Lecture 13 Wavelet transformation II. Fourier Transform (FT) Forward FT: Inverse FT: Examples: + + +...
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Transcript of Lecture 13 Wavelet transformation II. Fourier Transform (FT) Forward FT: Inverse FT: Examples: + + +...
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