Wavelet Transform

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

, || )(


dadbxbaWaC

xf ba )(),(11

)( ,2

CdC 0

)(2

Admissibility condition.

88


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


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


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


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


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



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


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


)],(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


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


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


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


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

Documents

Wavelet Transform