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Overview
Introduction to image compression
Wavelet transform concepts
Subband Coding Haar Wavelet
Embedded Zerotree Coder
References
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Introduction to image compression
Why image compression?
Ex: 3504X2336 (full color) image :
3504X2336 x24/8 = 24,556,032 Byte
= 23.418 Mbyte
Objective
Reduce the redundancy of the image data
in order to be able to store or transmit datain an efficient form.
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Introduction to image compression
For human eyes, the image will still seems tobe the same even when the Compressionratio is equal 10
Human eyes are less sensitiveto those highfrequency signals
Our eyes will average fine details within the
small area and record only the overallintensity of the area, which is regarded as alowpassfilter.
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Quick Review
The time-frequency plane for STFT is uniform
Constant resolution
at all frequencies
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Continuous Wavelet Transform
FT &STFT use wave to analyze signal WT use wavelet of finite energy to analyze
signal
Signal to be analyzed is multiplied to awavelet function, the transform is computedfor each segment.
The width changes with each spectralcomponent
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Continuous Wavelet Transform
Performing the inner product of the childwavelet andf(t), we can attain the waveletcoefficient
We can reconstructf(t) with the wavelet
coefficient by
dttftfw bababa )()(, ,,,
2,,)(
1)(
a
dadbtw
Ctf baba
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Continuous Wavelet Transform
Adaptivesignal analysis
-At higher frequency , the windowis narrow,value of amust be small
The time-frequency plane for WT(Heisenberg)
multi-resolution
diff. freq.
analyze with diff.
resolution
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window a
Low freq. large
High freq. small
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Discrete Wavelet Transform
Advantage over CWT: reduce the computationalcomplexity(separate into H & L freq.)
Inner product off(t)and discrete parameters a& b
If a0=2,b0=1, the set of the wavelet
Znm,, 000 mm anbbaa
n)-t2(2)(
Znm,)n-t()(
2/
,
002/
0,
mm
nm
mmnm
t
baat
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Discrete Wavelet Transform
The DWT coefficient
We can reconstructf(t) with the waveletcoefficient by
dtnbtatfattfw mm
nmnm ))(()()(),( 002/
0,,
)()(,,
twtfnm
m nnm
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Subband Coding
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WT compression
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2-point Haar Wavelet(oldest & simplest)
h[0] = 1/2, h[1] = 1/2,
h[n] = 0 otherwise
g[n] = 1/2 for n= 1, 0
g[n] = 0 otherwise
n
g[n]
-3 -2 -1 0 1 2 3
n
h[n]
-3 -2 -1 0 1 2 3
-then
1,
2 2 1
2L
x n x nx n
1,
2 2 1
2H
x n x nx n
(Averageof 2-point) (differenceof 2-point)
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Haar Transform
2-steps
1.Separate Horizontally
2. Separate Vertically
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2-Dimension(analysis)
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Diagonal
HorizontalEdge
Vertical
Edge
Approximation
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Haar Transform
A B C D A+B C+D A-B C-D
L H
(0,0) (0,1) (0,2) (0,3) (0,0) (0,1) (0,2) (0,3)
(1,0) (1,1) (1,2) (1,3) (1,0) (1,1) (1,2) (1,3)
(2,0) (2,1) (2,2) (2,3) (2,0) (2,1) (2,2) (2,3)
(3,0) (3,1) (3,2) (3,3) (3,0) (3,1) (3,2) (3,3)
Step 1:
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Haar Transform
Step 2:
A C A+B C+D
B D LL HL
L H
A-B C-D
LH HH
(0,0) (0,1) (0,2) (0,3) (0,0) (0,1) (0,2) (0,3)
(1,0) (1,1) (1,2) (1,3) (1,0) (1,1) (1,2) (1,3)(2,0) (2,1) (2,2) (2,3) (2,0) (2,1) (2,2) (2,3)
(3,0) (3,1) (3,2) (3,3) (3,0) (3,1) (3,2) (3,3)
L H LH HH
LL HL
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LL1 HL1LL2 HL2
HL1
LH2 HH2
LH1 HH1 LH1 HH1
LL3 HL3HL2
HL1LH3 HH3
LH2 HH2
LH1 HH1
First level Second level
Third level
Most importantpart of the image
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Example:
68 103 6 19 326 -38 6 19
76 79 -4 -7 16 -32 2 -7
2 -3 4 1 2 -3 4 1
-10 5 -2 -9 -10 5 -2 -9
20 15 30 20 35 50 5 1017 16 31 22 33 53 1 9
15 18 17 25 33 42 -3 -8
21 22 19 18 43 37 -1 1
Original image O 1st
horizontal separation
1stvertical separation 2ndlevel DWT result
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OriginalImage
LH
HL
HH
LL
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LL2 HL2
LH2 HH2
LH
HL
HH
LH
HL
HH
HL2
LH2 HH2
LL3 HL3
HH3LH3
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Embedded Zerotree Wavelet Coder
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Structure of EZW
Root: a
Descendants: a1, a2, a3
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3-level Quantizer(Dominant)
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sp
sn
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EZW Scanning Order
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LL3 HL3
HL2
HL1
LH3 HH3
LH2
HH2
LH1 HH1
scan order of the transmission band
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Scanning Order
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sp: significant positivesn: significant negative
zr: zerotree rootis: isolated zero
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Example:
Get the maximumcoefficient=26
Initial threshold :
1. 26>16sp
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Each symbol needs 2-bit: 8 bits
The significant coefficient is 26,
thus put it into the refinementlabel : Ls= {26}
To reconstruct the coefficient: 1.5T0=24 Difference:26-24=2
Threshold for the 2-level
quantizer: The new reconstructed value:
24+4=28
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2-level Quantizer(For Refinement)
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New Threshold: T1=8
iz zr zr sp sp iz iz14-bit
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Important feature of EZW
Its possible to stop the compressionalgorithm at any time and obtain anapproximate of the original image
The compression is a series of decision, thesame algorithm can be run at the decoder toreconstruct the coefficients, but according tothe incoming but stream.
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References
[1] C.Gargour,M.Gabrea,V.Ramachandran,J.M.Lina, A short introduction to
wavelets and their applications,Circuits and Systems Magazine, IEEE, Vol. 9,No. 2. (05 June 2009), pp. 57-68.
[2] R. C. Gonzales and R. E. Woods, Digital Image Processing. Reading, MA,Addison-Wesley, 1992.
[3] NancyA. Breaux and Chee-Hung Henry Chu,Wavelet methods for
compression, rendering, and descreening in digital halftoning, SPIEproceedings series, vol. 3078, pp. 656-667, 1997 .
[4] M. Barlaud et al., "Image Coding Using Wavelet Transform" IEEE Trans. onImage Processing1, No. 2, 205-220 (April, 1992).
[5] J. M. Shapiro, Embedded image coding using zerotrees of waveletcoefficients,IEEE Trans. Acous., Speech, Signal Processing, vol. 41, no. 12,pp. 3445-3462, Dec. 1993.