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Introduction to image compression Why image compression? Ex: 3504X2336 (full color) image : 3504X2336 x24/8 = 24,556,032 Byte = Mbyte Objective Reduce the redundancy of the image data in order to be able to store or transmit data in an efficient form. 3 EE lab.530
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Presenter : r98942058 余芝融
1EE lab.530
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 data in an efficient form.
3EE lab.530
Introduction to image compression For human eyes, the image will still
seems to be the same even when the Compression ratio is equal 10
Human eyes are less sensitive to those high frequency signals
Our eyes will average fine details within the small area and record only the overall intensity of the area, which is regarded as a lowpass filter.
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Quick Review Fourier Transform
Does not give access to the signal’s spectral variations
To circumvent the lack of locality in time → STFT
dtetfF tj
)()(
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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
a wavelet function, the transform is computed for each segment.
The width changes with each spectral component
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Continuous Wavelet Transform Wavelet: finite interval function with zero mean(suited to analysis transient signals)
Utilize the combination of wavelets(basis func.) to analyze arbitrary function
Mother wavelet Ψ(t):by scaling and translating the mother wavelet, we can obtain the rest of the function for the transformation(child wavelet, Ψa,b(t)))(1)(, a
bta
tba
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Continuous Wavelet Transform Performing the inner product of the
child wavelet and f(t), we can attain the wavelet coefficient
We can reconstruct f(t) with the wavelet coefficient by
dttftfw bababa )()(, ,,,
2,, )(1)(
adadbtw
Ctf baba
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Continuous Wavelet Transform
Adaptive signal analysis -At higher frequency , the window is narrow,
value of a must 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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Gaussian Window for S-Transform
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High Frequency
Low Frequency
Time Shifted
SKC-2009
Discrete Wavelet Transform Advantage over CWT: reduce the
computational complexity(separate into H & L freq.)
Inner product of f(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,
mmnm
mmnm
t
baat
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Discrete Wavelet Transform
The DWT coefficient
We can reconstruct f(t) with the wavelet coefficient by
dtnbtatfattfw mmnmnm ))(()()(),( 00
2/0,,
)()( ,, twtf nmm n
nm
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Subband Coding
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WT compression
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
½ ½
n
h[n]
-3 -2 -1 0 1 2 3
½
-½ then
1,
2 2 12L
x n x nx n
1,
2 2 12H
x n x nx n
(Average of 2-point)
(difference of 2-point) 17EE lab.530
Haar Transform
2-steps 1.Separate Horizontally 2. Separate Vertically
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2-Dimension(analysis)
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Diagonal
Horizontal
Edge
VerticalEdge
Approximation
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 TransformStep 2:A C A+
BC+D
B D LL HL
L HA-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 HL1 LL2 HL2 HL1LH2 HH2
LH1 HH1 LH1 HH1
LL3 HL3HL2
HL1LH3 HH3
LH2 HH2
LH1 HH1
First level
Second level
Third level
Most important part of the image
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Example:
68 103 6 19 326 -38 6 1976 79 -4 -7 16 -32 2 -72 -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 1 915 18 17 25 33 42 -3 -821 22 19 18 43 37 -1 1
Original image O
1st horizontal separation
1st vertical separation
2nd level 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
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
EZW Scanning Order
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LL3 HL3HL2
HL1LH3 HH3
LH2HH2
LH1 HH1
scan order of the transmission band
EZW Scanning Order
EE lab.530 30scan order of the transmission
coefficient
Scanning Order
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sp: significant positivesn: significant negativezr: zerotree rootis: isolated zero
Example: Get the maximum coefficient=26 Initial threshold :
1. 26>16 →sp 2. 6<16 & 13,10,6, 4 all less than 16→zr 3. -7<16 & 4,-4, 2,-2 all less than
16→zr 4. 7<16 & 4,-3, 2, 0 all less than
16→zrEE lab.530 32
16 226
2log0 T
Each symbol needs 2-bit: 8 bits The significant coefficient is 26, thus put it into the refinement label : 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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44/0 T
2-level Quantizer(For Refinement)
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New Threshold: T1=8
iz zr zr sp sp iz iz→ 14-bit
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Important feature of EZW It’s possible to stop the compression
algorithm at any time and obtain an approximate of the original image
The compression is a series of decision, the same algorithm can be run at the decoder to reconstruct the coefficients, but according to the 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,” SPIE proceedings series, vol. 3078, pp. 656-667, 1997 .
[4] M. Barlaud et al., "Image Coding Using Wavelet Transform" IEEE Trans. on Image Processing 1, No. 2, 205-220 (April, 1992).
[5] J. M. Shapiro, “Embedded image coding using zerotrees of wavelet coefficients,” IEEE Trans. Acous., Speech, Signal Processing, vol. 41, no. 12, pp. 3445-3462, Dec. 1993.
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