Prof. Vidya Manian. INEL 6209 (Spring 2010) ECE, UPRM

Transcription

1 Wavelets and Multiresolution l Processing Chapter 7 Prof. Vidya Manian Dept. ofelectrical andcomptuer Engineering INEL 6209 (Spring 2010) ECE, UPRM Wavelets 1

2 Overview Background Multiresolution expansion Wavelet transform in one dimension i Fast wavelet transform Wavelet transforms in two dimensions INEL 6209 (Spring 2010) ECE, UPRM 2

3 Fourier basis functions are sinusoids, wavelets are based on small waves, of varying frequency and limited duration Fourier only frequency information, no temporal information Wavelets: information of frequency when and where Signal processing and analysis using wavelets (Mallat[1987]) INEL 6209 (Spring 2010) ECE, UPRM 3

4 Multiresolution Subband coding from signal processing Quadrature mirror filtering from digital speech recognition Pyramidal image processing Features that might go undetected at one resolution may be easy to detect at another INEL 6209 (Spring 2010) ECE, UPRM 4

5 Background INEL 6209 (Spring 2010) ECE, UPRM 5

6 Image pyramids Representing images at more than one resolution Collection of decreasing resolution images arrangedin the shapeof a pyramid Base level J is of size 2 J x 2 J or N x N J=log2N, apex level l0 is of size 1x11 General level j is of size 2 j x 2 j where 0 j J Restrict to P reduced resolution approximations of the original image INEL 6209 (Spring 2010) ECE, UPRM 6

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8 Level j 1 approximation outputs are used to build approximation pyramid Level j prediction residual output is used to build complementary prediction residual pyramid Step 1: compute a reduced resolution approximation of the level j input image. Place the resulting approximation at level j 1 Step 2: create an estimate of level j input image from above approx. the resulting prediction image has same dimensions as the level j input image Step 3: compute difference bt between the prediction image of step 2 and input of step 1. Place this result in level j of the prediction residual pyramid INEL 6209 (Spring 2010) ECE, UPRM 8

9 Example INEL 6209 (Spring 2010) ECE, UPRM 9

10 Subband coding Image is decomposed into a set of bandlimited components called subbands The subbands can be reassembled to reconstruct the original image INEL 6209 (Spring 2010) ECE, UPRM 10

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15 Example INEL 6209 (Spring 2010) ECE, UPRM 15

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18 The Haar transform INEL 6209 (Spring 2010) ECE, UPRM 18

19 Haar scaling functions INEL 6209 (Spring 2010) ECE, UPRM 19

20 Haar wavelet functions INEL 6209 (Spring 2010) ECE, UPRM 20

21 Wavelet series expansion of y=x 2 using Haar wavelets INEL 6209 (Spring 2010) ECE, UPRM 21

22 CWT and Fourier spectrum INEL 6209 (Spring 2010) ECE, UPRM 22

23 Fast wavelet transform analysis bank INEL 6209 (Spring 2010) ECE, UPRM 23

24 2 scale FWT analysis bank and frequency splitting characteristics INEL 6209 (Spring 2010) ECE, UPRM 24

25 Example:d2 scale FWT of {1,4, 3,0} using Haar scaling and wavelet vectors INEL 6209 (Spring 2010) ECE, UPRM 25

26 Inverse FWT synthesis filter bank INEL 6209 (Spring 2010) ECE, UPRM 26

27 2 scale FWT inverse synthesis bank INEL 6209 (Spring 2010) ECE, UPRM 27

28 Example INEL 6209 (Spring 2010) ECE, UPRM 28

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30 2D FWT INEL 6209 (Spring 2010) ECE, UPRM 30

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32 Symlets or symmetrical wavelets Have aeeast least asymmetry y and highest number of vanishing moments for a given compact support Low pass reconstruction filter g0(n)=hφ(n) ) for 0 n 7 Wavelets in image processing Step 1. compute a 2D wavelet transform of an image Step 2. Alter the transform Step 3. Compute the inverse transform INEL 6209 (Spring 2010) ECE, UPRM 32

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34 Orthonormal 4 th order symlet filter coefficients (Daubechies[1992]) INEL 6209 (Spring 2010) ECE, UPRM 34

35 Wavelet based edge detection Dilineate signal and background Zeroing horizontal details isolate the vertical edges INEL 6209 (Spring 2010) ECE, UPRM 35

36 Wavelet based edge detection INEL 6209 (Spring 2010) ECE, UPRM 36

37 Wavelet based noise removal Step 1. choose wavelet and number of levels (scales)p, for decomposition. Compute FWT of the noisy image Step 2. threshold detail coefficients. Select and apply threshold h to dtil detail coefficients i from scales J 1 to J P. Hard thresholding : setting to zero the elements whose absolute values are < the threshold and then scaling the nonzero coefficients toward 0. Step 3. compute the IWT using original approx. coefficients at level J P and modified detail coefficients for levels J 1 to J P INEL 6209 (Spring 2010) ECE, UPRM 37

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39 FWTs using the wavelet toolbox [lod,hid,lor,hir]=wfilters(wname) lor HiR]=wfilters(wname) [f1,f2]=wfilters(wname, type) Waveinfo(wfamily) f il [phi,psi,xval]=wavefun(wname,iter) [lod,hid,lor,hir]=wfilters( haar ) Waveinfo( haar ) INEL 6209 (Spring 2010) ECE, UPRM 39

40 F=magic(8); Manipulating transform decomposition vector [c1,s1]=wavedec2(f,3, haar ); Size(c1) approx=appcoef2(c1,s1, haar ) horizdet2=detcoef2( h,c1,s1,2); newc1=wthcoef2( h wthcoef2( h,c1,s1,2); Newhorizdet2=detcoef2( h,newc1,s1,2); INEL 6209 (Spring 2010) ECE, UPRM 40

41 [cs]=wavedec2(x [c,s]=wavedec2(x,n,lod,hid) [c,s]=wavedec2(x,n,wname) F=magic(4) i() [c1,s1]=wavedec2(f,1, haar ) [c2,s2]=wavedec2(f,2, haar ) INEL 6209 (Spring 2010) ECE, UPRM 41

42 Displaying wavelet decomposition F=imread( vase vase.tif tif ) coefficients [c,s]=wavefast(f,2, db4 ) Wavedisplay(c,s,); figure; wavedisplay(c,s,8); Figure; wavedisplay(c,s, 8); INEL 6209 (Spring 2010) ECE, UPRM 42

43 Inverse FWT G=waverec2(cswname); G=waverec2(c,s,wname); G=waverec2(c,s,loR,hiR) INEL 6209 (Spring 2010) ECE, UPRM 43

44 Wavelets in image processing wavelet directionality and edge detection f=imread( A.tif ); imread(a.tif Imshow(f) [cs]=wavefast(f [c,s]=wavefast(f,1, sym4 ); figure; wavedisplay(c,s, 6); [nc,y]=wavecut( a,c,s); figure;wavedisplay(nc,s, 6); edges=abs(waveback(nc,s, sym4 )); figure; imshow(mat2gray(edges)); INEL 6209 (Spring 2010) ECE, UPRM 44

45 Using wavezero to generate increasingly smoothed hdversions f=imread( Atif ); f=imread(a.tif [c,s]=wavefast(f,4, sym4 ); Wavedisplay(c,s,20); [c,g8]=wavezero(c,s,1, sym4 ); [c,g8]=wavezero(c,s,2, sym4 ); [c,g8] g8]=wavezero(c,s,3, sym4 ); [c,g8]=wavezero(c,s,4, sym4 ); INEL 6209 (Spring 2010) ECE, UPRM 45

46 Progressive reconstruction f=imread( Strawberries.tif ); [cs]=wavefast(f [c,s]=wavefast(f,4, jpeg9.0); 0 ); wwavedisplay(c,s,8); f=wavecopy( a,c,s); figure; imshow(mat2gray(f)); [c,s]=waveback(c,s, jpeg9.7,1); (,,jpg,); f=wavecopy( a,c,s); [c,s]=waveback(c,s, jpeg9.7,1); f=wavecopy( a,c,s); figure; imshow(mat2gray(f)); [c,s]=waveback(c,s, jpeg9.7,1); f=wavecopy( a,c,s); figure; imshow(mat2gray(f)); [c,s]=waveback(c,s, jpeg9.7,1); f=wavecopy( a,c,s); cs) figure; imshow(mat2gray(f)); [c,s]=waveback(c,s, jpeg9.7,1); f=wavecopy( a,c,s); figure; imshow(mat2gray(f)); INEL 6209 (Spring 2010) ECE, UPRM 46