Lecture 13 Wavelet transformation II

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

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

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

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

STFT: Partition of the space-frequency plane

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

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

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

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

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

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

Haar: Scaling function and Wavelets

Daubechies wavelets of order 2 Scaling function Wavelet function

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

Haar wavelet transform

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, ] {-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=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, ] {-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

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

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)

Wavelet transform as Subband filtering