# (Inverse continuous wavelets Transform)

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```					Inverse Continuous Wavelets Filter Banc System
Ig Hyun CHO, Seung Houn LEE, Dong Han YOON Abstract
This paper is contributed to the inverse continuous wavelets filter banc system. The inverse continuous wavelets transform permits to reconstruct a signal if the admissibility condition is satisfactory. Then we can obtain the new time-scale plan which is composed by a real amplitude and the scale information of a signal. This plan can be considered as an analysis filter banc system. On the other hands the original signal can be reconstructed by the simple summation of the plan. This process can be considered as a synthesis filter banc system. In this work we have been tried to establish the inverse continuous wavelets filter banc system and applied to remove noise in Ultrasonic signal.

I. The inverse continuous wavelets Transform The mathematical base of a continuous wavelet transform has been established by Grossman and J.Morlet. They demonstrated that the original signal could be reconstructed mathematically [Ref1]. If the function f1(x) , f2(x,) and b,a(x)  L2(), b,a(x)  ,  ( x)  1  ( x  b ) the continuous b, a a a wavelet transform and inverse continuous wavelets transform define as follow. [Ref2]

~ f (t )  FFT 1 a [ F ( )   (a )   (a )] (7)
The inverse continuous wavelets transform has a strict condition for perfect reconstruction of signal f(t). This means that the wavelets admissibility condition must be satisfactory. In frequency domain the admissibility condition becomes

( )  0

(8)

W f (b, a) 
1 Cg
 

1  t b f (t ) *  dt a   a 

(1)

f (t ) 

 

 

1 f  t  b  dadb Wb,a (t )   a  a  a
2

(2)

Cg  



 ( )





d  

By adopting dual wavelets , equation (1) is given by [Ref4].

f (t ) 

a   a  



1 ~ t b Wb,f2 j (t )  ( )db a a  (t ) ~  b,a (t )  j  b,a 2   (2 j  )
 
j  

This means that the wavelets spectrum has not energy near the zero frequency. The wavelets is band pass filter and because of admissibility condition, if the signal f(t) has strong energy near the zero frequency, the perfect reconstruction is not impossible. Thus strong low frequency component must be extracted before wavelets application. If the signal has not strong energy near the zero frequency, the stable reconstruction of the signal by inverse continuous wavelets transform is possible. The next figure shows superposition of the original signal and reconstruction signal by the inverse continuous wavelets transform. The next figure shows reconstruction results. The Original signal and reconstruction signal are superposed. In this calculation, the signal to noise is approximately  1.02 x103 . The signal to noise is given by below formula.

(4) (5)

S / N  20 log 10 ( 2 signal /  2 recocnstrction )

(9)

Equation (4) can be represented by convolution if t equals b.

1 t 1~ t f (t )  a [ f (t )   ( )]   ( )  a Wb,fa (t ) 1 (6) a a a a
Thus in frequency domain equation (5) becomes



Department of Electronic Engineering, Graduate School Kumoh National Institute of Technology 3B SYSTEM  Electronic Engineering, Kumoh National Institute of Technology


1

4
1.8

2

1.6 1.4

0

1.2 1.0

f(t), fr(t)

|(f)|

-2

0.8 0.6

-4

0.4 0.2

-6

f(x) frecontruction(x)
0 2

S/N ratio= -1.02x10

-3

0.0 -0.2 -30 -20 -10 0 10 20 30

4

6

8

10

t

f

Fig.1 Superposition of the original signal f(x) and reconstruction signal by ICWT. Such as in Fig.1, the physical difference between the original signal and the reconstruction signal can be neglected.

Fig 3. Marr wavelets spectrum, N=1000, dt=0.02, a=1, 0.5, 0.25, 0.125, 0.0625 Marr wavelet is deigned directly in frequency domain thus it satisfies perfectly admissibility condition. The other method for construction wavelet is time domain design. This method is generally adopted. For frequency analysis, Gabor wavelet given by Eq.11 can be utilized.

II. Wavelets and Fine scale sampling low The inverse continuous wavelet transform can be effectuated in frequency domain utilizing Fourier Transform as Eq.(7). The next figure shows the flow chart of continuous wavelets transform and the inverse continuous wavelets transform in frequency domain.

 (t ) 

1 ( 2 )1 / 4

e



t2 2 2

e

i 2f 0 t

(11)

Generally, if we design the wavelet in time domain, the admissibility condition is not satisfied numerically. But by choosing the oscillation and the size of wavelet, we can minimize the energy of wavelet near zero frequency. The precision of reconstruction depends on the admissibility condition also scale parameter sampling. In the case of discrete wavelets transform the wavelet is orthornormal, thus by dyadic sampling the reconstruction is possible. On the other hands, in the continuous wavelet transform, dyadic sampling is not sufficient to reconstruct signal because of the rapid diminution of scale sampling point number at a scale level. In this work we propose new scale sampling method as bellow.

IFFT

Fig.2 The Flowchart of the continuous wavelets transform The stable reconstruction of signal needs strictly admissibility condition of wavelet. In order to satisfy perfectly the admissibility condition, the wavelet can be designed in frequency domain. This wavelet called Marr wavelet [Ref7].

a  2 j *

(12)

1 b,a (u )  ( j | u 2 |) m exp[  | u 2 |2 / 2] (10) m!
The coefficient m is the positive integer which control wavelet oscillation and the coefficient u is the frequency of t.

This method satisfies the admissibility condition and compensates scale sampling point number between two dyadic level. In this case wavelet scale parameter a is sampled as follow

amin  2 j * amax ,  *   1 log 2 ( amin )
j amax

(13)

Next figure shows the fine dyadic sampling result of scale parameter.

2

20

15

a=2 amax

10

-j

In analysis filter banc application, we can divide frequency band by the scale parameter sampling. We can choice iteration number j arbitrary and obtain the diverse frequency band information with minimizing frequency interference. The next figure shows non stationary signal and wavelet scalogram.

5

0

0

20

40

60

80

100

iteration ( j=100)

Fig 4 Fine scale sampling ( N=1024, j=100, amax=20, amin=0.02) In this case we can add the scale sampling point between two dyadic scale successively. Thus we can obtain sufficient sampling point for a stable signal reconstruction. III. Multi channel filter banc system From Eq.7 we can obtain new time-scale plan which can be defined by

(a) Scalogram of non stationary signal

(b) wavelet plan of non stationary signal Fig.6 The continuous wavelets transform of non stationary signal (Gabor wavelets) Next figure shows the analysis filter banc application results. The scale bandwidth is divided by 10.
1.0 0.5

1 t 1~ t Wb,fa1 (t )  f (t )   ( )   ( ) a a a a
Then the reconstruction of signal is given by

(14)

f (t )  a W

f 1 b,a

(t )

(15)

f(a1)

0.0

-0.5

-1.0

0

2

4

6

8

10

t

This means that the reconstruction signal can be obtained by the simple summation of time-scale space. Thus the time-scale space has all scale information of signal with real amplitude. The collection of signal, which compose the new time scale plan , f (ai ) ,(i=1,2…n-1) have a difference frequency each other. Thus these signals can be considered as the result of analysis filter banc process. We can obtain all frequency range information of signal. On the other hands, the reconstruction process f (t )  f (a i ) corresponds to synthesis filter

(a) a=0.02~0.031
2.0 1.5 1.0 0.5

f(a2)

0.0 -0.5 -1.0 -1.5 -2.0 0 2 4 6 8 10

t

(b) a=0.313~0.061
1.5 1.0 0.5

f(a3)

0.0

-0.5

-1.0

-1.5 0 2 4 6 8 10

t



(c) a=0.0653~0.112
1.0 0.5

i

banc process. The next figure shows ICWT multi channel filter banc system.
f(a1)

f(a4)

0.0

-0.5

-1.0

0

2

4

6

8

10

t

(d) a=0.1194~0.233
1.0 0.5

ANALYSIS FILTER BANC

f(a5)

f(a2)

SYSNTHESIS FILTER BANC

0.0

-0.5

-1.0

Wf(b,a)

Wf-1(b,a)

f(a3)

fr(t)
1.0 0.8 0.6 0.4

0

2

4

6

8

10

t

(e) 0.2234~0.427
f(t) f(an)
0.2

f(a6)

0.0 -0.2 -0.4 -0.6 -0.8 -1.0 0 2 4 6 8 10

t

(f) a=0.456~0.834 Fig.5 ICWT multi channel filter banc system

3

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1.0 0 2 4 6 8 10

t

(g) 0.8913~1.742
1.0 0.8 0.6 0.4 0.2

can be utilized. But the noise frequency influences all frequency range thus every noise can not be extracted. In this work we utilize the inverse continuous wavelet transform to remove digital noise. The next figure shows the time-scale plan which obtained from the inverse continuous wavelet transform.

f(a7) f(a8)

0.0 -0.2 -0.4 -0.6 -0.8 -1.0 0 2 4 6 8 10

t

(h) 1.7418~3.183
1.0 0.8 0.6 0.4 0.2

①
Time-scale Filter

f(a9)

0.0 -0.2 -0.4 -0.6 -0.8 -1.0 0 2 4 6 8 10

t

(g) a=3.40~6.221
1.0 0.8 0.6 0.4 0.2

②

③

f(a10)

0.0 -0.2 -0.4 -0.6 -0.8 -1.0 0 2 4 6 8 10

t

(h) a=6.651~13 Fig.7 Analysis Filter banc application results. This method permits to obtain all frequency range information of signal with minimizing frequency interference. The multi channel filter banc theory has been developed widely in speech and image coding system using Fourier Transform or Discrete Wavelet Transform which is known as multi resolution analysis. In this work, we have been developed the multi channel filter banc system using the inverse continuous wavelet transform and proposed simple application to remove digital noise of ultrasonic wave. IV. Ultrasonic signal Denoising by Time-scale filter Ultrasonic wave is used frequently to investigate a material internal state. The next figure shows ultrasonic wave which traverses polymer material.
100 80 60

Fig.9 Wavelet plan for Ultrasonic wave In the Fig.9, ①shows the small scale component of the signal which represents digital noise, and ② ③ are input and output signal information. To remove digital noise, we introduce the time-scale filter. Time-scale filter is defined as

1, (a  a  a2 , t1  t  t2 ) w(b, a)   (b, a) 1 0

(16)

This function has a rectangular type and its size is defined by time and scale. In Fig.9 the time scale filter is shown. Then the time scale filtering can be defined as

h(t )  W 1 (b, a)   (b, a)
a a

(17)

The next figure shows the denoising result for the output signal g(t).
8 6 4 2 0 -2 -4 -6

f(t)

20
amplitude

g(t)

0 -20 -40 -60 -80 -100 0 10 20 30 40

g(t)

40

x10
-8 0 10 20 30 40

-6

t(sec)

t
8 6

(a) Before denoising
After Denoising by ICWT

x10

-7

4 2 0 -2 -4

t

Fig.8 Ultrasonic wave. f(t):Input signal, g(t): Output signal. In figure 8, the strong absorption of material attenuate input signal f(t) and then the output signal g(t) information become very weak. Furthermore the digital noise is an obstacle to investigate the shape of output signal g(t). In order to determine ultrasonic parameter such as the velocity and attenuation precisely, the denoising process is needed in strong absorbed material. In order to reduce the digital noise, low pass filtering

g(t)

-6

x10
-8 0 10 20 30 40

-6

t

(b) After denoising Fig.10 Digital noise removing by ICWT filter banc The ICWT denoising method minimizes frequency interference which are origin of singularity of signal and existence of near frequency component. In time

4

scale plan ,these frequency components disperse in scale direction thus the discontinuity of noise is not influenced in frequency direction. Thus all noise component extracted by time-scale filter.

Speech and Signal Processing, ICASSP-98, Seattle, vol. III (May 12-15, 1998), pp. 1533-1536.
[13] Candes, E.J. and Donoho, D.J. Curvelets, Multiresolution Representation, and Scaling Laws, Wavelet Applications in Signal and Image Processing VIII, SPIE, 4119 (2000). [14] B. TORRESSANI-" Analyse Continue par Ondelettes" -Ed. SAVOIRS ACTUELS, Inter-Edition /CNRS Editions,1995, pp 26-28.

V. Conclusion
The inverse continuous wavelet filter banc system is treated. If the admissibility condition of wavelet is satisfied and signal do not have strong energy near zero frequency, the signal can be reconstructed stably. Then the process can be considered as multi filter banc system. We can obtain all frequency range information of signal minimizing frequency interference. On the other hands, we can obtain also the new time scale plan which is obtained by convolution with the continuous wavelet transform and dual wavelet. This plan has the real amplitude information of signal at each scale. Thus we can obtain total signal information at each scale and local information introducing the time-scale filter. In this work, we introduced ultrasonic wave to apply the inverse continuous wavelets transform for removing digital noise. In this case the digital noise is removed effectively without distortion of signal form.

7. Reference [1] A.Grossman and J.Morlet,” Decomposition of Function into Wavelets of Constant Shape and Related Transform” , Mathematics & Physics, Lecture on Recent Results, Vol.1, Edited by L.Streit, World Scienific, Singapore,1986 [2] LEE SeungHoun, Yun DongHan, “Introduction to wavelet Transform” Ed. JINHAN, pp.299. [3] A.Munoz, R.Ertle, M.Unser,”Continuous wavelets Transform with Arbitary scale and O(N) Complexity”, Signal processing ,Vol.82, no.5, pp.749~757,May.2002 [4] M.Unser, Akram.Aldroubi, Steven P.Schiff,”Fast Implementation of the Continuous Wavelets Transform with Integer Scale”, IEEE Transformation on Signal Processing, Vol.42, No.12, December.1994 [5] John.Shadowsky,” Investigating Signal Charactristics Using the Continuous Wavelets Transform” , JOHNS HOPKINS APL TECHNICAL DIGEST VOLUME 17, No.3 pp.258-269, 1996 [6] C.E.Heil, D.F.Walnut,” Continuous and Discrete Wavelets Transform” ,SIAM Review, Vol.31,No.4, pp.628~667,December,1989 [7] Jon.Kirby,” Gravity, Topography and the Continuous wavelets Transform” , Curtin University of Technology, Perth.Western Australia. [8] R.K.Young,” Wavelets Theory and its applicaion” ,Kluver, Boston, 1993 [9] Kaiser.G, “ A Friendly Gulde to Wavelets” Boston, Birkhauser 1994 [10] Vertteri and C.Herley,” Wavelets and Filter Banks, Theory and design” , IEEE Transaction on Signal Processing, Vol.40, N0.9, pp.2207~2232, 1992 [11] C.S. Burrus, R.A. Gopinath, and H. Guo, Introduction to Wavelets and Wavelet Transforms, PrenticeHall, Upper Saddle River, NJ (1998) , pp. 268

[12] James M. Lewis and C.S. Burrus, "Approximate Contnuous Wavelet Transform with an Application to Noise Reduction," Proceedings of the IEEE International Conference on Acoustics,

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