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Lecture 13 Wavelet transformation II

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Fourier Transform (FT) Forward FT: Inverse FT: Examples: + + + Slide from Alexander Kolesnikov ’s lecture notes

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Two test signals: What is difference? x(t)=cos( 1 t)+cos( 2 t)+cos( 3 )+cos( 4 t) x 1 (t)=cos( 1 t) x 2 (t)=cos( 2 t) x 3 (t)=cos( 3 t) x 4 (t)=cos( 4 t) x 1 (t) x 2 (t) x 3 (t) x 4 (t) a) b) 1 = 10 2 = 20 3 = 40 4 =100 Slide from Alexander Kolesnikov ’s lecture notes

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Spectrums of the test signals a) b) Signals are different, spectrums are similar Signals are different, spectrums are similar Why? Slide from Alexander Kolesnikov ’s lecture notes

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Short-Time Fourier Transform (STFT) Window h(t) Signal in the window Result is localized in space and frequency: Why? Input signal

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STFT: Partition of the space-frequency plane

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Problems with STFT Uncertainity Principle: Improved space resolution Degraded frequency resolution Improved frequency resolution Degraded space resolution Problem: the same and t throught the entire plane! STFT is redundant representation Not good for compression

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Solution: Frequency Scaling Smaller frequency make the window more narrow Bigger frequency make the window wider More narrow time window for higher frequencies here s is scaling factor

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New partition of the space-frequency plane Coordinate, t Frequency,

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New partition of the plane Discrete wavelet transform Short-time Fourier transform Wavelet functions are localized in space and frequency Hierarchical set of of functions

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Frequency vs Time

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FT vs WT From one domain to another domain.

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Scale and shift Scale Shift

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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.

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Scale and frequency

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Example of Wavelet functions Haar Ingrid Dauhechies

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Biorthogonal

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Example of Wavelets Coiflets Symlets

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Examples of Wavelet functions Morlet Mexican Hat Meyer

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Decomposition: approximation and detail One-level decomposition Multi-level decomposition

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Haar wavelets

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Scaling function and Wavelets 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. where

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Haar: Scaling function and Wavelets

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Daubechies wavelets of order 2 Scaling function Wavelet function

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Discrete wavelet transform Wavelets details Low-resolution approx. NB! k j j1j1

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Haar wavelet transform

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Haar wavelet transform: Example Input data : X={x 1,x 2,x 3,…, x 16 } 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}

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Inverse Haar wavelet transform: Example Inverse Haar wavelet transform : (s,d) (a,b) 1) a=s+d/2 2) b=s d/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}

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Wavelet transform as Subband Transform To be continued...

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Wavelet Transform and Filter Banks

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h 0 (n) is scaling function, low pass filter (LPF) h 1 (n) is wavelet function, high pass filter (HPF) is subsampling (decimation)

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Inverse wavelet transform Synthesis filters: g 0 (n)= (-1) n h 1 (n) g 1 (n)= (-1) n h 0 (n) is up-sampling (zeroes inserting)

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Wavelet transform as Subband filtering

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