- Email: [email protected]
Ultrasonic flaw detection in NDE of highly scattering materials using wavelet and Wigner–Ville transform processing
Ultrasonics 42 (2004) 847–851 www.elsevier.com/locate/ultras
Ultrasonic flaw detection in NDE of highly scattering materials using wavelet and Wigner–Ville transform processing M.A. Rodrıguez a
a,*
, J.L. San Emeterio b, J.C. L azaro c, A. Ramos
b
ETSI Telecomunicacion, Universidad Politecnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain b Instituto de Acustica, CSIC, Calle Serrano 144, 28006 Madrid, Spain c ETSI Informatica, UNED, Calle Juan del Rosal 16, 28040 Madrid, Spain
Abstract In ultrasonic non-destructive evaluation of highly scattering materials the backscattering noise may attain peak values greater than the searched flaw pulse and the mean value of noise spectrum is very similar to the searched echo spectrum. Several specific methods have been proposed for the reduction of this type of noise, but the comparison of the performance of different methods is still an open problem. In this paper, we make a comparison among some methods based on simultaneous representations in time and frequency/scale domains of the ultrasonic traces. Synthetic and experimental traces are de-noised using a discrete wavelet processor with decomposition level-dependent threshold selection and a method that combines Wigner–Ville transform and filtering in the time–frequency domain. The results are comparatively evaluated in terms of signal to noise ratio and probability of detection. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Ultrasonic NDE; Grain noise reduction; Wavelet; Wigner–Ville transform
1. Introduction The precise detection of echographic pulses buried in coherent noise is a problem of major importance in ultrasonic non-destructive testing. In order to reduce this type of noise, several methods have been proposed including split-spectrum processing, wavelet transform processing, and techniques based on time–frequency distributions [1–7]. In this paper, a discrete wavelet processor with decomposition level-dependent threshold selection and a method that combines Wigner–Ville transform (WVT) and filtering in the time–frequency domain are used for noise reduction and for detection of a single ultrasonic flaw echo contaminated by coherent grain noise. Synthetic noise registers with an incrusted flaw signal have been processed using both methods. These synthetic ultrasonic traces have been generated by using a frequency domain model that includes frequency dependent material attenuation and frequency dependent scattering. In addition, some experimental pulse-echo NDT traces, obtained from the inspection of a carbon *
Corresponding author. Tel.: +34-9638-79309. E-mail address: [email protected] (M.A. Rodrıguez).
0041-624X/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2004.01.063
fibre reinforced plastic composite block with artificial flat-bottom holes, have been processed. The results provided by the specific Wavelet and Wigner–Ville processors used in this work are comparatively evaluated in terms of signal-to-noise ratio (SNR) and probability of detection (PD).
2. Synthetic and experimental ultrasonic traces Synthetic noise registers are frequently used for the evaluation of signal processing algorithms. In this paper we use a previously developed structural noise model [8]. Single scattering, frequency dependent material attenuation, frequency dependent scattering, and a Gaussian distribution of the scatters are assumed. The received signal Y ðf Þ, including the flaw echo and noise, is modeled in the frequency domain as Y ðf Þ ¼ ððA expðj2pf sD Þ þ N1 ðf Þf 2 ÞH ðf ÞÞ expða0 f 4 Þ þ N2 ðf Þ
ð1Þ
where f is frequency, H ðf Þ is the transmit–receive frequency response of the piezoelectric ultrasonic transducer, N1 ðf Þ represents the scatters distribution which produces the grain noise, and a0 is the attenuation
848
M.A. Rodrıguez et al. / Ultrasonics 42 (2004) 847–851 6
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40
50
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Fig. 1. Initial synthetic trace (a) and results of signal processing: with WVT (b), with wavelet SURE (c), with wavelet MINIMAX (d) and with wavelet kk (k ¼ 3) method (e).
factor. The noise generated by the scatters presents a frequency band very similar to the echo one. The
0
5
10 Time (µs)
15
20
Fig. 2. Experimental ultrasonic trace (a) and results of signal processing: with WVT (b), with wavelet SURE (c), with wavelet MINIMAX (d) and with wavelet kk (k ¼ 3) method (e).
coherent noise received from the flaw region is assumed to be slightly lesser than in the absence of flaw [9] and this is implemented by means of a temporal Gaussian
M.A. Rodrıguez et al. / Ultrasonics 42 (2004) 847–851
window. N2 ðf Þ is additional Gaussian white noise which can be related to the ultrasonic and measurement systems. The flaw echo is modeled as a reflection arriving at time sD by means of the delayed delta function, with A as a weighting factor. Several parameters in this model permit its tuning to different inspection conditions. Noise registers in this work have been generated with a sampling frequency fs ¼ 64 MS/s, attenuation factor a0 ¼ 1:8 1026 s4 and N ¼ 4096 points. Coherent and white noises normalized in amplitude have been added. The flaw signal of amplitude determined by A is added at the central position of the noise register, being A ¼ F =rn where rn is the standard deviation of the particular noise register and F is an index of the input SNR. Several sets of 500 ultrasonic traces with the factor F varying from 2 to 4 have been generated in this way. In particular, 8 sets of 500 traces have been generated with F ¼ 2, 2.5, 2.75, 3, 3.25, 3.5 3.75 and 4. Fig. 1a is an example of a synthetic ultrasonic trace generated with an amplitude given by F ¼ 2:5. Several experimental signals were acquired from a carbon fibre reinforced plastic (CFRP) composite block with flat-bottom holes using a Panametrics transducer (310S) of 5 MHz and the Panametrics Ultrasonic Analyser 5052UA [8]. The ultrasonic traces were acquired by means of a digital oscilloscope, Tektronix TDS 744 of 2GS/s, and data length of 5000 samples, which were transferred via GPIB to a computer. Fig. 2a shows one of these experimental traces.
3. Algorithm description Time-frequency and time-scale methods provide a two-dimensional representation of signals in both time and frequency domains, noting that scale is directly related to frequency. This representation permits to simultaneously exploit the time and the frequency characteristics of the NDE ultrasonic signals which are of finite duration and band limited. 3.1. Wigner–Ville transform processing WVT is a quadratic transform with important properties for ultrasonic applications: time–frequency constant resolution and high energy concentration. The general definition of the WVT of a signal xðtÞ is [10] Z 1 s s j2pf s WVTx ðt; f Þ ¼ xa t þ xa t ds ð2Þ e 2 2 1 where xa ðtÞ is the analytic signal of xðtÞ. The processing begins with the analytic signal calculation of the trace by using the Hilbert transform. After this the Wigner–Ville representation is obtained by using the Boashash method [11]. In the following step, a filtering in the time–frequency domain is performed, try-
849
ing to select and emphasize the relevant frequency content of the searched echo. The process is based in the differences that exist between the time–frequency shape of the echo-pulse and the grain noise one. The echo produces a regular representation in time and frequency while the noise is more irregular, although the mean shapes of the echo and the coherent noise are very similar. In addition, the echo presents an interesting property in the Wigner–Ville domain: in most of the cases it is positive for any pair of time–frequency values. Common ultrasonic pulses can be modeled by a kind of functions which produce positive WVT’s [12], while the noise does not belong to this kind of functions and its WVT produces positive and negative values. Thus, by means of the suppression of the parts with negative values in the WVT, the noise influence is attenuated [7]. The signal reconstruction can be done with a simple sum in the frequency axis after the attenuation of the noise components. The WVT does not preserve phase information, thus the reconstructed signal is related to the original signal envelope. 3.2. Wavelet transform processing The wavelet transform Ws ða; bÞ of a signal xðtÞ is given by Ws ða; bÞ ¼
Z
1
xðtÞW 1
tb a
dt
ð3Þ
where Wð Þ is the mother wavelet and a, b the dilatation and translation coefficients respectively. Wavelet de-noising procedures can be summarized as (i) wavelet transform of the noisy register; (ii) pruning and/or thresholding of the coefficients in the transformed domain; (iii) reconstruction of the de-noised signal by inverse transform. In previous works [4,8,9], the influence of several processing parameters for grain noise reduction has been studied. In this work we use Daubechies [13] DB6 as mother wavelet, soft type and level dependent thresholding. Three methods for threshold value selection are used [14]. Each method is independently applied to each decomposition level. SURE and Minimax threshold selection rules are described in reference [15]. An alternative threshold selection rule, previously proposed [9], is also used. This new threshold kki is determined from the mean and standard deviation of the squared wavelet coefficients kki ¼ ðli þ ksi Þ1=2 ;
li ¼
Ni X
x2ij =Ni ;
j¼1
s2i ¼
Ni X
ðx2ij li Þ2 =ðNi 1Þ
j¼1
where i is the decomposition level.
ð4Þ
850
M.A. Rodrıguez et al. / Ultrasonics 42 (2004) 847–851
In this work, this kki threshold has been used with k ¼ 3, and applied to the two scales with the maximum average of the squared coefficients. The remaining detail decomposition levels are pruned.
4. Simulation results Several sets of synthetic traces generated as described previously have been de-noised using the described Wigner–Ville and Wavelet processors. Nine sets of 500 traces, characterized by the factor of amplitude F , have been processed (including F ¼ 0, without ultrasonic flaw signal). The results have been evaluated by means of two parameters: SNR and PD. The SNR is defined as the quotient of the peak value of the trace determined in a time window around the zone of the trace where the signal was incrusted (target zone) divided by the standard deviation of the whole trace. This target time window is centred with the incrusted signal and has a length double of this signal. The same definition is used for the raw input traces and the output processed traces: SNR ¼
peak value target zone trace standard deviation
threshold ¼ C trace standard deviation
ð6Þ
where C is a constant that varies between 0.1 and 10 with increments of 0.1, and determinates the selectivity of the final detection. Table 1 shows the SNR mean values obtained after processing each set of traces using the different methods. Each set is identified by the amplitude factor F in the first column. The SNR of the initial (non-processed) sets of traces are shown in column 2. Columns 3–6 show the SNR obtained after processing. It can be noticed that Wigner–Ville and Wavelet processors using SURE and Minimax threshold selection rules produce similar results, with Wigner–Ville slightly better in all cases. Minimax performs slightly better than SURE for high initial SNR. In all cases the new proposed threshold selection rule [9], based on the energy of the wavelet coefficients in the transform domain, performs notably better. Fig. 3 shows the PD computed for the sets of traces with F ¼ 2:5, 3, 3.5 and 4. The PD of initial (non-processed) traces are also shown. A clear correlation can be observed between the results obtained by using SNR and PD as evaluation indexes.
ð5Þ
This SNR definition, which is a good measure of performance when a single flaw is present, is similar to the definition used in [2,3,6]. In these works, the standard deviation of the trace excluding the flaw or target region was used as the denominator in Eq. (5). Nevertheless, when the de-noising procedure is very severe the trace portion outside the flaw region may provide a very small standard deviation and therefore the obtained SNR can be abnormally high. For this reason, the standard deviation of the whole trace is used in this work. The PD is estimated as the quotient of the number of traces exceeding a given threshold with the total number of traces. In our case, as the total number of cases in each set is 500, we obtain the PD values with an accuracy of ±0.2%. The thresholds are selected and calculated using the standard deviation of the traces by means of the expression:
5. Conclusions Several sets of 500 ultrasonic traces contaminated with coherent grain noise have been processed using time–frequency and time-scale transforms. In the case of discrete wavelet transform processing, different parameters (mother wavelet, type of thresholding, decomposition level, threshold selection rule) determinate the performance of the de-noising procedure. In this work, DB6, soft thresholding, maximum decomposition level 12, and level dependent thresholds have been used. In the case of WVT processing a single flaw echo is processed and cross-terms do not perturb the results as in the case of multiple echo location. Results summarized in Table 1 for mean values of SNR show that for wavelet processing, SURE and Minimax threshold selection rules produce similar results, Minimax being
Table 1 Comparison of mean values of SNR of initial and processed ultrasonic traces Factor F
Initial
Wigner–Ville
Wavelet SURE
Wavelet Minimax
Wavelet kk (k ¼ 3)
0 2 2.5 2.75 3 3.25 3.5 3.75 4
3.2212 3.5768 3.8420 4.0076 4.1891 4.3853 4.5881 4.7967 5.0086
3.8342 5.0989 5.9036 6.3166 6.7030 7.0477 7.3353 7.5556 7.7170
3.7701 4.5946 5.1590 5.5219 5.9090 6.2887 6.6436 6.9414 7.2251
3.7861 4.4225 5.0105 5.3973 5.8355 6.3000 6.7967 7.2519 7.6212
3.9095 6.6394 8.1376 8.7953 9.3678 9.8064 10.194 10.501 10.679
M.A. Rodrıguez et al. / Ultrasonics 42 (2004) 847–851 1
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slightly better for high initial SNR. Wigner–Ville processing performs slightly better than the previous ones in all cases. The new threshold selection method, based on the energy of the wavelet coefficients, performs the best in all cases with a notable SNR enhancement. The results displayed in Fig. 3 for the different curves of PD present a close correlation with the results for SNR, showing the same tendencies. Further work is needed to check the tendencies presented in this paper with other processing parameters. Other performance comparisons of methods for grain noise reduction have been previously presented [5,16], but additional work is also needed to complete the analysis of the efficiency of different algorithms.
0.8 0.6
Acknowledgements
0.4
This work was supported by the Spanish Ministry of Science and Technology (R&D Project DPI200200441) and CYTED-CNPq (Research Project PULSETS).
0.2 0 1
(c)
References
0.8 0.6 0.4 0.2 0 1
(d)
0.8 0.6 Original Wigner-Ville
0.4
SURE MiniMax k=3
0.2 0 0
2
4
6
8
10
Detection threshold constant: C
Fig. 3. Probability of detection PD of initial and processed ultrasonic traces using different processing techniques: (a) F ¼ 2:5; (b) F ¼ 3; (c) F ¼ 3:5; and (d) F ¼ 4.
[1] V.L. Newhouse, N.M. Bilgutay, J. Saniie, E.S. Furgason, Ultrasonics 20 (1982) 59–68. [2] P. Karpur, P.M. Shankar, J.L. Rose, V.L. Newhouse, Ultrasonics 25 (1987) 204–208. [3] P.M. Shankar, U. Bencharit, N.M. Bilbutay, J. Saniie, Mater. Evaluat. 46 (1988) 1100–1104. [4] A. Abbate, J. Koay, J. Frankel, S.C. Schroeder, P. Das, IEEE Trans. Ultrason. Ferroelec. Freq. Contr. 44 (1997) 14–25. [5] M.A. Malik, J. Saniie, Proc. of the IEEE 1996 Ultrasonic Symp., 1996, pp. 701–704. [6] P. Karpur, O.J. Canelones, Ultrasonics 30 (1992) 351–357. [7] M.A. Rodrıguez, L. Vergara, Appl. Signal Process. 3 (1996) 88– 94. [8] J.C. Lazaro, J.L. San Emeterio, A. Ramos, J.L. Fernandez, Ultrasonics 40 (2002) 263–267. [9] J.C. Lazaro, J.L. San Emeterio, A. Ramos, Proc. of the IEEE 2002 Ultrasonic Symposium, 2002, pp. 756–759. [10] T.A.C.M. Claasen, W.F.G. Mecklenbr€auker, Philips J. Res. 35 (1980) 217–250, 276–300 and 372–389. [11] B. Boashash, P.J. Black, IEEE Trans. Acoustic, Speech Signal Process. 35 (1987) 1611–1618. [12] M.A. Rodrıguez, NDT&E Int. 36 (2003) 441–445. [13] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics, vol. 61, SIAM, Philadelphia, 1992. [14] I.M. Johnstone, B.W. Silverman, J. Roy. Statist. Soc. 59 (1987) 319–351. [15] D.L. Donoho, I.M. Johnstone, J. Am. Statist. Assoc. 90 (1995) 1200–1224. [16] X. Li, N.M. Bilgutay, Proc. 1992 Int. Conf. Communication Technology, 1992, 14B.01.1-6.
Ultrasonic flaw detection in NDE of highly scattering materials using wavelet and Wigner–Ville transform processing M.A. Rodrıguez a
a,*
, J.L. San Emeterio b, J.C. L azaro c, A. Ramos
b
ETSI Telecomunicacion, Universidad Politecnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain b Instituto de Acustica, CSIC, Calle Serrano 144, 28006 Madrid, Spain c ETSI Informatica, UNED, Calle Juan del Rosal 16, 28040 Madrid, Spain
Abstract In ultrasonic non-destructive evaluation of highly scattering materials the backscattering noise may attain peak values greater than the searched flaw pulse and the mean value of noise spectrum is very similar to the searched echo spectrum. Several specific methods have been proposed for the reduction of this type of noise, but the comparison of the performance of different methods is still an open problem. In this paper, we make a comparison among some methods based on simultaneous representations in time and frequency/scale domains of the ultrasonic traces. Synthetic and experimental traces are de-noised using a discrete wavelet processor with decomposition level-dependent threshold selection and a method that combines Wigner–Ville transform and filtering in the time–frequency domain. The results are comparatively evaluated in terms of signal to noise ratio and probability of detection. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Ultrasonic NDE; Grain noise reduction; Wavelet; Wigner–Ville transform
1. Introduction The precise detection of echographic pulses buried in coherent noise is a problem of major importance in ultrasonic non-destructive testing. In order to reduce this type of noise, several methods have been proposed including split-spectrum processing, wavelet transform processing, and techniques based on time–frequency distributions [1–7]. In this paper, a discrete wavelet processor with decomposition level-dependent threshold selection and a method that combines Wigner–Ville transform (WVT) and filtering in the time–frequency domain are used for noise reduction and for detection of a single ultrasonic flaw echo contaminated by coherent grain noise. Synthetic noise registers with an incrusted flaw signal have been processed using both methods. These synthetic ultrasonic traces have been generated by using a frequency domain model that includes frequency dependent material attenuation and frequency dependent scattering. In addition, some experimental pulse-echo NDT traces, obtained from the inspection of a carbon *
Corresponding author. Tel.: +34-9638-79309. E-mail address: [email protected] (M.A. Rodrıguez).
0041-624X/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2004.01.063
fibre reinforced plastic composite block with artificial flat-bottom holes, have been processed. The results provided by the specific Wavelet and Wigner–Ville processors used in this work are comparatively evaluated in terms of signal-to-noise ratio (SNR) and probability of detection (PD).
2. Synthetic and experimental ultrasonic traces Synthetic noise registers are frequently used for the evaluation of signal processing algorithms. In this paper we use a previously developed structural noise model [8]. Single scattering, frequency dependent material attenuation, frequency dependent scattering, and a Gaussian distribution of the scatters are assumed. The received signal Y ðf Þ, including the flaw echo and noise, is modeled in the frequency domain as Y ðf Þ ¼ ððA expðj2pf sD Þ þ N1 ðf Þf 2 ÞH ðf ÞÞ expða0 f 4 Þ þ N2 ðf Þ
ð1Þ
where f is frequency, H ðf Þ is the transmit–receive frequency response of the piezoelectric ultrasonic transducer, N1 ðf Þ represents the scatters distribution which produces the grain noise, and a0 is the attenuation
848
M.A. Rodrıguez et al. / Ultrasonics 42 (2004) 847–851 6
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0.1 0.05
0
0
-0.5
-0.05 -0.1
-1 0
10
20
30
40
50
60
Time (µs)
Fig. 1. Initial synthetic trace (a) and results of signal processing: with WVT (b), with wavelet SURE (c), with wavelet MINIMAX (d) and with wavelet kk (k ¼ 3) method (e).
factor. The noise generated by the scatters presents a frequency band very similar to the echo one. The
0
5
10 Time (µs)
15
20
Fig. 2. Experimental ultrasonic trace (a) and results of signal processing: with WVT (b), with wavelet SURE (c), with wavelet MINIMAX (d) and with wavelet kk (k ¼ 3) method (e).
coherent noise received from the flaw region is assumed to be slightly lesser than in the absence of flaw [9] and this is implemented by means of a temporal Gaussian
M.A. Rodrıguez et al. / Ultrasonics 42 (2004) 847–851
window. N2 ðf Þ is additional Gaussian white noise which can be related to the ultrasonic and measurement systems. The flaw echo is modeled as a reflection arriving at time sD by means of the delayed delta function, with A as a weighting factor. Several parameters in this model permit its tuning to different inspection conditions. Noise registers in this work have been generated with a sampling frequency fs ¼ 64 MS/s, attenuation factor a0 ¼ 1:8 1026 s4 and N ¼ 4096 points. Coherent and white noises normalized in amplitude have been added. The flaw signal of amplitude determined by A is added at the central position of the noise register, being A ¼ F =rn where rn is the standard deviation of the particular noise register and F is an index of the input SNR. Several sets of 500 ultrasonic traces with the factor F varying from 2 to 4 have been generated in this way. In particular, 8 sets of 500 traces have been generated with F ¼ 2, 2.5, 2.75, 3, 3.25, 3.5 3.75 and 4. Fig. 1a is an example of a synthetic ultrasonic trace generated with an amplitude given by F ¼ 2:5. Several experimental signals were acquired from a carbon fibre reinforced plastic (CFRP) composite block with flat-bottom holes using a Panametrics transducer (310S) of 5 MHz and the Panametrics Ultrasonic Analyser 5052UA [8]. The ultrasonic traces were acquired by means of a digital oscilloscope, Tektronix TDS 744 of 2GS/s, and data length of 5000 samples, which were transferred via GPIB to a computer. Fig. 2a shows one of these experimental traces.
3. Algorithm description Time-frequency and time-scale methods provide a two-dimensional representation of signals in both time and frequency domains, noting that scale is directly related to frequency. This representation permits to simultaneously exploit the time and the frequency characteristics of the NDE ultrasonic signals which are of finite duration and band limited. 3.1. Wigner–Ville transform processing WVT is a quadratic transform with important properties for ultrasonic applications: time–frequency constant resolution and high energy concentration. The general definition of the WVT of a signal xðtÞ is [10] Z 1 s s j2pf s WVTx ðt; f Þ ¼ xa t þ xa t ds ð2Þ e 2 2 1 where xa ðtÞ is the analytic signal of xðtÞ. The processing begins with the analytic signal calculation of the trace by using the Hilbert transform. After this the Wigner–Ville representation is obtained by using the Boashash method [11]. In the following step, a filtering in the time–frequency domain is performed, try-
849
ing to select and emphasize the relevant frequency content of the searched echo. The process is based in the differences that exist between the time–frequency shape of the echo-pulse and the grain noise one. The echo produces a regular representation in time and frequency while the noise is more irregular, although the mean shapes of the echo and the coherent noise are very similar. In addition, the echo presents an interesting property in the Wigner–Ville domain: in most of the cases it is positive for any pair of time–frequency values. Common ultrasonic pulses can be modeled by a kind of functions which produce positive WVT’s [12], while the noise does not belong to this kind of functions and its WVT produces positive and negative values. Thus, by means of the suppression of the parts with negative values in the WVT, the noise influence is attenuated [7]. The signal reconstruction can be done with a simple sum in the frequency axis after the attenuation of the noise components. The WVT does not preserve phase information, thus the reconstructed signal is related to the original signal envelope. 3.2. Wavelet transform processing The wavelet transform Ws ða; bÞ of a signal xðtÞ is given by Ws ða; bÞ ¼
Z
1
xðtÞW 1
tb a
dt
ð3Þ
where Wð Þ is the mother wavelet and a, b the dilatation and translation coefficients respectively. Wavelet de-noising procedures can be summarized as (i) wavelet transform of the noisy register; (ii) pruning and/or thresholding of the coefficients in the transformed domain; (iii) reconstruction of the de-noised signal by inverse transform. In previous works [4,8,9], the influence of several processing parameters for grain noise reduction has been studied. In this work we use Daubechies [13] DB6 as mother wavelet, soft type and level dependent thresholding. Three methods for threshold value selection are used [14]. Each method is independently applied to each decomposition level. SURE and Minimax threshold selection rules are described in reference [15]. An alternative threshold selection rule, previously proposed [9], is also used. This new threshold kki is determined from the mean and standard deviation of the squared wavelet coefficients kki ¼ ðli þ ksi Þ1=2 ;
li ¼
Ni X
x2ij =Ni ;
j¼1
s2i ¼
Ni X
ðx2ij li Þ2 =ðNi 1Þ
j¼1
where i is the decomposition level.
ð4Þ
850
M.A. Rodrıguez et al. / Ultrasonics 42 (2004) 847–851
In this work, this kki threshold has been used with k ¼ 3, and applied to the two scales with the maximum average of the squared coefficients. The remaining detail decomposition levels are pruned.
4. Simulation results Several sets of synthetic traces generated as described previously have been de-noised using the described Wigner–Ville and Wavelet processors. Nine sets of 500 traces, characterized by the factor of amplitude F , have been processed (including F ¼ 0, without ultrasonic flaw signal). The results have been evaluated by means of two parameters: SNR and PD. The SNR is defined as the quotient of the peak value of the trace determined in a time window around the zone of the trace where the signal was incrusted (target zone) divided by the standard deviation of the whole trace. This target time window is centred with the incrusted signal and has a length double of this signal. The same definition is used for the raw input traces and the output processed traces: SNR ¼
peak value target zone trace standard deviation
threshold ¼ C trace standard deviation
ð6Þ
where C is a constant that varies between 0.1 and 10 with increments of 0.1, and determinates the selectivity of the final detection. Table 1 shows the SNR mean values obtained after processing each set of traces using the different methods. Each set is identified by the amplitude factor F in the first column. The SNR of the initial (non-processed) sets of traces are shown in column 2. Columns 3–6 show the SNR obtained after processing. It can be noticed that Wigner–Ville and Wavelet processors using SURE and Minimax threshold selection rules produce similar results, with Wigner–Ville slightly better in all cases. Minimax performs slightly better than SURE for high initial SNR. In all cases the new proposed threshold selection rule [9], based on the energy of the wavelet coefficients in the transform domain, performs notably better. Fig. 3 shows the PD computed for the sets of traces with F ¼ 2:5, 3, 3.5 and 4. The PD of initial (non-processed) traces are also shown. A clear correlation can be observed between the results obtained by using SNR and PD as evaluation indexes.
ð5Þ
This SNR definition, which is a good measure of performance when a single flaw is present, is similar to the definition used in [2,3,6]. In these works, the standard deviation of the trace excluding the flaw or target region was used as the denominator in Eq. (5). Nevertheless, when the de-noising procedure is very severe the trace portion outside the flaw region may provide a very small standard deviation and therefore the obtained SNR can be abnormally high. For this reason, the standard deviation of the whole trace is used in this work. The PD is estimated as the quotient of the number of traces exceeding a given threshold with the total number of traces. In our case, as the total number of cases in each set is 500, we obtain the PD values with an accuracy of ±0.2%. The thresholds are selected and calculated using the standard deviation of the traces by means of the expression:
5. Conclusions Several sets of 500 ultrasonic traces contaminated with coherent grain noise have been processed using time–frequency and time-scale transforms. In the case of discrete wavelet transform processing, different parameters (mother wavelet, type of thresholding, decomposition level, threshold selection rule) determinate the performance of the de-noising procedure. In this work, DB6, soft thresholding, maximum decomposition level 12, and level dependent thresholds have been used. In the case of WVT processing a single flaw echo is processed and cross-terms do not perturb the results as in the case of multiple echo location. Results summarized in Table 1 for mean values of SNR show that for wavelet processing, SURE and Minimax threshold selection rules produce similar results, Minimax being
Table 1 Comparison of mean values of SNR of initial and processed ultrasonic traces Factor F
Initial
Wigner–Ville
Wavelet SURE
Wavelet Minimax
Wavelet kk (k ¼ 3)
0 2 2.5 2.75 3 3.25 3.5 3.75 4
3.2212 3.5768 3.8420 4.0076 4.1891 4.3853 4.5881 4.7967 5.0086
3.8342 5.0989 5.9036 6.3166 6.7030 7.0477 7.3353 7.5556 7.7170
3.7701 4.5946 5.1590 5.5219 5.9090 6.2887 6.6436 6.9414 7.2251
3.7861 4.4225 5.0105 5.3973 5.8355 6.3000 6.7967 7.2519 7.6212
3.9095 6.6394 8.1376 8.7953 9.3678 9.8064 10.194 10.501 10.679
M.A. Rodrıguez et al. / Ultrasonics 42 (2004) 847–851 1
(a)
0.8 0.6 0.4 0.2 0 1
(b)
851
slightly better for high initial SNR. Wigner–Ville processing performs slightly better than the previous ones in all cases. The new threshold selection method, based on the energy of the wavelet coefficients, performs the best in all cases with a notable SNR enhancement. The results displayed in Fig. 3 for the different curves of PD present a close correlation with the results for SNR, showing the same tendencies. Further work is needed to check the tendencies presented in this paper with other processing parameters. Other performance comparisons of methods for grain noise reduction have been previously presented [5,16], but additional work is also needed to complete the analysis of the efficiency of different algorithms.
0.8 0.6
Acknowledgements
0.4
This work was supported by the Spanish Ministry of Science and Technology (R&D Project DPI200200441) and CYTED-CNPq (Research Project PULSETS).
0.2 0 1
(c)
References
0.8 0.6 0.4 0.2 0 1
(d)
0.8 0.6 Original Wigner-Ville
0.4
SURE MiniMax k=3
0.2 0 0
2
4
6
8
10
Detection threshold constant: C
Fig. 3. Probability of detection PD of initial and processed ultrasonic traces using different processing techniques: (a) F ¼ 2:5; (b) F ¼ 3; (c) F ¼ 3:5; and (d) F ¼ 4.
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