2D Haar Wavelet Transform for Image Compression

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2D Haar Wavelet Transform for Image Compression

• Geometric Modeling Project
• Young Lee

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What is Wavelet transform?

• Approximation using superposition of functions
  using wavelet prototype function
• i.e. Fourier series
• f(x) Sn (an cos(nx) bn sin(nx))

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What to do with wavelets?

• History is short
• Lots of applications data analysis/process, data
  compression, denoising, feature extraction, image
  segmentation, storing, synthesizing, fractal, etc

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Background

• The limit of Fourier Transform no time
  resolution Not good for non-stationary signals
• Time-frequency joint representation is necessary
  gt solution Wavelet transform
• Scale is used instead of frequency
• Represented in time-scale space
• Principle of uncertainty frequency vs. time

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Important properties of wavelet

• Admissibility condition
• ? ?(w) 2 / w dw lt ?
• ?(w) 2 w0 0
• ? ?(t) dt 0
• band-pass filter
• Vanishing moment
• ? ?(t) tm dt 0 m 0,1,,M-1
• Gives fast decay of the wave

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Wavelet prototype function a.k.a. Mother wavelet
Y(t)

• Haar ?(t) 1 (0 t 1), -1 (-1 t 0)
  
• 0 otherwise
• Mexican hat ?(t) (1-t2) e-t2/2

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Mexican Hat Haar
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Continuous wavelet transform (CWT)

• Continuous Wavelet Function
• ? a,b (t) a-1/2 ?((t-b)/a)
• a dilation factor, b translation factor
• a-1/2 Energy conservation weighting value when
• E ? ?(t) 2 dt
• Continuous Wavelet Transform
• T(a,b) ? x(t) ?a,b (t) dt

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• Inverse wavelet transform
• x(t) ? ? T(a,b) ?a,b (t) da db/a2
• Original signal is recovered from its wavelet
  transform by integrating over all scales and
  locations, i.e. a and b

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Discrete Wavelet Transform (DWT)

• Dyadic grid for efficient discretization
• Discrete Wavelet Function
• Common choices for DWT wavelet parameters
• a and b is 2 and 1
• ?m,n(t) 2-m/2 ?(2-m t n )
• Property
• ? ?m,n(t) ?m,n(t) dt 1 if m m, n n
• 0 otherwise

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• Discrete wavelet transform
• Tm,n ? x(t) ?m,n(t) dt
• Inverse wavelet transform
• x(t) Sm,n Tm,n ?m,n(t) when -?ltm,nlt?
• Arbitrary signal can be reconstructed using
  wavelet coefficients
• Problem number of m,n is infinite

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Scaling function a.k.a. Father wavelet

• Scaling function
• ?(t) Sm,n Tm,n ?m,n(t)
• On a dyadic grid
• ?m,n(t) 2-m/2 ?(2-m t n)
• Properties
• ? ?(t) dt 1
• Low-pass filter

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Multiresolution formulation

• Scaling and wavelet function
• ?m1,n(t) 2-m/2 Sk ck ?m,2nk(t)
• ?m1,n(t) 2-m/2 Sk bk ?m,2nk(t)
• Approximation and wavelet coefficients
• Sm1,n 2-m/2 Sk ck Sm,2nk
• Tm1,n 2-m/2 Sk bk Sm,2nk

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Signal reconstruction using multiresolution
formulation

• xm(t) Sn Sm,n ?m,n(t)
• dm(t) Sn Tm,n ? m,n(t)
• xm-1(t) xm(t) dm(t) ? xm(t) xm-1(t) - dm(t)
• x(t) xm0(t) Smm0dm(t)
• x(t) Sn Sm0,n ?m0,n(t) Smm0Sn Tm,n ?m,n(t)
• SM,0 ?M,0(t) SmMSn Tm,n ?m,n(t)

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Energy conservation

• Energy of Input signal
• E Sn (S0,n)2
• Energy of final decomposition
• E (SM,n)2 SmMSn(Tm,n)2
• Energy of each stage
• E Si (W(m)i )2
• gt All remains constant

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2D Haar wavelet transform

• 2D scaling function
• ?(t1,t2) ?(t1)?(t2) ltgt ½1 11 1
• 2D horizontal wavelet
• ?h(t1,t2) ?(t1)?(t2) ltgt ½1 -11 -1
• 2D vertical wavelet
• ?v(t1,t2) ?(t1) ?(t2) ltgt ½1 1-1 -1
• 2D diagonal wavelet
• ?d(t1,t2) ?(t1) ?(t2) ltgt ½1 -1-1 1

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Schematic diagram in comparison

• 1D decomposition
• S0
• S1,n T1,n
• S2,n T2,n

• 2D decomposition
• S0
• S1,n Tv1,n Th1,n Td1,n
• S2,n Tv2,n Th2,n Td2,n

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Decomposition of Lena image
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Wavelet coefficient histogram at each scale
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Energy conservation in 2D

• E Si,j (X0 i,j )2 Si,j (W(m)i,j )2
• Energy is well represented by just a few wavelet
  coefficients
• gt Image compression has taken place !!

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Reconstruction in 2D

• Xm-1 Xm Dm
• X0 XM SmM Dm
• when
• Dm Dmv Dmh Dmd
• Dmv Tmv ?mv , Dmh Tmh ?mh , and
• Dmd Tmd ?md
• XM SM ?M

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Reconstruction of Lena image

• X8 X7 X6

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• X5 X4 X3

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• X2 X1 X0

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Image compression procedure

• Choose a wavelet type and decomposition level
• Compute decomposition of signal
• Remove coefficients less than given threshold
• Reconstruct signal

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Image compression

• Compression rate 25

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Image compression

• Compression rate 90

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Image compression

• Compression rate 95

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Image compression

• Compression rate 98

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Advantages and disadvantagesof Haar Wavelet

• Advantage
• Simple and fast
• DisadvantageHarsh aliases horizontal and
  vertical discontinuities
• gt Solution Right choice of wavelet - smoother,
  overlapping kind of wavelets

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Daubechies Wavelets

• Dont have explicit analytic formulation
• Satisfy wavelet criteria
• Have compact support
• Smooth to some degree

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Conclusion

• Wavelet transform is an excellent tool for data
  compression

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• Questions?