22
22
)(2
2
2
uguf
dxexgdsesf
dsdxexgesf
dsdxexgsf
dsdtetsgsf
dtedstsgsfdstsgsfFxgxfF
uxjusj
uxjsuj
xsuj
utj
utj
dtyxtytx
dtyxtytx
)()()(*)(
)()()(*)(
0 , )(, 2 aRbaRLf
1010
Forward / Inverse TransformForward / Inverse Transform [4/5] [4/5]Forward / Inverse TransformForward / Inverse Transform [4/5] [4/5]
a
bxaxba 2/1
, || )(
)(ˆ)(ˆ),(
)(ˆ)(ˆ)(*)(),(
),(),(
)(*)()()()()(),]([),(
)(0)(
)(
0,2
0,0,
2
0,0,,,
,
2/12/1
0,
uufdbebaW
uufbbfFbaWF
dbebaWbaWF
bbfdxbxxfdxxxffbafWbaW
xa
bxa
a
bxabx
aubj
aa
ubj
aababa
baa
dtyxtytx
dtyxtytx
)()()(*)(
)()()(*)(
)(ˆ*)(ˆ),()],]([[)],([ 0,2 uufdbebaWbafWFbaWF aubj
Theoremcwt_003
Proof
0 , )(, 2 aRbaRLf
1111
Forward / Inverse TransformForward / Inverse Transform [5/5] [5/5]Forward / Inverse TransformForward / Inverse Transform [5/5] [5/5]
a
bxaxba 2/1
, || )(
dadbxbaWaC
xf ba )(),(11
)( ,2
dbdaxbaWaC
ufFxf
dbdaubaWaC
uf
dbdauaa
baWaaC
dbdaeaubaWaaC
uf
Cufdau
ufdbdaeaubaWaa
daa
auufdadbeaubaW
aa
a
auufdbeaubaW
aa
auufaauufdbebaW
ba
ba
baubj
ubj
ubj
ubj
aubj
)(),(11
)](ˆ[)(
)(ˆ),(11
)(ˆ
)(ˆ1
),(11
)(ˆ),(11
)(ˆ
)(ˆ)(ˆ
)(ˆ)(ˆ),(1
)(ˆ)(ˆ)(ˆ),(
1
)(ˆ)(ˆ)(ˆ),(
1
)(ˆ)(ˆ)(ˆ*)(ˆ),(
,21
,2
,2/12/12
2/1
2
22/1
2
22/1
2
22/1
2/1
0,2
Theoremcwt_004
Proof
0 , )(, 2 aRbaRLf
1212
Wavelet TransformWavelet TransformMorlet Wavelet - Stationary SignalMorlet Wavelet - Stationary SignalWavelet TransformWavelet TransformMorlet Wavelet - Stationary SignalMorlet Wavelet - Stationary Signal
xex x
2ln
2cos)(
2
signal Original f
[f]Wψ
[f]Wa
1ψ2
1313
Wavelet TransformWavelet TransformMorlet Wavelet - Transient SignalMorlet Wavelet - Transient SignalWavelet TransformWavelet TransformMorlet Wavelet - Transient SignalMorlet Wavelet - Transient Signal
signal Original
xex x
2ln
2cos)(
2
f
[f]Wa
1ψ2
[f]Wψ
1414
Wavelet TransformWavelet TransformMorlet Wavelet - Transient SignalMorlet Wavelet - Transient SignalWavelet TransformWavelet TransformMorlet Wavelet - Transient SignalMorlet Wavelet - Transient Signal
signal Original
[f]Wψ [f]Wψ
[f]Wa
1ψ2
[f]Wa
1ψ2
[f]Wa
1ψ2
xex x
2ln
2cos)(
2
f
1515
Wavelet TransformWavelet TransformMorlet Wavelet - Non-visible Oscillation [1/Morlet Wavelet - Non-visible Oscillation [1/3]3]Wavelet TransformWavelet TransformMorlet Wavelet - Non-visible Oscillation [1/Morlet Wavelet - Non-visible Oscillation [1/3]3]
][fWa
11ψ2
][fWa
12ψ2
xex x
2ln
2cos)(
2
210)0.01(x1 1000e(x)f
9,11 xif x)5sin(2)(
11,,9 xif (x)(x)f
1
12 xf
f
(x)f1
(x)f2
Scalogram
Scalogram
1616
Wavelet TransformWavelet TransformMorlet Wavelet - Non-visible Oscillation [Morlet Wavelet - Non-visible Oscillation [22//33]]Wavelet TransformWavelet TransformMorlet Wavelet - Non-visible Oscillation [Morlet Wavelet - Non-visible Oscillation [22//33]]
(x)f1
(x)f2
xex x
2ln
2cos)(
2
Scalogram
Scalogram
][fW 1ψ
][fW 2ψ
][fWa
11ψ2
][fWa
12ψ2
1717
Wavelet TransformWavelet TransformMorlet Wavelet - Non-visible Oscillation [Morlet Wavelet - Non-visible Oscillation [33//33]]Wavelet TransformWavelet TransformMorlet Wavelet - Non-visible Oscillation [Morlet Wavelet - Non-visible Oscillation [33//33]]
xex x
2ln
2cos)(
2
][fW 1ψ
Scalogram
][fWa
11ψ2
(x)f2
][fW 2ψ
Scalogram
][fWa
12ψ2
(x)f1
1818
Wavelet TransformWavelet TransformHaar Wavelet - Stationary SignalHaar Wavelet - Stationary SignalWavelet TransformWavelet TransformHaar Wavelet - Stationary SignalHaar Wavelet - Stationary Signal
signal Original
otherwise 0
1x2
1 if 1
2
1x0 if 1
)(x
[f]Wψ
[f]Wa
1ψ2
1919
Wavelet TransformWavelet TransformHaar Wavelet - Transient SignalHaar Wavelet - Transient SignalWavelet TransformWavelet TransformHaar Wavelet - Transient SignalHaar Wavelet - Transient Signal
signal Original
otherwise 0
1x2
1 if 1
2
1x0 if 1
)(x
[f]Wψ [f]Wψ
[f]Wa
1ψ2
[f]Wa
1ψ2
[f]Wa
1ψ2
2020
Wavelet TransformWavelet TransformMexican Hat - Stationary SignalMexican Hat - Stationary SignalWavelet TransformWavelet TransformMexican Hat - Stationary SignalMexican Hat - Stationary Signal
2
2
2
2
22
1)(
x
ex
x
1
signal Original
[f]Wψ
[f]Wa
1ψ2
2121
Wavelet TransformWavelet TransformMexican Hat - Transient SignalMexican Hat - Transient SignalWavelet TransformWavelet TransformMexican Hat - Transient SignalMexican Hat - Transient Signal
signal Original
1σ [f]Wψ
1σ [f]Wa
1ψ2
2
2
2
2
22
1)(
x
ex
x
1
0.5σ [f]Wψ 0.25σ [f]Wψ
0.5σ [f]Wa
1ψ2
0.25σ [f]Wa
1ψ2
2222
Wavelet TransformWavelet TransformMorlet WaveletMorlet WaveletFourier/WaveletFourier/Wavelet
Wavelet TransformWavelet TransformMorlet WaveletMorlet WaveletFourier/WaveletFourier/Wavelet
f
[f]Wψ
F[f]
[f]Wa
1ψ2
b)1,(a [f]Wψ
b)20,(a [f]Wψ
b)10,(a [f]Wψ
Fourier
Wavelet
xex x
2ln
2cos)(
2
2323
Wavelet TransformWavelet TransformMorlet WaveletMorlet WaveletFourier/WaveletFourier/Wavelet
Wavelet TransformWavelet TransformMorlet WaveletMorlet WaveletFourier/WaveletFourier/Wavelet
Fourier
Wavelet
xex x
2ln
2cos)(
2
f
F[f]
[f]Wψ [f]W
a
1ψ2
2424
CWT - Correlation 1CWT - Correlation 1CWT - Correlation 1CWT - Correlation 1
)()()()()(),(0,,0,, afaba RxxfxxfbaW
)()()()()( *, tytxdttytxR yx
CWT
Cross-correlation
CWT W(a,b) is the cross-correlation at lag (shift) between f(x) and the wavelet dilated to scale factor a.
2525
CWT - Correlation 2CWT - Correlation 2CWT - Correlation 2CWT - Correlation 2
2,
22 ||)(||||)(|| |),(| xxfbaW ba
)()(
||)(||||)(|| |),(|
,
2,
22
xfx
xxfbaW
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).
2626
CWT - Correlation 3CWT - Correlation 3CWT - Correlation 3CWT - Correlation 3
)],(Re[2||)(|| ||)(|| ||)()(|| 2,
22, baWxxfxxf baba
The global maximum of the real part of W2(a,b)provides a measure of the fit between f(x) and the corresponding ab(x)
ab(x) closest to f(x) for that value of pair (a,b)for which Re[W(a,b)] is a maximum.
)],(Re[2||)(|| ||)(|| ||)()(|| 2,
22, baWxxfxxf baba
-ab(x) closest to f(x) for that value of pair (a,b)for which Re[W(a,b)] is a minimum.
2727
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 position/time and frequency selectivity;that is, it is able to localize events both in position/time and in frequency.
Time:The segment of f(x) that influences the value of W(a,b) for any (a,b)is that stretch of f(x) that coinsides with the interval over which ab(x)has the bulk of its energy.This windowing effect results in the position/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).
2828
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.
2929
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
3030
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
3131
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 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, …).
3232
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)
a=1/2
a=1
a=2
3333
Filtering / CompressionFiltering / CompressionFiltering / CompressionFiltering / Compression
)(xf ),]([ bafW
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
3434
CWTCWT - DWT - DWTCWTCWT - DWT - DWT
dxxfxfbafWbaW baba )()(),]([),( ,,
dadbxbaWaC
xf ba )(),(11
)( ,2
CdC 0
)(2
a
bxaxba 2/1
, || )(
CWT
DWT
m
m
anbb
aa
00
0
nxx mmnm 22 )( 2/
,
m
m
nb
a
2
2
1 2 00 ba
Binary dilationDyadic translation
Dyadic Wavelets
voicea called group, one as processed are of pieces v
octaveper voicesofnumber 2
nm,
/10
va v
3535
Mexican HatMexican HatMexican HatMexican Hat
2
2
2
x2
2π1 e
σ
x2Ψ(x)
1σ
3636
Rotation - ScalingRotation - Scaling2 dim2 dimRotation - ScalingRotation - Scaling2 dim2 dim
cosθsinθ
sinθcosθR
y
x
s0
0sS
Rotation
Scaling
3737
Translation - Rotation - ScalingTranslation - Rotation - Scaling3 dim3 dimTranslation - Rotation - ScalingTranslation - Rotation - Scaling3 dim3 dim
1000
0100
00cosθsinθ
00sinθcosθ
R z
1000
0s00
00s0
000s
z
y
x
S
Rotation
Scaling
1000
t100
t010
t001
Tz
y
xTranslation
),,()) 1 zyxT(θz)Ry,x,(TR(θ z
3838
Mexican HatMexican Hat - 3 Dim - 3 DimMexican HatMexican Hat - 3 Dim - 3 Dim
2
2
2σ
x2
2π1 e
σ
x2Ψ(x)
1σ
cosθsinθ
sinθcosθR
2
y
2x
a
10
0a
1
A
ARRP T
y
xr
y
x
b
bb
brPbrT
a
T
y
brPbr
2
1
a2π
1b,a
e2)r(Ψx
y
x
a
aa
2a 1a yx
3939
EndEnd