Ponencia presentada en el XIX Congreso Internacional de Acústica (ICA2007), Madrid, 2-7 Sep 2007.-- PACS: 43.60.Hj.Wavelet processing has become a popular and well-established technique that offers good
results and great flexibility for noise reduction. The basic wavelet de-noising scheme consists of a thresholding of the Discrete Wavelet Transform (DWT) coefficients. Nevertheless, some
improvements over this basic non-linear scheme can be obtained by the use of Undecimated Wavelet Transforms (UWT), also known as redundant, shift-invariant or stationary wavelet transforms, because they provide redundant and translation invariant representations. The introduction of redundancy produces less efficient signal representations but it can be useful in
de-noising. Ultrasonic traces are frequently contaminated with structural or grain noise, which can not be removed by classical time averaging or band-pass filtering. An analysis of wavelet de-noising of some individual traces is presented. In addition, a significant number of synthetic ultrasonic signals are also processed for a statistical study and comparison of UWT and DWT de-noising in terms of the signal-to-noise ratio of the resulting traces.This work has been supported by the Spanish MEC, R&D Project CICYT Ref. DPI2005-00124 of the I+D National Plan and grant FPI BES-2004-5296.Peer reviewe

Pardo, Emilia

Ramos, Antonio

Rodríguez, Miguel A.

San Emeterio, José Luis

Digital.CSIC

                                      19th INTERNATIONAL CONGRESS ON ACOUSTICS                          MADRID, 2-7 SEPTEMBER 2007                                                                                  SHIFT INVARIANT WAVELET PROCESSING OF ULTRASONIC TRACES    PACS: 43.60.Hj  Pardo, Emilia1; San Emeterio, Jose Luis1; Rodriguez, Miguel Angel2; Ramos, Antonio1 1Instituto de Acústica, CSIC, Serrano 144, 28006-Madrid.  empargo@ia.cetef.csic.es 2ETSIT, Univ. Politecnica Valencia, Camino Vera s/n, 46022 Valencia, Spain. marodrig@upvnet. upv.es    ABSTRACT Wavelet processing has become a popular and well-established technique that offers good results and great flexibility for noise reduction. The basic wavelet de-noising scheme consists of a thresholding of the Discrete Wavelet Transform (DWT) coefficients. Nevertheless, some improvements over this basic non-linear scheme can be obtained by the use of Undecimated Wavelet Transforms (UWT), also known as redundant, shift-invariant or stationary wavelet transforms, because they provide redundant and translation invariant representations. The introduction of redundancy produces less efficient signal representations but it can be useful in de-noising. Ultrasonic traces are frequently contaminated with structural or grain noise, which can not be removed by classical time averaging or band-pass filtering. An analysis of wavelet de-noising of some individual traces is presented. In addition, a significant number of synthetic ultrasonic signals are also processed for a statistical study and comparison of UWT and DWT de-noising in terms of the signal-to-noise ratio of the resulting traces.   INTRODUCTION  Grain noise or structural noise is a common problem in ultrasonic non destructive evaluation NDE. This type of noise presents a frequency band very similar to that of the echoes of interest. Figure 1 shows an example of a synthetic grain noise register obtained from a previously developed noise generator. There is a clear overlapping of the frequency spectra of the signal and noise. As a consequence of this spectral overlapping, the usual techniques employed for white noise reduction are not effective. The specific methods proposed for the reduction of this type of correlated noise are mainly based on obtaining spatial or frequency diversity. Among the frequency diversity methods, Split Spectrum Processing (SSP) has had especial significance [1]. De-noising based on wavelet processing constitutes a well established theoretical framework, which was initially developed by Donoho and Johnstone [2-5]. The discrimination between signal and noise in the wavelet domain follows the idea that the signal, with certain waveform structure, is concentrated on a few wavelet coefficients, while the noise, with a more random structure, is spread over a higher number of coefficients. According to this idea, the basic de-noising process can be summarized in three main steps: (i) wavelet transform of the noisy register; (ii) pruning and/or thresholding of the coefficients in the transformed domain; and (iii) reconstruction of the de-noised signal by the inverse transform. Wavelet processing, which offers great flexibility and potential capabilities for removing noise from signals, has been used during the last years for ultrasonic grain noise reduction [6-9]. In this work, the discrete wavelet transform (DWT) and shift-invariant wavelet transforms are used for noise reduction of synthetic ultrasonic traces generated by using a frequency domain model which includes frequency dependent material attenuation and frequency dependent scattering [7-9]. An analysis of different wavelet de-noising schemes for ultrasonic grain noise reduction, using decomposition level dependent thresholds appropriate for correlated noise, is presented.      19th INTERNATIONAL CONGRESS ON ACOUSTICS – ICA2007MADRID 2 Figure 1. Time and frequency representations of a synthetic ultrasonic grain noise register (in blue) and the clean echo signal to be inserted (in red) in order to construct an ultrasonic synthetic trace.  SHIFT INVARIANT WAVELET-BASED DE-NOISING TECHNIQUES The basic wavelet de-noising consists of a thresholding of the discrete wavelet transform (DWT) coefficients. The dyadic discrete wavelet transform (DWT) is obtained from one of the most popular discretizations of the Continuous Wavelet Transform CWT, sampling the dilation coefficient by 2j and the translation coefficient by 2jn, as follows: Zn,jdtn2t21)t(x)n2,2(CWT)n,j(DWTjjjjxx Î÷øöçèæ -== ò¥+¥- y           (1) This produces an orthogonal, non-redundant wavelet decomposition. The DWT coefficients can be computed by the algorithm of Mallat, consisting of a digital filter bank tree combined with decimation blocks [10]. Figure 2.a shows the complete DWT de-noising scheme based on this algorithm. Hˆ  and Gˆ  represent respectively the low-pass and high-pass decomposition filters, and H and G the corresponding reconstruction filters. The operator (2?) represents decimation, and (2?) is an insertion of zeros. Some improvements over this basic non-linear scheme can be obtained by the use of the Undecimated Wavelet Transform (UWT). The redundancy introduced by this transform produces less efficient signal representations, but it can be useful in some applications, for example in de-noising [11-13]. The most immediate way to implement the UWT is the à trous algorithm, that is essentially the same as Mallat’s but removing downsampling, and upsampling in the filter responses [14]. Figure 2.b shows the UWT de-noising scheme based on this algorithm. An alternative implementation of the shift invariant transform is the cycle-spinning scheme [11]. It consists of applying the basic Mallat algorithm to circularly-shifted versions of the input sequence. For each circular shift, an independent reconstruction can be done, and improved results can be obtained by averaging the different reconstructions. Both methods, à trous dual filter bank and averaged cycle-spinning, are equivalent and would lead to identical results if the wavelet coefficients were the same. But in this case, the coefficients are different because the thresholds applied are different: the à trous option selects a unique threshold for all the coefficients at the same level, while the cycle-spinning option selects independent thresholds for every circularly-shifted wavelet decomposition. This leads to two different shift invariant wavelet denoising processors, that have been specifically developed in this work, and have been applied to the problem of ultrasonic grain noise reduction. 0 10 20 30 40 50 60-4-2024time (ms)0 0.5 1 1.5 2 2.5 305001000frequency (MHz)   19th INTERNATIONAL CONGRESS ON ACOUSTICS – ICA2007MADRID 3     Figure 2.(a) Block diagrams of a classical DWT de-noising processor based on the algorithm of Mallat, and (b) a UWT de-noising processor based on the algorithm à trous, for decomposition levels j = 1, 2.   WAVELET PROCESSING OF ULTRASONIC TRACES  Several processing parameters condition the overall performance of wavelet de-noising. The most significant are: a) type of wavelet transform; b) type of mother wavelet; c) type of thresholding; d) threshold selection rule.   In this work we will use the Discrete Wavelet Transform and Undecimated Wavelet Transforms. The two versions of shift invariant processing described before will be used, one based on the à trous implementation of the UWT, and the other on the average cycle-spinning scheme. The influence of the type of wavelet transform in the results will be analysed, keeping the other parameters constant. Daubechies db6 mother wavelet [10], soft thresholding and Minimax [2] level-dependent [5] threshold selection will be used. The maximum decomposition level is selected as J=7, and zero padding is used in the border treatment.   The efficiency in noise reduction will be evaluated in terms of the signal-to-noise ratio defined as:  ( ) ( )÷÷÷øöçççèæ×= åå==ntNintnstNists NitNitlogSNR1212 )()(10          (2)  where ts(i) are the amplitudes of the trace points located in a time window around the desired echo (Nts points), and tn(i) are the amplitudes of the points in the rest of the trace (Ntn points).  Figure 3.a shows a synthetic ultrasonic trace obtained by inserting a single flaw signal into a synthetic grain noise register. A grain noise generator that includes frequency-dependent material attenuation and scattering [7-9] was used. The amplitude of the inserted signal is determined by a factor F that allows varying the initial SNR. The ultrasonic trace presented in figure 3.a was obtained with a factor F = 3.25, presenting an initial SNR = 3.48 dB. The results of processing this trace, with the parameters previously specified, are shown in Figure 3.b, 3.c, 3.d, for DWT, à trous UWT and cycle-spinning UWT schemes, respectively. The resulting SNR after processing are 9.47dB, 18.78dB and 16.91dB. A better performance of the UWT processors can be clearly appreciated.  GHGHGHGH22222222sINsOUTThresholdingG1H1G2H2G1H1G2H2sINsOUTThresholding  (a)  (b)    19th INTERNATIONAL CONGRESS ON ACOUSTICS – ICA2007MADRID 4 Figure 3. Synthetic ultrasonic trace (a) and results of DWT (b) à trous UWT (c) and cycle-spinning UWT (d) de-noising.   A statistical analysis, based on nine sets of 500 ultrasonic traces, obtained with different values of the factor F was performed with the processing parameters previously specified. Figure 4 shows the mean and standard deviation of the SNR values as a function of the factor F, for the nine sets of synthetic ultrasonic traces, after the wavelet de-noising procedure. The initial SNR is also included in each figure. The mean SNR curves show that the different de-noising methods provide improvements over the initial traces, with a moderated rise as F increases. On the other hand, the standard deviation curves tend to diminish as F increases, indicating less dispersed SNR values and therefore more stable de-noising.  Comparing the different processing schemes, the plots show that shift invariant de-noising obtains better performance than DWT, with higher mean values and lower standard deviation in the SNR of the resulting traces.    CONCLUSIONS  The performances of two shift invariant wavelet processors, based on the à trous algorithm and the cycle-spinning procedure have been evaluated for grain noise reduction using synthetic ultrasonic traces. For the set of processing parameters used, both shift invariant de-noising schemes provide better results than the DWT based procedure. A statistical analysis, performed for a set of synthetic traces, shows that shift invariant processing provides higher mean values and lower standard deviation of the resulting signal-to-noise ratios. This indicates a general tendency to higher quality of the traces and greater robustness of the UWT processing in relation to the classical DWT de-noising.   ACKNOWLEDGEMENTS This work has been supported by the Spanish MEC, R&D Project CICYT Ref. DPI2005-00124 of the I+D National Plan and grant FPI BES-2004-5296.   -505(a) Synthetic noisy trace SNR = 3.48 dB-101(b)        DWT    SNR = 9.47 dB-101(c)     A TROUS   SNR = 18.78 dB0 10 20 30 40 50 60-101time(ms)(d) CYCLE SPINNING SNR = 16.91 dB   19th INTERNATIONAL CONGRESS ON ACOUSTICS – ICA2007MADRID 5   Figure 4. Mean value and standard deviation of the SNR (dB) for nine sets of 500 synthetic ultrasonic traces, as a function of the factor F.    REFERENCES:  [1] V.L. Newhouse, N.M. Bilgutay, J. Saniie, E.S. Furgason, “Flaw-to-grain echo enhancement by split-spectrum processing,” Ultrasonics, vol. 20, pp. 59-68, 1982. [2] D.L. Donoho and I.M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika, vol. 81, pp. 425-455, 1994. [3] D.L. Donoho, I.M. Johnstone, G. Kerkyacharian and D. Picard, “Wavelet shrinkage: Asymptotia?”, J. of the Royal Statistical Soc., Series B, vol. 57, pp. 301-369, 1995. [4] D.L. Donoho and I.M. Johnstone, “Adapting to unknown smoothness via wavelet shrinkage,” J. Amer. Statist. Assoc., vol. 90, pp. 1200-1224, 1995. [5] I.M. Johnstone and B.W. Silverman, “Wavelet threshold estimators for data with correlated noise,” Journal of the Royal Statistical Society, vol. 59, pp. 319-351, 1997 [6] A. Abbate, J. Koay, J. Frankel, S.C. Schroeder and P. Das, “Signal detection and noise suppression using a wavelet transform signal processor,” IEEE Trans Ultrason Ferroelectr Freq Control, vol. 44, pp. 14–25, 1997. [7] J.C. Lazaro, J.L. San Emeterio, A. Ramos, J.L. 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Shift invariant wavelet processing of ultrasonic traces

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