Presenter : r98942058 余芝融

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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 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

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

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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

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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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