Image Compression

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DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF
JOENSUU JOENSUU, FINLAND

• Image Compression
• Lecture 17
• Wavelet Transform
• Alexander Kolesnikov

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Fourier Transform (FT)

• Forward FT

• Inverse FT

• Examples

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Two test signals What is difference?
a)
?1 10? ?2 20? ?3 40? ?4100?
x(t)cos(?1 t)cos(?2t)cos(?3)cos(?4t)
b)
x1(t)cos(?1t) x2(t)cos(?2t) x3(t)cos(?3t) x4(t)
cos(?4t)
x1(t) x2(t) x3(t) x4(t)
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Spectrums of the test signals
a)
Signals are different, spectrums are similar
b)
Why?
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What is wrong with the Fourier Transform?
Lets consider twoi basis functions sin(?t) and
?(t)
Support region In space In frequency
sin(?t) ? 0 ?(t)
0 ? The basis function sin(?t) is not
localized in space (time)! The ?(t) (sample) is
not localized in frequency
Lets introduce basis function which is compact
in space (time) and frequency domains Gabor
? Short-Time Fourier Transform (STFT)
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Short-Time Fourier Transform (STFT)
Input signal
Window h(t)
Signal in the window
Result is localized in space and frequency Why?
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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
STFT is redundant representation?Not good for
compression
Problem the same ?? and ?t throught the entire
plane!
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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
Frequency, ?
Coordinate, t
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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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Scale Illustration
Coast line

• Coarse scale flying
• in a jet at 5 km

• Medium scale bird
• flight at 100 m

• Fine scale beach walk

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Wavelet transform wavlet mother function
Two properties of mother wavelet function
mother wavelet
baby wavelets
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Wavelet transform wavelet mother function
How to obtain a set of wavelet functions?
Translation (?) and dilation (scaling, s)
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Scaling (stretching or compressing)
s1
s0.5
s0.25
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Translation (shift)
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Wavelet functions translation and dilation
s2-7
s2-6
s2-6
s2-6
s2-5
s2-4
s2-3
s2-3
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Examples of mother wavelets
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Haar wavelets
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Scaling function and Wavelets
Scaling function
Wavelet function
where
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 j1.
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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
Low-resolution approx.
Wavelets details
j1
j
NB!
k
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Haar wavelet transform
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Haar wavelet transform
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Haar wavelet transform Example
Input data Xx1,x2,x3,, x16 Haar wavelet
transform (a,b)?(s,d) where 1) scaling
function s(ab)/2 (smooth, LPF) 2) Haar
wavelet d(a-b) (details, HPF)
X10,13, 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
121.25 -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) asd/2 2) bs?d/2
X10,13, 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
121.25 -1.25, 1.75 -2,0.5,-0.5 -3, -3,
-3, -2, -1,-2,-1
Y 121.25 -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
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Wavelet transform as Subband Transform
To be continued...