Transcript Discrete Wavelet Transform on image compression

DISCRETE WAVELET TRANSFORM
ON IMAGE COMPRESSION
Presenter : r98942058 余芝融
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Overview
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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.
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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

F ( )   f (t )e j t dt

 Does not give access to the signal’s spectral
variations
 To circumvent the lack of locality in time
→STFT
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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
t b
 a ,b (t ) 
(
)
a
a
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Continuous Wavelet Transform
 Performing the inner product of the child
wavelet and f(t), we can attain the wavelet
coefficient

wa ,b   a ,b , f (t )    a ,b f (t )dt

 We can reconstruct f(t) with the wavelet
coefficient by
1
f (t ) 
C


dadb
  wa,b a,b (t ) a 2
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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
 Low freq.
 High freq.
a
large
small
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Gaussian Window for S-Transform
High Frequency
Time Shifted
Low Frequency
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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
a  a0m , b  nb0a0m
m,n  Z
 If a0=2,b0=1, the set of the wavelet
 m,n (t )  a  (a t - nb0 )
m/2
0
m
0
m, n  Z
 m,n (t )  2  (2 t - n)
m/2
m
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Discrete Wavelet Transform
 The DWT coefficient
wm,n  f (t ), m,n (t )  a0m / 2  f (t ) (a0m (t )  nb0 )dt
 We can reconstruct f(t) with the wavelet
coefficient by
f (t )   wm,n m,n (t )
m
n
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Subband Coding
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WT compression
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2-point Haar Wavelet(oldest & simplest)
g[n] = 1/2 for n = −1, 0
h[0] = 1/2, h[−1] = −1/2,
g[n] = 0 otherwise
h[n] = 0 otherwise
g[n]
½
½
-3
-1
0
-2
½
h[n]
1
2
3 n
-3
-2
-1
0
1
2
3 n
-½
then
x1, L  n 
x  2n  x  2n  1
2
(Average of 2-point)
x1, H  n 
x  2n  x  2n  1
2
(difference of 2-point)
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Haar Transform
 2-steps
1.Separate Horizontally
2. Separate Vertically
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2-Dimension(analysis)
Approximatio
n
Horizontal
Edge
Vertical
Edge
Diagonal
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Haar Transform
Step 1:
A
B
C
D
A+B C+D
L
A-B
C-D
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)
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Haar Transform
Step 2:
A
C
B
D
A+B
C+D
LL
A-B
L
H
HL
C-D
LH
HH
LL
HL
(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
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LL1
HL1
LH1
HH1
LL2
LH2 HH2
LH1
First level
Most important
part of the image
HL2
HL1
HH1
Second level
LL3
HL3
LH3
HH3
LH2
HL2
HL1
HH2
LH1
HH1
Third level
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Example:
20
15
30
20
35
50
5
10
17
16
31
22
33
53
1
9
15
18
17
25
33
42
-3
-8
21
22
19
18
43
37
-1
1
1st horizontal separation
Original image O
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
1st vertical separation
2nd level DWT result
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Original
Image
LL
HL
LH
HH
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LL2 HL2
HL
LH2 HH2
LH
LL3
HL3
HH
HL2
LH3 HH3
HL
LH2 HH2
LH
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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)
sp
sn
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EZW Scanning Order
LL3
LH3
HL3
HL2
HH3
HL1
LH2
HH2
LH1
HH1
scan order of the transmission band
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EZW Scanning Order
scan order of the transmission coefficient
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Scanning Order
sp: significant positive
sn: significant negative
zr: zerotree root
is: isolated zero
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Example:
 Get the maximum
coefficient=26
 Initial threshold :
T0  2
log2 2 6
 16
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→zr
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 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: T0 / 4  4
 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 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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