·Table of Contents
·Methods and Instrumentation
Denoising Method of AE Signal by Using Wavelet Transform
JingRong Zhao Yukuan Ma and Hong Gao
JILIN University of Technology, E-mail: firstname.lastname@example.org
Changchun Huiyuan Test Technology Co.Ltd. P.R.China
We introduce a new algorithm of AE signal denoising by using wavelet analysis in this paper. Owing to the different singularities of the signal and noise, the local maxima of their dyadic wavelet transform modulus are different too. According to the multi-scale behaviors of wavelet transform of signal and random noise in AE data, the algorithm of AE denoising based on wavelet analysis is proposed. The results show the method is valid.
Key words: AE (acoustic emission) signal, wavelet analysis, multi-scale propagation behavior,Demising
The AE technique plays more and more important role in NDT field, especially in material researching, pressure vessel evaluating and intensity watching and measuring for the component of plane. However the research for signal processing technique in AE detection is still a key problem in theory research and engineering application of AE detection technique. In recent years, all digital AE instruments have come out, these provided guarantee on hardware. So it is necessary for us starting from software to find new method for AE signal processing, especially to seek a new digital signal processing way for noise removing. Wavelet analysis is a significant branch of applied mathematics, which has developed in recent years. It is called mathematical microscope, for it can be applied to remove noise from signal, coding and encoding, edge detection, data compress, mode identification and so on. Wavelet analysis has been used in signal processing more and more widely, and it had been used for removing noise from signal in seismic data. Wavelet theory demonstrates that Lipschitz regularity of signal can be calculated its multi-scale behaviors of wavelet transform on modulus maxima figures. When we process signal using wavelet transform, because of the different Lipschitz regularity of signal and random noise, the change rules of the extremum point are different along with the change of scales. So according to this we can remove the extremum point corresponding to noise and then reconstruct signal. Thus purpose of denoising is achieved. In this paper we just take these into account, using wavelet analysis in AE detection, put forward a new algorithm of AE signal denoising.
II. PRINCIPLE OF DENOISNG BASED ON WAVELET ANALYSIS
A.Theory of wavelet transform
The wavelet transform of a function f(t)ÎL2(R)is defined by
L2(R) denotes the Hibert space of measurable, square-integrable functions. The function is said to be a wavelet if and only when the following condition is satisfied.
There is the dilation of Y (t) by the scale factor s.
In order to be used expediently in practice,s is scattered as discrete binary system ,i.e. Let ,then the wavelet is ,its wavelet transform is
Hence its contrary transform is
There x(t) satisfies .
Being dispersed in time domain farther, a discrete wavelet transform can be obtained. It exist an effective and fast algorithm
there W2,f is the wavelet transform coefficients of f(t)ÎL2(R)*s2,f approximates to f(t) on the scale 2j×Hj,Gj are the discrete filters gained by inserting(2j- 1) zeros into every two samples of H,G. And the relation between G and H is:
B. The relation between signal singularity and signal local maxima of wavelet transform modules
Singularities of signal and random noise are different. Their dyadic wavelet transform maximum modules have variable rules by variable scales. A remarkable property of wavelet transform is its ability to characterize the local regularity of functions.
Usually the local regularity of function is often measured with Lipschitz regularity.
Let n be a positive integer and n<a£n+1. A function f(x) is said to be Lipschitz a, at x0, if and only if there exists two constants A and h0>0 ,and a polynomial of order n, Pn(x),such that for h>h0
Literature  indicated that a function f(x) that is continuously differentiable at a point is Lipschitz 1 at this point. If the derivative of f(x) is bounded but discontinuous at x0,f(x) is still Lipschitz 1 and x0, and following Definition 1, we consider that if f(x) is Lipschitz not singular at x0.
Let f(x)ÎL2(R)and[a,b] an interval of R. Let 0<a<1. For any e>0, f(x) is uniformly Lipschitz a over [a+e,b-e], if and only if for any e>0 there exists a constant Ae such that for xÎ[a+e,b-e] and s>0,
In common use of dyadic scale of wavelet transform,equations above becomes
here j is scale,if Lipschitz regularity of function is positive at x,then from equations (9) one can prove that along with scale j increasing,the maxima of wavelet transform modules of wavelet transform is increased,if Lipschitz regularity is negative, then the case is opposite.
For Broad Band random noise, its characteristic of wavelet transform was described in .
Let n(x) is a real and wide stationary random white noise ,with mean square s2, is its dyadic wavelet transform. Let wavelet y(x) be a real function,then also is a random process,its mean square is
Due to the wavelet function has a compact support, is a constant,s2 is a constant too, the average amplitude of is in the inverse ratio of 2j.
At a given scale 2j,wavelet transform is a random process of variable x,if suppose n(x) be gauss noise white,then is a gauss process either. Literature  made a deduction of its average density of maximum modules equal to
here y(1)(x) and y(1)(x) are first and second derivative of y(x). This shows average density of gauss white noise is in the inverse ratio of scale 2j,that is to say the larger scale is ,the sparser its average density is. Hence equations (11),(12) become one of important characteristics to distinguish the maximum modules spread behavior of wavelet transform of signal and noise at different scale.
III. DENOISING ALGORITHM OF AE SIGNAL
As has been said before, coefficients of signal wavelet transform increased with scale increase. However because of the negative singularity of white noise, its amplitude and density and mean square decrease along with scale increasing. So if the amplitude and average density of wavelet transform local maximum modules increase rapidly with the scale decreasing, then it shows that the corresponding singular point has a negative Lipschitz regularity, and the maximum modules mostly dominated by white noise should be eliminated for denoising. But if the amplitude of signal maximum modules is larger than that of white noise at some points, well then on the large scale according to the maximum modules spreading behavior of wavelet transform of signal and noise at different scale, one can distinguish which maximum modules points are dominated by signal. But on some small scales where the SNR (signal-to-noise) are low relatively, the position and amplitude of local maximum modules are mostly dominated by noise, hence it is hardly to directly use information about the maximum modules at this scale to detect signal.
Therefore we design denoising algorithm of AE signal as follows:
- First transform the signal with noise by discrete binary wavelet, and choose the large scale to make the maximum modules of signal dominant. While the scale can't be too large to lost certain important local singularity of signal.
- Second seek the points of maximum modules corresponding to coefficients of wavelet transform at every scale.
- Deal with the maximum modules points at the largest scale 2J as follows:
- Seek the maximal amplitude of maximum modules point,let it be A.
- Because the amplitude and average density of wavelet transform local maximum modules of noise decrease at binary rate with scale increasing, then it shows that on the largest the maximum modules mostly dominated by signal. But some points with lower amplitude maybe the points those spread from the maximum modules dominated by noise at the lower scales, these mainly lie on SNR and selected scale. Therefore we set a threshold following:
In the equation above, N is noise power set before, J is the largest scale selected, Z is a constant, let it be 2. Using the threshold above one can get rid of the amplitude of maximum modules points under T0 on 2J, for considering these maximum modules points dominated by noise.
- For every maximum modules point x0 at scale 2J ,using a arithmetic to search towards its line of maximum modules, that is to search the pickup points corresponding to x0, and for scale 2j, j<J remove the points those aren't on the line maximum modules. Practical course of action is as follows:
- At the scale 2J let the fore-and-aft maximum module points be x1 and x2, the corresponding pickup point of x1 is x'1, then search the corresponding pickup point x0 at (x'1,x2)
- "x'0Î (x'1,x2), if x'0=x0, and have the same sign,then x'0 is the corresponding pickup point of x0.
- If these kind of points are inexistence, one can seek them to left and right direction with the bound of x0,if at(x'1,x2), of the points with same sign satisfies and , then x'0 is pickup point of x0,signed left direction seeking sing=0°
- Let x'0 be pickup point of x0, if at x'0 the amplitude is twice as that at x0, then x'0 and x0 should be removed as maximum modules of noise, or reserved as a pair point (x0,x'0).
- If sign=0,then the maximum modules points at scale 2j , j < j-1 can be searched in (x'1,x0).
- Repeat the steps till the scale 2j.
- In the reserved points, if exist,then will be reserved as the points on the line of maximum modules. While points those dissatisfied the conditions above will be removed.
- Remove all points of maximum modules at first scale 21. According to the corresponding maximum modules points from 2j to 22, to estimate corresponding Lispchitz a and smoothness parameter s ,then work out the first maximum modules point by a and s over again. Meanwhile, its position is as the same as the corresponding maximum modules point at the scale of 22
- Using the kept maximum modules points reconstruct the signal with the alternative projection algorithm.
IV.Content and results of experiment
Based on the denoising algorithm above, we
make a computer simulative experiment on 586
PC computer with C language. Fig.1 shows noisy
signal, which obtained on digital acoustic emission
instrument.The acoustic emission experiment was
simulated on complex test block made of aluminum
plane of 330x165x2mm(0.5mm AE source
of fracture lead). Here we choose three orders
Daubeachies wavelet as basic wavelet function.
Fig.2 shows the module maxima of wavelet transform of noisy signal. It shows that at the higher scales, many module maxima disappeared at corresponding lower scales. So we can conclude that these module maxima points are dominated by noise, which should be removed. While the other module maxima points correspond to the transformation of signal, which should be kept. The results of denoising is shown as Fig.3. The original signal and the processed signal are both given in Fig.3. It shows that the influence of noise was decreased greatly and the effect of waveform resuming of AE signal is fine. And the error between the reconstruct signal and the original is in acceptable range.
Fig 2: module maxima of wavelet transform of noisy signal
Fig 1: noisy signal
Fig 3: original signal and reconstruct signal
- Mallat S,Hwang W L, Singularity Detection and Processing with Wavelet. IEEE Trans on
Information Theory , 1992, 38(2) pp 617~643
- Peng Yuhua,Wang Webing, Application of Wavelet Analysis for Detecting the Arrival Time
of Scattered Wave and Denoising. Acta Electronica Sinica. Vol.21, No.7 pp113~116