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.
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duke.edu/pub/brani/papers/wav4kids[A].ps.z
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on Applied Mathematics. SIAM, Vol. 16, pp. 195-199.
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Manual, Available by HTTP at http://playfair.stanford.edu/reports/wavelab/wavelabref.pdf,
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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