FINAL_Discrete Wavelet Transform on Image Compression

8/13/2019 FINAL_Discrete Wavelet Transform on Image Compression

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Overview

Introduction to image compression

Wavelet transform concepts

Subband Coding Haar Wavelet

Embedded Zerotree Coder

References

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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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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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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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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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44/0 T


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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.

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FINAL_Discrete Wavelet Transform on Image Compression