Lecture 13 Wavelet transformation II

Fourier Transform (FT)

• Forward FT:

dtetxX ti )()(~

deXtx ti)(

~)( 2

1• Inverse FT:

• Examples: )(2)(~

)( 000

dteeXetx tititi

1)()(~

)()( dtetXttx ti

++ +

Slide from Alexander Kolesnikov ’s lecture notes

Two test signals: What is difference?

x(t)=cos(1 t)+cos(2t)+cos(3)+cos(4t)

x1(t)=cos(1t)x2(t)=cos(2t)x3(t)=cos(3t)x4(t)=cos(4t)

x1(t) x2(t) x3(t) x4(t)

a)

b)

1= 102= 203= 404=100

Slide from Alexander Kolesnikov ’s lecture notes

Spectrums of the test signals

a)

b)

Signals are different, spectrums are similar

Signals are different, spectrums are similar

Why?Why?

Slide from Alexander Kolesnikov ’s lecture notes

Short-Time Fourier Transform (STFT)

dethxtX i)()(),(~

Window h(t)

Signal in the window

Result is localized in space and frequency: Why?Result is localized in space and frequency: Why?

Input signal

STFT: Partition of the space-frequency plane

ktk

2

Problems with STFT

Uncertainity Principle: 1 t

Improved space resolution Degraded frequency resolution

Improved frequency resolutionDegraded space resolution

t

Problem: the same and t throught the entire plane!Problem: the same and t throught the entire plane!

STFT is redundant representationNot good for compression

Solution: Frequency Scaling

• Smaller frequency make the window more narrow

• Bigger frequency make the window wider

1~Const

1

t

t

)(~

)/( sHsh

More narrow time window for higher frequencies

here s is scaling factor

New partition of the space-frequency plane

Coordinate, t

Frequency,

New partition of the plane

Discrete wavelet transformShort-time Fourier transform

• Wavelet functions are localized in space and frequency• Hierarchical set of of functions

Frequency vs Time

FT vs WT

• From one domain to another domain.

Scale and shift

• Scale

• Shift

Five steps to calculate WT

1. Take a wavelet and compare it to a section at the start of the original signal.

2. Calculate a number, C, that represents how closely correlated the wavelet is with this section of the signal.

3. Shift the wavelet to the right and repeat steps 1 and 2 until you’ve covered the whole signal.

4. Scale (stretch) the wavelet and repeat steps 1 through 3.

5. Repeat steps 1 through 4 for all scales.

Scale and frequency

Example of Wavelet functions

• Haar

• Ingrid Dauhechies

Biorthogonal

Example of Wavelets

• Coiflets

• Symlets

Examples of Wavelet functions

• Morlet

• Mexican Hat

• Meyer

Decomposition: approximation and detail

• One-level decomposition

• Multi-level decomposition

Haar wavelets

)2(2),( 2 ktkj jj

Scaling function and Wavelets

k

ktkht )2()(2)( 0

k

ktkht )2()(2)( 1 Wavelet function:

Scaling function:

The functions (t) and (t) are orthonormal

The most important property of the wavelets:To obtain WT coefficients for level j we can process

WT coefficients for level j+1.

The most important property of the wavelets:To obtain WT coefficients for level j we can process

WT coefficients for level j+1.

)1()1()( 01 kNhkh k where

Haar: Scaling function and Wavelets

)12(2

1)2(

2

12)(

)12(2

1)2(

2

12)(

ttt

ttt

)1()1()( 01 kNhkh k

Daubechies wavelets of order 2

Scaling function Wavelet function

k

ktkht )2()(2)( 0

k

ktkht )2()(2)( 1

Discrete wavelet transform

1

0

00 )2(2)()2(2)()( 22j

jj

j

k

jj

jj

k

ktkdktkstf

Wavelets detailsLow-resolution approx.

NB!NB!

k

j

j1

Haar wavelet transform

Haar wavelet transform

)(0 kh )(1 kh

Haar wavelet transform: Example

Input data: X={x1,x2,x3,…, x16}

Haar wavelet transform: (a,b)(s,d)

where:

1) scaling function s=(a+b)/2 (smooth, LPF)

2) Haar wavelet d=(a-b) (details, HPF)

X={10,13, 11,14, 12,15, 12,14, 12,13, 11,13, 10,11} [11.5,12.5, 13.5,13, 12.5,12, 10.5] {-3, -3, -3, -2, -1,-2,-1} [12, 13.25, 12.25, 10.5] {-2,0.5,-0.5} {-3, -3, -3, -2, -1,-2,-1} [12.625, 11.375] {-1.25, 1.75} {-2,0.5,-0.5} {-3, -3, -3, -2, -1,-2,-1} [12]{1.25} {-1.25, 1.75} {-2,0.5,-0.5} {-3, -3, -3, -2, -1,-2,-1}

Inverse Haar wavelet transform: Example

Inverse Haar wavelet transform: (s,d) (a,b)

1) a=s+d/2

2) b=sd/2

Y= [12]{1.25} {-1.25, 1.75} {-2,0.5,-0.5} {-3, -3, -3, -2, -1,-2,-1} [12.625,11.375] {-1.25, 1.75} {-2,0.5,-0.5} {-3, -3, -3, -2, -1,-2,-1} [12, 13.25, 12.25, 10.5] {-2,0.5,-0.5} {-3, -3, -3, -2, -1,-2,-1} [11.5,12.5, 13.5,13, 12.5,12, 10.5] {-3, -3, -3, -2, -1,-2,-1} {10,13, 11,14, 12,15, 12,14, 12,13, 11,13, 10,11}

X={10,13, 11,14, 12,15, 12,14, 12,13, 11,13, 10,11} [11.5,12.5, 13.5,13, 12.5,12, 10.5] {-3, -3, -3, -2, -1,-2,-1} [12, 13.25, 12.25, 10.5] {-2,0.5,-0.5} {-3, -3, -3, -2, -1,-2,-1} [12.625, 11.375] {-1.25, 1.75} {-2,0.5,-0.5} {-3, -3, -3, -2, -1,-2,-1} [12]{1.25} {-1.25, 1.75} {-2,0.5,-0.5} {-3, -3, -3, -2, -1,-2,-1}

Wavelet transform as Subband Transform

To be continued...

Wavelet Transform and Filter Banks

Wavelet Transform and Filter Banks

h0(n) is scaling function, low pass filter (LPF)

h1(n) is wavelet function, high pass filter (HPF)

is subsampling (decimation)

Inverse wavelet transform

Synthesis filters: g0(n)=(-1)nh1(n)

g1(n)=(-1)nh0(n)

is up-sampling (zeroes inserting)

Wavelet transform as Subband filtering