Wavelets and Multiresolution Processing Jen-Chang Liu, Spring 2006 Copyright notice: Some images are...
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![Page 1: Wavelets and Multiresolution Processing Jen-Chang Liu, Spring 2006 Copyright notice: Some images are from Matlab help.](https://reader035.fdocuments.us/reader035/viewer/2022062313/56649d355503460f94a0cbed/html5/thumbnails/1.jpg)
Wavelets and Multiresolution Processing
Jen-Chang Liu, Spring 2006
Copyright notice: Some images are from Matlab help
![Page 2: Wavelets and Multiresolution Processing Jen-Chang Liu, Spring 2006 Copyright notice: Some images are from Matlab help.](https://reader035.fdocuments.us/reader035/viewer/2022062313/56649d355503460f94a0cbed/html5/thumbnails/2.jpg)
Preview Fourier transform
Basis functions are sinusoids
Wavelet transform 小波 Basis functions are small waves, of varying freq
uency and limited duration
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Signal representation (1) Fourier transform
dueuFxf uxj 2)()(
Sinusoid has unlimited duration
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Signal representation (2) Wavelet transform
dttpositionscalepositionscaleCtf ),,(),()(
A wavelet has compact support (limited duration)
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Scaling (1) What is the scale factor?
Ex#1: Plot the above diagrams (hint: plot command)
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Scaling (2) Scaling for wavelet function
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Shift Shift for wavelet function
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Steps to compute a continuous wavelet transform
Take a wavelet and calculate its similarity to the original signal
Shift the wavelet and repeat
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Steps to compute a continuous wavelet transform (2)
Scale the wavelet and repeat
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Scale and frequency
Rapid changeHigh frequency
Slow changeLow frequency
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Continuous wavelet analysis
Matlab command wavemenu Continuous wavelet 1-D File => Load Signal(toolbox/wavelet/wavedemo/noissin.mat) db4, scale 1:48 Zoom in details (wavelet display button)
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Discrete wavelet transform
Continuous wavelet transform: calculate wavelet coefficient at every possible scale and shift
Discrete wavelet transform: choose scale and shift on powers of two (dyadic scale and shift) Fast wavelet transform exist Perfect reconstruction
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Filtering structure for wavelet transform
S. Mallat[89] derived the subband filtering structure for wavelet transform
DetailApproximation
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Multi-level decomposition Wavelet decomposition tree
High passfilters
Low passfilters
HL
22
HL
22
HL
HL
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Two-dimensional wavelet transform
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MATLAB: 2d SWT (Stationary Wavelet Transform)
load noiswom [swa, swh, swv, swd]=swt2(X, 1, 'db1');
Ex#2: show the swa, swh, swv, swd
A0=iswt2(swa, swh, swv, swd, 'db1'); err=max(max(abs(X-A0))); nulcfs=zeros(size(swa)); A1=iswt2(swa, nulcfs, nulcfs, nulcfs, 'db1');
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DWT with downsampling
Twice of the original data
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DWT using Matlab wavemenu Choose wavelet 2-D Load image ->
toolbox/wavelet/wavedemo/wbarb.mat
Bior3.7, level 2 Square and tree
mode
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Ex#3: DWT of iris image Download the iris16.bmp Download the iris normalization sample code Generate the normalized iris image
Truncate to 56x512 image, save as .mat file Use db2, 4 level wavelet analysis in the wave
menu tool
64
512
56
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Matlab: one-level DWT functions
load wbarb Single level decomposition [cA1, cH1, cV1, cD1]=dwt2(X,'bior3.7'); Construct from approximation or details A1=upcoef2('a', cA1, 'bior3.7', 1); A1=idwt2(cA1, [],[],[], 'bior3.7', size(X)); Xfull=idwt2(cA1,cH1,cV1,cD1, 'bior3.7'); Ex#4: reconstruct from cH1, cV1, and cD1
respectively and show them all
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Matlab: multilevel DWT [C, S]=wavedec2(X, 2, 'bior3.7');
C
SBookkeeping matrix
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Matlab: multilevel DWT (2) cA2=appcoef2(C,S,'bior3.7', 2); cH2=detcoef2('h',C,S,2); EX#5: Show all cA2, cH2, cV2, cD2,
cH1, cV1, cD1
Reconstruction X0=waverec2(C,S,'bior3.7');