RETRACTED: Matching pursuit-based approach for ultrasonic flaw detection

ARTICLE IN PRESS

Signal Processing 86 (2006) 962–970 www.elsevier.com/locate/sigpro

Matching pursuit-based approach for ultrasonic flaw detection

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N. Ruiz-Reyesa,, P. Vera-Candeasa, J. Curpia´n-Alonsoa, J.C. Cuevas-Martineza, F. Lo´pez-Ferrerasb a

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Electronics and Telecommunication Engineering Department, University of Jae´n, Linares, Jae´n, Spain b Signal Theory and Communications Department, University of Alcala´, Alcala´, Madrid, Spain Received 27 October 2003; received in revised form 18 July 2005 Available online 24 August 2005

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Abstract

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In this paper, a fast and efficient matching pursuit-based approach is proposed to detect ultrasonic flaw echoes in strongly scattering materials. Matching pursuit requires a dictionary composed of elements well matched with the ultrasonic pulse echoes obtained from the transducer. The method proposed in this paper utilizes time-shifted Morlet functions as dictionary elements because they are well matched with the ultrasonic pulse echoes obtained from the transducer used in the experiments. Our method is fast enough to be used in the signal processing stage of real-time inspection systems. Computer simulation was performed to verify flaw detection for diverse ultrasonic waves embodied in high-level synthetic grain noise. Flaw detection is also experimentally verified using ultrasonic traces acquired from a carbon fibre reinforced plastic material. Numerical results show meaningful SNR improvements for low input SNR (below 0 dB). r 2005 Elsevier B.V. All rights reserved.

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Keywords: Matching pursuit; NDT; Morlet pulse; Flaw echo; Grain noise; Ultrasonic signal processing

1. Introduction

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Flaw detection by ultrasonic non-destructive evaluation or testing (NDE or NDT) has been Corresponding author. Tel.: +34 953648554;

fax: +34 953648508. E-mail addresses: [email protected] (N. Ruiz-Reyes), [email protected] (P. Vera-Candeas), [email protected] (J. Curpia´n-Alonso), [email protected] (J.C. Cuevas-Martinez), [email protected] (F. Lo´pez-Ferreras).

proven to be an effective means to assure the quality of materials. In the analysis of backscattered ultrasonic signals, the microstructure of the tested materials can be considered as an unresolved and randomly distributed set of reflection centres. The back-scattered ultrasonic signal is the result of reflecting the transmitted acoustic pulse with these reflection centres. This noise-like signal of structural origin (ultrasonic grain noise) is time-variant and, unfortunately, in some cases presents a frequency band very similar to that of

0165-1684/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2005.07.019

ARTICLE IN PRESS N. Ruiz-Reyes et al. / Signal Processing 86 (2006) 962–970

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space. We define the over-complete dictionary as a family D ¼ fgi ; i ¼ 0; 1; . . . ; Lg of vectors in H, such as kgi k ¼ 1. The problem of choosing functions gi ½n
that best approximate the analysed signal x½n
is computationally very complex. Matching pursuit is an iterative algorithm that offers sub-optimal solutions for decomposing signals in terms of expansion functions chosen from a dictionary, where l 2 norm is used as the approximation metric because of its mathematical convenience. When a well-designed dictionary is used in matching pursuit, the non-linear nature of the algorithm leads to compact adaptive signal models. In each step of the iterative procedure, vector gi ½n
which gives the largest inner product with the analysed signal is chosen. The contribution of this vector is then subtracted from the signal and the process is repeated on the residual. At the mth iteration the residue is ( x½n
; m ¼ 0; m (1) r ½n
¼ rmþ1 ½n
þ aiðmÞ giðmÞ ½n
; ma0;

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the echoes issuing from the flaws to be detected. Therefore, it cannot be cancelled by classical time averaging or matched band-pass filtering techniques. Many signal processing techniques have been utilized for signal-to-noise ratio (SNR) improvement in ultrasonic NDT of highly scattering materials. The most popular one is the split spectrum processing (SSP) [1–3], because it makes possible real-time ultrasonic test for industrial applications, providing quite good results. Alternatively to SSP, wavelet transform (WT) based denoising/detection methods have been proposed during recent years [4–8], yielding usually to higher improvements of SNR at the expense of an increase in complexity. Adaptive time-frequency analysis by basis pursuit (BP) [9,10] is a recent technique for decomposing a signal into an optimal superposition of elements in an overcomplete waveform dictionary. This technique and some other related techniques have been successfully applied to denoising ultrasonic signals contaminated with grain noise in highly scattering materials [11,12], as an alternative to the WT technique, the computational cost of the BP algorithm being the main drawback. In this paper, we propose a novel matching pursuit-based signal processing method for improving SNR in ultrasonic NDT of highly scattering materials, such as steel and composites. Matching pursuit is used instead of BP to reduce the complexity. Despite its iterative nature, the method is fast enough to be real-time implemented. The performance of the proposed method has been evaluated using both computer simulation and experimental results, even when the input SNR (SNRin) is lower than 0 dB (the level of echoes scattered by microstructures is above the level of flaw echoes).

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2. Matching pursuit

Matching pursuit was introduced by Mallat and Zhang [13]. Let us suppose an approximation of the ultrasonic backscattered signals x½n
as a linear expansion in terms of functions gi ½n
chosen from an over-complete dictionary. Let H be a Hilbert

where aiðmÞ is the weight associated to optimum atom giðmÞ ½n
at the mth iteration. The weight am i associated to each atom gi ½n
2 D at the mth iteration is introduced to compute all the inner products with the residual rm ½n
:

am i ¼

hrm ½n
; gi ½n
i hrm ½n
; gi ½n
i ¼ hgi ½n
; gi ½n
i kgi ½n
k2

¼ hrm ½n
; gi ½n
i.

ð2Þ

The optimum atom giðmÞ ½n
(and its weight aiðmÞ ) at the mth iteration are obtained as follows: giðmÞ ½n
¼ arg min krmþ1 ½n
k2 gi 2D

2 m ¼ arg max jam i j ¼ arg max jai j. gi 2D

gi 2D

ð3Þ

The computation of correlations hrm ½n
; gi ½n
i for all vectors gi ½n
at each iteration implies a high computational effort, which can be substantially reduced using an updating procedure derived from Eq. (1). The correlation updating procedure [13] is performed as follows: hrmþ1 ½n
; gi ½n
i ¼ hrm ½n
; gi ½n
i  aiðmÞ hgiðmÞ ½n
; gi ½n
i.

ð4Þ

ARTICLE IN PRESS N. Ruiz-Reyes et al. / Signal Processing 86 (2006) 962–970

0.5

0

-0.5

-1

-1 2

(a)

0

D

-0.5

0.5

3

time (s)

4 x 10-6

2

(b)

3

time (s)

4 x 10-6

Fig. 1. Comparison: (a) flaw signal; (b) Morlet pulse.

bandwidth of the transducer response. S i is a constant to achieve kgi k ¼ 1 and f s is the sampling frequency of the ultrasonic signal. Note that the method is valid for any kind of transducer if a pulse adapted to the transducer response is defined. It can be defined by an analytic expression, as it happens in the case of the Morlet pulse, or can be synthesized by experimentation. In both cases, the dictionary would be composed of time-shifted versions of the pulse analytically or experimentally defined. Next, we focus on describing our matching pursuit-based method to obtain an enhanced ultrasonic signal. Matching pursuit searches for the optimum Morlet function within the dictionary (which gives the higher correlation with the ultrasonic signal) at each iteration. A residual signal is obtained by subtracting the contribution of the optimum Morlet pulse from the previous residue (see Eq. (1)). In line with this iterative approach, an enhanced signal can be defined as follows: 8 x½n
; m ¼ 0; > < mþ1 m (6) xe ½n
¼ xe ½n
þ aiðmÞ giðmÞ ½n
> : G g ½n
; ma0; m iðmÞ

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According to matching pursuit, our method has to define the dictionary and the way to obtain an enhanced signal. Since real-time implementation is claimed for usefulness, the complexity of the proposed method must be reduced as much as possible. Because matching pursuit is an iterative algorithm, the goal can be achieved if both the amount of operations at each iteration and the number of iterations to reach an adequate SNR increase are reduced. If the dictionary is based on a pulsed function that mimics the echoes coming from the flaws to be detected, matching pursuit can efficiently extract flaw echoes in only a few iterations, because grain echoes have random amplitude and phase, while flaw echoes always have the same phase and higher amplitude than grain echoes. A small and simple dictionary is desirable, too. Analysing the impulse response of the transducer used in the experiments, an ðL ¼ N þ MÞ size dictionary composed of discrete-time shifted Morlet functions is defined. Here, N is the length of the ultrasonic data register and M the length of the discrete Morlet function. Such a pulsed function is well correlated with the flaw echoes, as shown in Fig. 1. The size of the dictionary has been selected to extend over all possible flaw echo positions. The dictionary elements (time-shifted Morlet pulses) can be expressed as

normalized amplitude

3. The proposed method to improve ultrasonic signals

1

1

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Correlations hgiðmÞ ½n
; gi ½n
i can be pre-calculated and stored, once over-complete set D has been determined. Therefore, according to (4), only computing correlations (hx½n
; gi ½n
i) at the first iteration is required.

normalized amplitude

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gi ½n
¼ S i eððniÞ=T m f s Þ
 fm  cos 2p ðn  iÞ þ fm , fs

ð5Þ

where i ¼ 1; . . . ; L and n ¼ i  M=2 þ 1; . . . ; i þ M=2. T m , f m and f s are the Morlet pulse parameters. They are related to the frequency and the

where G m is the gain associated to optimum function giðmÞ ½n
at the mth iteration. In order to enlarge this echo in the enhanced signal, this gain must be higher than optimum weight aiðmÞ when the optimum function corresponds to the flaw

ARTICLE IN PRESS N. Ruiz-Reyes et al. / Signal Processing 86 (2006) 962–970

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(8)

m

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The higher parameter a is the most probable selected vector giðmÞ ½n
at the mth iteration corresponds to a flaw echo and vice versa. Nevertheless, energy calculations are highly time consuming due to multiplications [14]. To reduce the complexity associated to the computation of parameter am , a new probability indicator is defined: a2iðmÞ . bm ¼ PiþM=2 ð n¼iM=2þ1 jrmþ1 ½n
jÞ2

In order to ascertain the complexity of the method, the operations required by the algorithm are next explained. At the first iteration, correlations (hx½n
; gi ½n
i) are calculated. At the next following iterations, our method performs:

ð7Þ

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(10)

Therefore, the new signal xe ½n
to enhance the visibility of flaw echoes is expressed as follows: 8 x½n
; m ¼ 0; > < mþ1 m x ½n
þ a g ½n
iðmÞ iðmÞ (11) xe ½n
¼ e > : bm a g ½n
; ma0: iðmÞ iðmÞ



It must be noted that residual energy E rmþ1 is computed once the contribution of the weighted optimum function is subtracted. Parameter am informs us about the probability that optimum vector giðmÞ ½n
at the mth iteration corresponds to the flaw signal. This probability, denoted by Pm , is related to parameter am as follows: Pm / eð1=a Þ .

Gm ¼ bm aiðmÞ .

N þ M  1 comparisons, corresponding to Eq. (3). 2M  1 multiplications and accumulations, corresponding to Eq. (4). M additions, one division and one multiplication, corresponding to Eq. (9). One multiplication, one addition and M multiplications and accumulations, corresponding to Eq. (11).

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E ðaiðmÞ giðmÞ Þ a2iðmÞ E giðmÞ ¼ E rmþ1 E rmþ1 a2iðmÞ ¼ PiþM=2 . mþ1 ½n
j2 jr n¼iM=2þ1

am ¼

similar results, bm is the simplest one. Then, to observe the above stated gain conditions, gain G m in Eq. (6) and probability indicator bm in Eq. (9) are related as follows:

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echo. On the other hand, the gain must be close to zero when the optimum function represents backscattered noise and, as a result, the method removes it in the enhanced signal. Therefore, the enhanced signal is obtained either by amplifying or attenuating optimum function giðmÞ ½n
at the mth iteration, whether this function corresponds or not to the flaw echo. Once the enhanced signal xe ½n
has been defined, which depends on gain Gm , computing this gain is required. For this purpose, we must consider that if a given ultrasonic backscattered signal contains a flaw echo, its energy will prevail in a localized part of the ultrasonic signal, since flaw echoes are usually bigger than grain echoes. Therefore, in order to achieve the above stated objective, we compute the following energy ratio in that localized part of the ultrasonic signal:

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(9)

Parameter bm implies to compute the sum of the residue absolute values instead of residue energy calculation. Although am and bm coefficients give



The above values have been obtained once correlations hgiðmÞ ½n
; gi ½n
i have been pre-calculated and stored. Note the use of parameter bm instead of parameter am reduces the algorithm complexity in M multiplications per iteration. After a few iterations, the enhanced signal xe ½n
has a higher SNR than the original one x½n
, as is shown in computer simulations and experimental results. The stopping criterion is the following: when the maximum value of weights am i computed at a given iteration is below a threshold, which depends on the minimum size of the flaw to be detected, matching pursuit stops at that iteration. When the number of iterations exceeds the upper bound, the increase in computational cost does not improve the SNR significantly, because the highest SNR improvement is obtained at the iteration in which the flaw signal is selected and amplified.

ARTICLE IN PRESS N. Ruiz-Reyes et al. / Signal Processing 86 (2006) 962–970

4. Computer simulations Grain noise (clutter or speckle) models are frequently used to generate synthetic noise registers for the evaluation of the performance of noise reduction algorithms [15,16]. This grain noise of structural origin is described as the superposition of backscattered signals from the grains boundaries. If material scattering and attenuation are considered, the noise signal can be described as [16] Y n ðoÞ ¼ H trans ðoÞH trans ðoÞ

ð12Þ

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! K X bk o2 2as zk o4 j2zk o=cl  e e , zk k¼1

not high enough and there is a lower chaos in the material microstructure, other distributions fit better, such as K or Gamma distributions. The material attenuation coefficient used was as ¼ 8  1028 m1 s4 . The sampling rate for the simulated signals was f s ¼ 100 MHz and the propagation velocity cl ¼ 6000 m=s. The grain noise signals then corresponds to a 0:5cl ðN=f s Þ ¼ 12:29 cm material path. The transducer response has 5 MHz centre frequency and 50% bandwidth. Therefore, the Morlet pulse parameters in Eq. (5) are: T m ¼ 0:025 ms, f m ¼ 5 MHz and fm ¼ p=2. The Morlet pulse is defined as a M ¼ 300 sample-length function, as depicted in Fig. 1. We have considered different flaw echo amplitudes and locations achieving SNRs between 5 and 15 dB. The minimum depth from where flaw signal starts is 4 cm. In order to quantify the enhancement in the ultrasonic signal, the SNR of both pre- and postprocessed signals (SNRin and SNRout , respectively) were evaluated and compared. SNR was computed as the ratio, expressed in dB, of the maximum flaw signal amplitude and the maximum noise amplitude. Fig. 2 shows three simulated signals, which correspond to: (a) no hole; (b) hole at 9:62 cm depth and (c) hole at 5:62 cm depth. Pre- and postprocessed signals are displayed in the same figure. Our method clearly enhances the visibility of the flaw echo and reduces the grain noise. Table 1 presents the SNR improvement and complexity results for 10 different ultrasonic signals (arbitrarily chosen), among 200 simulated, where the flaw echoes were embedded in different locations. The SNR improvement is about 19:2 dB and the average number of iterations is 24.

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A few iterations are required to achieve a noticeable SNR enhancement, as will be shown in computer and experimental results.

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where H trans ðoÞ is the electromechanical transducer response, zk the position of the kth grain echo, as the material attenuation coefficient, cl the propagation velocity, K the number of grain backscattered signals and bk the scattering amplitude from the kth scatterer. The flaw signal was simulated by one single Morlet pulse according to the model defined by Eq. (13) Y f ðoÞ ¼ H trans ðoÞH trans ðoÞ 4

ðe2as zf o ej2zf o=cl Þ,

ð13Þ

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where zf is the flaw position. Superimposing the normalized grain noise and the flaw echo signal at a known flaw location results in the simulated signal. N ¼ 4096 sampleslength grain noise signals were generated by superimposing K ¼ 2000 grain echoes with Gaussian-distributed amplitudes at the randomly distributed positions generated by an uniform pdf. We assume the number of grains within the material is high and there is no regularity, neither in the grain size nor in the distances between grains. In such case, it can be considered that grain noise amplitudes can be modelled by a Gaussian distribution [17,18]. When the number of grains is

5. Experimental results A carbon fibre reinforced parallelepiped plastic (CFRP) block of 120 mm thickness was machined on one of its plain surfaces to produce flat-bottom holes (FBH) at different depths along a straight line. Experimental echo traces were obtained using a circular ultrasonic transducer for longitudinal waves, 6:35 mm in diameter, 5 MHz of nominal

ARTICLE IN PRESS N. Ruiz-Reyes et al. / Signal Processing 86 (2006) 962–970

normalized amplitude

0.5 0 -0.5 0

1

3

-0.5 -1 3

4 x 10-5

time (s)

0 -0.5 -1 0

1

2

time (s)

3

2

3

4 x 10-5

0.5 0

-0.5 -1

1

4

2

3

4 x 10-5

3

4 x 10-5

time (s)

1

0.5 0

-0.5 -1

0

1

x 10-5

2

time (s)

R

(c)

1

0

normalized amplitude

1 0.5

1

TE

0

(b)

-1

time (s)

0.5

2

-0.5 0

1

1

0

x 10-5

time (s)

0

0.5

4

normalized amplitude

normalized amplitude

2

1

D

-1

(a)

normalized amplitude

Post processed signals

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normalized amplitude

Pre processed signals

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Fig. 2. Processed results using simulated signals: (a) no hole; (b) hole at 9:62 cm depth; (c) hole at 5:62 cm depth.

Table 1 SNR improvement and complexity results of the proposed method for 10 different simulated signals SNRin (dB)

SNRout (dB)

SNRimprovement ¼ SNRin  SNRout (dB)

Iterations

1 2 3 4 5 6 7 8 9 10

11.34 12.61 10.99 6.17 7.13 9.68 8.73 5.30 7.23 13.43

8.33 5.59 8.34 7.85 5.96 9.89 8.52 14.65 19.09 3.60

19.67 18.20 19.33 14.02 13.09 19.57 17.25 19.95 26.32 17.03

20 19 19 36 22 22 21 24 27 24

8.86

10.45

19.19

23.4 5.08

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Trial Number

Mean Std. Dev.

ARTICLE IN PRESS N. Ruiz-Reyes et al. / Signal Processing 86 (2006) 962–970

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Sync Output

CH2 CH1 Osciloscope

Panametrics

GPIB

Probe GPIB PC Holes

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Fig. 3. Figure presenting the experimental setup.

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high-pass filter ¼ 300 kHz. The ultrasonic traces were acquired by means of a digital oscilloscope, Tektronix TDS 744 of 2 GS/s and data length of 4000 samples, which were transferred via GPIB to a computer for further processing. The signals were acquired with a sampling frequency of 100 MHz. The experimental setup is shown in Fig. 3. Many experimental signals were recorded scanning with the contact transducer the composite block over the plane opposite to the one where the holes were drilled, in order to obtain input signals with and without hole indications. Fig. 4 shows four experimental signals, which correspond to: (a) no hole; (b) hole at 6:75 cm depth; (c) hole at 4:75 cm depth and (d) two holes at 4.75 and 6:75 cm depth. Pre- and post-processed signals are displayed in the same figure. The gain G m applied to each flaw echo depends on the flaw echo amplitude determined by

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frequency and 50% bandwidth. The probe was placed in contact and driven in pulse-echo operation with an ultrasonic analyser. The selected pulser/receiver parameters were: damping resistance ¼ 200 O, energy position ¼ 2 (810 pF for the HV discharge capacitor), receiver gain ¼ 26 dB, cut-off frequency of the receiver

Pre-processed signals

0

2

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normalized amplitude

0

1

-1 0

(c)

1

2

3 x 10-5

1 0

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(b)

-1 0 1

1

2

3 x 10-5

(d)

1

2

3 x 10-5

1

2

3 x 10-5

1

2

3 x 10-5

1

2 time (s)

3 x 10-5

0 -1 0 1 0 -1 0 1 0

0 -1 0

-1 0 1

3 x 10-5

normalized amplitude

-1 0 1

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0

(a)

Post-processed signals

1

1

1

2 time (s)

3 x 10-5

-1 0

Fig. 4. Processed results using experimental signals: (a) no hole; (b) hole at 6:75 cm depth; (c) hole at 4:75 cm depth; (d) two holes at 4.75 and 6:75 cm depth.

ARTICLE IN PRESS N. Ruiz-Reyes et al. / Signal Processing 86 (2006) 962–970

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Table 2 SNR improvement and complexity results of the proposed method for 10 different experimental signals SNRin (dB)

SNRout (dB)

SNRimprovement ¼ SNRin  SNRout (dB)

Iterations

1 2 3 4 5 6 7 8 9 10

15.24 10.03 15.08 14.47 15.65 12.57 13.43 10.84 14.33 14.75

1.36 11.00 2.47 3.80 0.43 5.48 4.34 9.68 4.76 3.91

16.60 21.03 17.55 18.27 16.08 18.05 17.77 20.52 19.09 18.66

22 34 23 20 23 19 27 16 21 22

Mean Std. Dev.

13.44

5.35

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22.7 4.90

The method is very efficient eliminating noise and increasing the visibility of ultrasonic flaw signals. Numerical results show SNR improvements of about 19 dBs with low computational cost. It has been possible to enhance flaw echoes in highly scattering materials (SNRin o0 dB), even when two adjacent flaw echoes appear in the same ultrasonic signal. The number of iterations needed to attain a noticeable SNR improvement is low, which permits its use in practical real-time ultrasonic NDT systems for industrial applications. Only 20 iterations are needed, on average, as has been demonstrated. A challenging issue for further research could be the generalization of the proposed approach to take into account the frequency-dependent absorption and scattering, which implies the traveling echo in the material goes through a change in frequency content. The flaw echo detection problem could be generalized to cover frequencyvarying pulse echoes.

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matching pursuit and the amount of grain noise close to the flaw echo. Therefore, this gain is not function of the depth of the hole, as can be seen at (b) and (c) in Fig. 4. The matching pursuit-based gain G m is highly dependent on the SNR at the flaw echo location, as shown at (d) in Fig. 4. Table 2 shows the SNR improvement and complexity results obtained for 10 experimental ultrasonic signals, all of them with SNRin o0 dB. The SNR improvement in these experiments is about 18:5 dB and the average number of iterations is 23. It is important to underline that even when two flaw echoes appear in the same temporal record the method performance is not degraded. In fact, the case of Fig. 4(d) corresponds to trial 7 in Table 2.

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Trial Number

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6. Conclusions

A novel matching pursuit-based method has been presented to improve the SNR in ultrasonic NDT of highly scattering materials. The method decomposes the ultrasonic signal into a suboptimal superposition of dictionary elements within an over-complete dictionary. A dictionary composed of discrete-time shifted Morlet functions have been defined and adapted to the frequency and bandwidth of the ultrasonic transducer used in the work.

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[18]