· Home · Table of Contents · Methods & Instrumentation | Noise reduction and data compression of pipeline magnetic flux leakage signal based on wavelet transformYang-Lijian, Fong-Haiying and Wong-YumeiSchool of Information Science and Engineering, Shenyang University of Technology,Shenyang, China. Contact |
A mass of data from pipeline magnetic flux leakage signals requires large storage space. Effective and real-time data compression technology is necessary. This paper offers the algorithm that applies bi-orthogonal spline wavelet on wavelet decomposition and reconstruction of MFL signals. It reduces measure noise effectively, and implements data compression and feature extraction at the same time. Example is provided to show the real-time compression and high signal-to-noise rate could be obtained.
IndexTerms: Bi-orthogonal Spline Wavelet, Multiresolution and multiscaling analysis, noise reduction, data compression, magnetic flux leakage (MFL)
Pipeline magnetic flux leakage signals were obtained by MFL method. Pipeline was magnetized when the MFL apparatus moved along with oil or gas in it. The local MFL signals were measured by probes, whose values corresponded to the conditions of pipeline erosion. Equidistant sampling was used for data acquisition. So the analysis of erosion signals can be regarded as time-frequency analysis after signals were normalized.
The data of signals were measured by multi-sensors, and quantized by A/D converter. Generally the pipeline is umpty to several hundred kilometers long, so the raw data of signal were bulky. Field signals are disturbed, moreover digital signal preprocessing and feature extraction are difficult. noise reduction and data compression are required. Whereas coding compression is unhelpful to on-line processing and hamper the real-time of measurement.
In this paper, bi-orthogonal spline wavelet was used for multi-resolution signal analysis. In pipeline MFL on-line detection system, measured noises are mainly high frequency signals corresponding to the small scale, and the noises are almost singular everywhere. It's energy reduced rapidly with the increase of scale. Noise signals are highly uncorrelated in different scales. However the wavelet transform of MFL signal assume high correlation, good smooth. In smaller several scales, the amplitude of signals didn't reduce with the scale increase. The local modulus maxima of consecutive scales nearly show up in the same position and have the same symbol. So the noise reduction and data compression are obtained at the same time based on the theory that noise has the different characteristics at different scales [1].
A. Selection of wavelet
In this paper, wavelet transform was used for multiscale signal analysis. Then the following factors are considered on selecting basic wavelets.
1. Orthogonality.
The sub-band data decomposed by orthogonal wavelet falling on the reciprocal orthogonal subspaces respectively. The correlation of each subband is reduced, but the FIR filters that are accurately reconstructed with orthogonality and linear phase don't exist. So select bi-orthogonal wavelet was selected by lowering conditions.
2. Compactly supported set.
The compactly supported basic wavelet is necessary. In this way, the finite length filter banks h_{i}(k)g_{0}(k) were gained and the intercept distortion of filters were avoided..
3. Symmetry.
Symmetrical filter banks have linear phase feature, so quantization error and edge distortion of reconstruction signals are diminished.
4. Regularity.
It is used for measuring the smoothing of wavelet function, and plays an important role in the minimum quantification error.
5. Vanished moment orders.
When the order is large enough, many data of high frequency under delicate scale may be neglect.
So two order bi-orthogonal spline wavelet is selected based on the above-mentioned factors. It is bi-orthogonal, compactly supported, linear phase and good smoothing. Furthermore, the computation is simple, and the real-time implementation is available.
B. The algorithm of wavelet decomposition and reconstruction
From a signal processing point of view, a wavelet is a bandpass filter. In the dyadic case, it is actually an octave band filter. Therefore the wavelet transform can be interpreted as a constant-Q filtering with a set of octave-band filters, followed by sampling at the respective Nyquist frequencies (corresponding to the bandwidth of the particular octave band) [2]. It is thus clear that by adding higher octave bands, one adds details, or resolution, to the signal. So let the number of resolutions be J, we are considering J octaves. Therefore, the frequency index j varies as 1,2,^{...}J corresponding to the scales 2^{1},2^{2},^{ ...}2^{J}. It takes a length N sequence X^{(0)} as input. The DWT algorithm can be succinctly represented as
(1) |
(2) |
Where n= ,^{...}, corresponding to the resolutions 1,2, ^{...}j^{...}J, h_{0}(k), h_{1}(k) are low-pass filter and high-pass filters derived from wavelet respectively, and l, m are the length of filters h_{0}(k), h_{1}(k). X^{ (j)}(n) is the discrete smoothing approximation output of low-pass filters; d^{ (j)}(n) is the discrete detail component output of high-pass filters. X contains exactly 2N elements, where each row corresponds to an octave. Thus, the zeroth row contains N elements (the input), the first row contains N/2 outputs, the second contains N/4 outputs and so on. d contains N elements, the first row contains N/2 output, the second contains N/4 outputs and so on. This algorithm is pictorially shown in Fig.1 (note the decimation by 2, which is reflected in (1) and (2) as the factor 2 in the index of X). This can be expressed by the following pseudocode [3]:
Begin {DWT}Signals reconstruction is the inversion of DWT. Its steps are shown in Fig.2. Firstly, interpolation is used to every branches, whereby the length of sequence before sampled was recovered, then reconstruction using the low-pass and high-pass filters g_{0,} g1 (to smooth the waveform patched zeroes). The equation (3) expresses the reconstruction of signals [4].
(3) |
Fig 1: the DWT block diagram. |
Fig 2: the reconstruction block diagram. |
In fig1,fig2, h_{0}(k)h_{1}(k) are low-pass filter and high-pass filter of wavelet decomposition, and g_{0}(k)g_{1}(k) are low-pass filter and high-pass filter of reconstruction respectively.
A. Multi-scale and multi-resolution analysis of MFL signals and feature extraction
The discrete MFL signals data were transformed by using the coefficients of filter banks of bi-orthogonal spline wavelet, which is shown in Fig.3. (Two order bi-orthogonal spline wavelet filter banks coefficients were listed in table 1.)
n | h_{0}(n) | h_{1}(n) | g_{0}(n) | g_{1}(n) |
0 | 0 | 0 | 0 | 0 |
1 | -0.17677669529664 | 0.35355339059327 | 0.35355339059327 | 0.17677669529664 |
2 | 0.35355339059327 | -0.70710678118655 | 0.70710678118655 | 0.35355339059327 |
3 | 1.06066017177982 | 0.35355339059327 | 0.35355339059327 | -1.06066017177982 |
4 | 0.35355339059327 | 0 | 0 | 0.35355339059327 |
6 | -0.17677669529664 | 0 | 0 | 0.17677669529664 |
Table 1: The coefficients of filter banks |
In Fig. 3, the frequency axis is divided into several frequencies and by scale parameters 2^{(j)}. The frequency bands d1, d2, d3, d4 are the detail components of four layers wavelet decomposition. Their rate of changes and peak values offered good feature of frequency domain. C1, C2, C3 and C4 are the smoothing approximations of four layers wavelet decomposition. From the fig 3, we can remark:
Fig 3: Bi-orthogonal spline wavelet decomposition of MFL signals.. |
As a result, the retained data assumed obviously physical signification, and at the same time data compression was implemented.
B. The signal-to-noise ratio (SNR) and the compression ratio (CR)
The calculation of SNR
(4) |
y(i) is the signals eliminated noise; N(i) is the noise [1].
Fig.4 (a) is the comparison of raw signals and reconstructed signals that data of channel d1 was removed when reconstructed, where the CR is 2, and SNR is 9.1dB. Fig.4 (b) is the comparison of raw signals and reconstructed signals that data of channel d1 and d2 are removed when reconstructed, where the CR is 4, and SNR is 8.8dB. Therefore, the CR is doubled when d2 is removed while the SNR hardly changed.
Fig 4: raw signal and reconstructed signals of data compression of bi-orthogonal spline wavelet.(abscissa is the distance,unit (mm);ordinate id the output of sensors, unit (V)) (a)raw signals and reconstructed signals after removed d1 (CR 2) (b)raw signals and reconstructed signals after removed d1,d2 (CR 4) |
Bi-orthogonal spline wavelet filter banks used in this paper are easily implemented in computer. It improved the real-time ability of signal pre-procession. On the other hand, bi-orthogonal spline wavelet assume linear phase which make reconstruction distortion very small and meet the needs of noise elimination, data compression and feature extraction of MFL signals. So noise reduction not only was obtained but also data compression was realized after MFL signal wavelet transformation.
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