80 - KMUTT · 2002. 6. 13. · 80 On the Experimentation of Data Approximation Using the Discrete...

8 0

On the Experimentation of Data Approximation Using the Discrete

Wavelet Transform Compared with the Discrete Cosine Transform

Manit Kiatkhamjaikajorn ’ Prapakorn Suwanna ’ and Manas Sangworasil ’King Mongkuts Institute of Technology Ladkrabang

Abstract

One advantage of the Wavelet transform is that often a large number of the detailcoefficients turn out to be very small in magnitude. Truncating or removing these small coeffi-cients from the representation introduces only small errors in the reconstruction and because ofthe Multiresolution analysis, they are localization errors. This paper propose the experimentationof data approximation using the wavelet transform. The reconstruction of signal and imageexamples by removing small coefficients from the transform domain are depicted. All processesusing MATLAB program for computing. Results of the proposed method are compared with thetradition Cosine transform method which supports high efficiency of the Wavelet transform.

’ Graduate Student, Lkpartment of J3ectronics

’ Lecturer, Department of Electronics

3 Associate Professor, Department of Electronics

1.1 fWWhb¶!~d6b~ (The Wavelet Transform) [l][Z]

?BJJ%$ L,(R) "ua Continuous-time energy function nl-slbel~sla~~~~~~lu~auwunls

(1)

$j,k(t)=2’ *$(2jI-k), j,k E Z (2)

v/cd = c &dJZ4(2t - n), n E z, g(n) E 1*(z) (5)"

2’-1 ./-I 2”-I

f(t) = ~oc,,,4,,m(d + c c 4,,!%,(~)n=i In=0

(6)

I&J c,, LLRZ d,, &I Coarse ~3t Detail expansion coefficients ~~XJ~I&I

84

Scaling or Father Wavelet

O” 1 Waar i

O*O:ini0 0.5 1

-0.1 O 0.5 I

_~--

-0.1 0 0.5 1

Mother Wavelet

Haar

0.2 --.Daubechies 4 -I

-0.2 10 0.5 1

0.1

0

-0.10 0.5 1

0.1

0

--“,Coiftet 1 ‘n 1

_o~~---.- _J-,,, _

0 0.5 1

@Jn 1 &wm:Y~~ WaVelet basis l~lJl& ~~O&~dhmml Scaling n’io Father Waveletm%d~~a~unm Mother Wavelet

8 5

Haar

IIJZ

l/Jz

Daubechies 4

0.482982913145

0.838518303738

0.224143888042

-0.129409522551

Symmlet 4

-0.075785714789

-0.029835527848

0.497818887832

0.803738751805

0.297857795808

-0.099219543577

-0.012803987282

0.032223100804

Coiflet 1

0.038580777748

-0.128989125398

-0.077181555498

0.807491841388

0.745887558934

0.228584285197

1.4 nisabnlasaa~ab~b~~~~a~ (The Discrete Wavelet Transform : DWT) [l]

cj,n, = C Mn - 2m) * Cj+*,n = Nmn)* c~+1 n l_2m. -”

(7)

(8)

2lJd 2 Analysis filter bank

86

'j(V,>

'j-3 (q_3)‘j-,(V,-z) 1 h 12 k

‘j-,(V,-,> h 12 JL

g12 +ä

g12 +

'j-3 (T-3)dj-3 (T-3)dj-2 (F-2)dj-l CT_,)

il]i 2 ~i’YISVElnisLLE)~392USt~~~0 9Coarse expansion coefficients clhh

C/+~,n = -(n-2d.c,,, + Cg(n-2&d./.m

pld 5 Synthesis filter bank

c,-3 - t 2 h&- 12 g 1 f2 hd

J - 2 -t2g Lf2hd b =J

J - 1 t2g

Column

(4 errory =/Im-m*= fO-fOfO-f() = fc:h f ( t h t / I h t)r=il+l

88

Percent relative L, error = II-f(t)-% x*00%

I/ /I(11)

f(t)2

l &~giru Blocks ~~n~swr&idudas 9 (Piecewise constants)

lh4%i 1 u’s : F(u)= gJf(j)cosI- (2j+1)(2u+l)a4N ) u=O,l,..., N - l (12)

24,v=o,1,..., N-l (13)

89

90

Blocks Bumps HeaviSine Doppler Lena

DCT type IV - 15.7288 40.5399 19.0280 3.0223 2.8801

Haar 3.4818* 23.0081 5.1358** 17.1220** 2.018Q**

Daubechies 4 8.8904 18.2545* 0.7950 8.2275 1.8489

8 8.8840 19.7748 0.2274 5.5887 1.8508

8 10.9925 21.3417 0.2985 4.5599 1.7385

10 11.5007 22.2702 0.3790 3.8823 1.8418

12 14.1484 22.3178 0.3858 3.4811 1.5400

14 13.2517 25.8058 0.4898 3.8844 1.8384

18 13.3528 29.5475** 0.8058 3.5138 1.8039

18 13.8354 28.4911 0.9158 3.8888 1.9705

20 14.8502** 28.1531 1.0838 4.2934 1.8781

Sysmmlet 4 9.8453 18.9010 o.l8Ql* 3.4211 1.5275

5 8.8383 21.8124 0.2733 3.1842 1.4889

8 11.5132 18.9979 0.2801 2.4702 1.3872

7 12.0880 20.2743 0.2580 2.3245 1.4288

8 10.8885 20.1785 0.3288 l.QQ88 ’ 1.3734*

Q 11.8188 22.5324 0.4281 1.8218 1.4887

10 11.8983 22.0830 0.4091 1.8384 1.3873

Coitlet 1 11.3750 18.9582 0.5989 5.2958 1.8820

2 9.5852 18.9778 0.2218 3.4148 1.5322

3 11.0388 18.8380 0.2145 2.3323 1.4850

4 11.4874 20.5522 0.2808 1.8727 1.4551

5 11.8028 21.7025 0.4137 1.8018* 1.5833

92

1.

2.

3.

4.

5.

6.

7.

8.

Fliege. N. J., 1994, Multirate digital signal processing : multirate systems, filter banks and

wavelets. John Wiley & Sons Ltd., Engliand. pp. 151-152 and pp. 251-271.

Vidakovic. B.. and Muller. P., 1994, Wavelets for Kids, Available by FTP at ftp://ftp.isds.

duke.edu/pub/brani/papers/wav4kids[A].ps.z

Young, R. K., 1993, Wavelet theory and its applications, Kluwer Academic Publishers,

pp. 1-2.

Cody, M. A., 1992. The fast Wavelet transform beyond Fourier transforms, Available by

FTP at ftp://ftp.dfw.net in directory /pub/users/mcody/fwt.ps.gs

Daubechies. I., 1992. “Ten Lectures on Wavelets”, CBMS-NSF Regional Conference Series

on Applied Mathematics. SIAM, Vol. 16, pp. 195-199.

Buckheit, J.. Chen, S., Donoho, D., Johnstone, I. and Scargle, J., 1995, WaveLub Reference

Manual, Available by HTTP at http://playfair.stanford.edu/reports/wavelab/wavelabref.pdf,

pp. 162.

Stollnitz, E. J.. Derose. A. D., and Salesin, D. H., 1995. “Wavelets for Computer Graphics :

A Primer, Partl”, IEEE Computer Graphics and Applications, 15(3) : pp. 76-84.

Elllott, D. F., 1987, Handbook of Digital Signal Processing : Engineering Applications,

Academic Press, Inc., California. pp. 486-492.

Blocks

64 biggest co%?cients by DCT ’6

4

.2

0

7‘L

84 biggest coeffic%hs by DWT : Ha&----’ ‘-’ .._. --.l

10 0.5 1

Bumps

-064 biggest cot icients by DCTff

16 ~._I_ ._ ^....__...__ , ".."_,""_"__.~..""_,~, /

I4

/

2

0

84 biggest coeffi%nts by DWT : 046 ir----- ““““, ...” .._. “--., “.̂ _,._,___ __” .,.

1

94

HeaviSine

I_.-- .._ . ..I

0 64 biggest cot! rcients by DCTfPI

84 biggest coeffi%nts by DWT : d 84 biggest coeffi%ts by DWT : Cd

0 0 .5 1

Doppler

064 biggest cot icients by DCTR 1

0.5

0

-0.50 0 .5 1

Blocks c

0 500 1000 1500 2000number of biggest coefficients retained

HeaviSiner I

L IO02tEg lo-‘O“05 1o-2o

5:

0 500 1000 1500 2000number of biggest coefficients retained

Bumps.-

---- -----7

, o-2o

0 500 1000 1500 2000number of biggest coefficients retained

Dopplerf------- “._”

g ,o-20.

z

\““X5,

,0-30/0 500 1000 1500 2000

number of biggest coefficients retained

The original iqage

5% of biggest coefficients by DCT

DWT vs. DCT1o9a

0 5000 10000number of coefficients retained

5% of biggest coefficients by DWT : 58

1001 i. /

150. I

200

2501 I __I

50 100 150 2’30 250