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> IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL< 1 Signal Matching Wavelet for Ultrasonic Flaw Detection in High Background Noise Guangming Shi, Senior Member, IEEE, Xuyang Chen, Student Member, IEEE, Xiaoxia Song, Fei Qi, Member, IEEE, Ailing Ding the noise. In the 1990s, many researchers used split spectrum Abstract—Wavelet transform (WT) is widely applied in processing (SSP) [1]-[3] to suppress the noise for UFD. ultrasonic flaw detection (UFD) system due to its property of However, the SSP does not possess the multiresolution analysis multiresolution time-frequency analysis. Those traditional property and thus is not suitable to deal with the non-stationary WT-based methods for UFD use wavelet basis with limited types ultrasonic echo signal in high background noise. The wavelet to match various echo signals (called wavelet-matching-signal), transform (WT), as a multiresolution time-frequency analysis so it is difficult for those methods to achieve the optimal match between echo signal and wavelet basis. This results in limited tool [4], is widely used to suppress the noise and detect the flaw detection ability in high background noise for those WT-based echo [5-16]. methods. In this paper, we propose a method of Continuous WT (CWT) mainly provides a theoretical signal-matching-wavelet (SMW) for UFD to solve this problem. direction for UFD applications [5]-[8], while the discrete WT Unlike the traditional UFD system, in the proposed SMW the (DWT) is feasible for practical UFD system due to its fast transmitted signal is designed to be a wavelet function for calculation and thus is more preferred [9]-[16]. Those matching a wavelet basis. This makes it possible to obtain the DWT-based methods for noise suppression in UFD are mainly optimal match between the echo signal and the wavelet basis. To divided into two categories. The first category applies achieve the optimal match from the aspect of energy, we derive thresholding scheme [9]-[12] in which only the coefficients three rules for designing transmitted signal and selecting wavelet basis. Further, the parameter selection in applying the proposed larger than a threshold are preserved, and then the signal is SMW to a practical UFD system is analyzed. In addition, a reconstructed with the preserved coefficients. The second low-rate DWT structure is designed to decrease the hardware category is pruning-based methods, which cut the coefficients cost, which facilitates the practical application of the proposed of the un-interesting scales (regarded as noise) and preserve SMW. The simulation results show that the proposed SMW can those of interesting scales in WT domain. [13]-[16]. Besides, to efficiently detect the flaw in high background noise even with improve the detecting performance in noise environment, some SNR being lower than -20 dB, outperforming the existing researchers proposed matching-wavelet-based methods methods by 5dB. [17]-[19]. In general, those existing WT-based methods [5-19] can be generalized to use a wavelet basis to match the echo of a Index Terms—Energy match, high background noise, signal transmitted signal, called wavelet-matching-signal. However, matching wavelet, ultrasonic flaw detection, wavelet transform under high background noise, the existing wavelet-based methods cannot solve the UFD problem efficiently. This is because, those methods do not take into account the match I. INTRODUCTION between the echo signal and wavelet basis during the LTRASONIC flaw detection (UFD) in high background construction of transmitted signal. It will result in a problem of U noise is in great demands in non-destructive evaluation of many industry applications, such as testing of using limited types of wavelet basis to match various echo signals in the processing of echo signals. This makes it difficult aeronautical materials, petroleum pipeline and automotive to achieve the optimal match and thus the detection engine. The key problem in these applications is to suppress the performance for weak signal in high background noise will not high background noise and separate the weak clean echo from be satisfactory for those industry applications mentioned above. To our knowledge, the acknowledged best result comes up Manuscript received April 19, 2009. This work is supported by NSFC (NO. 61033004, 60736043, and 61070138), and by a grant from Ph.D. Programs when the input signal-to-noise-ratio (SNR) of echo signal Foundation of Ministry of Education of China (No. 200807010004). reaches -15dB [6]. Guangming Shi, Xuyang Chen, Xiaoxia Song, and Fei Qi are with Institute of To address the UFD problem under high background noise, Intelligent Information Processing, School of Electronic Engineering, Xidian this paper proposes a method of signal-matching-wavelet University, Xi’an, 710071, China (E-mail: gmshi@xidian.edu.cn, (SMW). Different from those methods of xychen@mail.xidian.edu.cn, xxsong@mail.xidian.edu.cn, and fred.qi@gmail.com). wavelet-matching-signal, the proposed SMW design the Xiaoxia Song is also with the School of Physical&Electronics, Shanxi Datong transmitted signal to be a wavelet function. The idea of SMW is University, Datong, 037009, China feasible according to the fact that an arbitrary transmitted Ailing Ding is with School of Information Engineering, Chang’an University, Xi’an, 710071, China signal can be designed by a controllable transducer [20], [21]. > IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL< 2 In this way, it is possible to achieve the optimal match between where s (t ) is the transmitted signal, τ i is the delay of the i-th the echo signal and wavelet basis and the detection flaw echo and α i s (t − τ i ) is the i-th flaw echo. The parameter performance could be further improved. Here, we study SMW from the perspective of optimal energy α i is determined by the size of the i-th flaw and the attenuation match due to the fact that the clean echo and the noise have of the material. Since α i denotes only an amplitude of the flaw different energy distributions in WT domain. To achieve the optimal energy match, we need to solve two problems, the echo, we use s (t − τ i ) to represent the i-th flaw echo in the concentration of flaw echo and the separation of flaw echo from following discussion. noise in WT domain. By analyzing the two problems, we derive three match rules for designing transmitted signal and B. Analysis of Echo Signal on Energy Distribution in selecting wavelet basis. Based on the match rules and the actual Wavelet Domain requirements in UFD, we analyze the selection of parameters Since the energy distribution of clean echo signal is different used in the proposed method. The above rules and requirements may need a multi-level wavelet decomposition to from that of noise in WT domain, it is necessary to analyze the get better performance of noise suppression, while it will lead energy distribution of an echo signal for noise suppression. to a high sampling rate and difficult implementation in DWT is an efficient tool for wavelet analysis which can be hardware. Thus, we design a polyphase-decomposition-based performed by a dyadic tree structure. By using DWT low-rate DWT structure to decrease the hardware cost and decomposition, an echo signal x(t ) can be represented by further facilitate the practical application. Simulation results Lmax illustrate that the proposed SMW can efficiently detect the flaw in high background noise even with SNR being lower than -20 x(t= ) ∑∑ d j =1 k x j ,k ⋅ψ j , k (t ) + ∑ aLmax , kφLmax , k (t ), k (3) dB, outperforming the existing methods by 5dB. j ∈ + , k ∈ , The rest of this paper is organized as follows. Section II gives the analysis of echo signal on energy distribution. Section III where j and k denote the level of scale and shift amount in proposes the signal matching wavelet. Section IV gives the DWT, respectively, and ψ j , k (t ) and φLmax , k (t ) are the wavelet design of SMW for UFD. Experimental results in Section V manifest excellent performances of UFD. We close in Section function and scale function of DWT, respectively. Lmax is the VI with conclusions. maximal decomposition scale, d jx, k is the wavelet coefficients of x(t ) , aLmax , k is the approximation coefficient of x(t ) at II. PROCEDURE FOR ANALYSIS OF ECHO SIGNAL ON ENERGY scale Lmax . Lmax should be selected large enough so that the DISTRIBUTION coefficients {d jx, k } contain most information of r (t ) . The coefficients {aLmax , k } usually contain some low-frequency A. Echo Signal Model interference and thus are removed to suppress the noise. In UFD system, the echo signal contains the flaw echo and According to (1), d jx, k can be further decomposed by the background noise, and the flaw information can be obtained by suppressing the noise and detecting the echo signal. Next, d= d r, k + d n, k , x j ,k j j (4) we introduce briefly the echo signal model. Let x(t ) ∈ L2 ( be the echo signal, r (t ) be the clean echo ) where d r, k and d n, k represent the wavelet coefficients of j j signal (namely flaw echo), and n(t ) be the background noise r (t ) and n(t ) at the j-th scale, respectively. The energy of the with Gaussian distribution, which is generated by a randomly clean echo signal and the noise in WT domain is determined by distributed scatterers in the diagnosed material. Then the echo the wavelet coefficients d r, k and d n, k . So, the analysis on the j j signal model is established as follows energy distribution of the echo signal is important to the flaw x= r (t ) + n(t ) . (t ) (1) detection. The existing methods, using a wavelet basis to match a In this paper, we study the flaw detection of metal material by signal, easily make the energy of d r, k and d n, k overlap under j j A-scan UFD system and suppose that the ultrasonic trace is frequency-independent, homogeneous and non-dispersive in high background noise, which is shown in Fig. 1 [6]. The the metal material. So, the clean echo signal r (t ) can be energy of the flaw echo locates in a region ‘A’, while the energy expressed by of the background noise is mainly distributed in a ‘long-narrow’ region ‘B’ at the lower scales in Fig. 1. Since the r (= t) ∑α s(t − τ ), i ≥ 1 , i i i (2) regions ‘A’ and ’B’ overlap in some region, it is very difficult to separate the clean echo from the noise. > IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL< 3 Fig. 1. Energy distribution of an echo signal in WT domain (overlap case). Fig. 2. Energy distribution of an echo signal in WT domain (non-overlap case). which is not beneficial to our following study, we use the delay interval (denoted by ts ) of the adjacent orthogonal wavelet at III. THE PROPOSED SIGNAL MATCHING WAVELET the 0-th scale to indirectly describe the support length. The waveform of the wavelet function and the parameter ts A. Idea of Signal Matching Wavelet together determine a unique transmitted signal. The distribution of clean echo signal in WT domain is For the wavelet basis {ψ j , k (t ), j , k ∈ used in a DWT } related to both the echo signal and the wavelet basis, while the structure, each basis function in {ψ j , k (t )} has its practical traditional UFD methods only focus on how to select the wavelet basis to deal with an echo signal. In other words, they support length. Similar to ts , we define tb as the time interval design or select wavelet basis to match an echo signal, which between ψ 0, k (t ) and ψ 0, k +1 (t ) .Based on the principle of makes inevitable confusion of clean echo and noise in energy DWT, the parameter tb is equal to the sampling interval of an distribution, and undermines the ability of noise suppression and flaw detection in high background noise. To avoid the input signal in the DWT structure. According to the above confusion of clean echo and noise, it is important to definition, the basis function ψ j , k (t ) can be expressed as intentionally control the clean echo, besides the selection of j wavelet basis. In view of this point, we propose a method of − ψ j ,k (t ) = 2 2 ψ (2− j t − k ⋅ tb ), j , k ∈ . SMW for UFD in high background noise. Unlike the traditional WT-based UFD detection methods, Note, we use ‘wavelet basis’ to represent the basis used in a the transmitted signal in SMW is designed controllably in DWT system without extra explanation in the rest of the paper. order that the echo signal matches properly a wavelet basis in Now, we analyze the optimal energy match between the flaw energy distribution. By designing the transmitted signal to echo and wavelet basis for designing the transmitted signal and match a wavelet basis, we can indirectly control the energy selecting the wavelet basis. distribution of echo signal in WT domain. Since both the echo As mentioned above, Fig. 1 shows an overlap case of the signal and wavelet basis can be controllable, the optimal match energy distribution of echo signal in WT domain. From Fig. 1, between flaw echo and wavelet basis is achievable, which will the energy distribution of the flaw echo has a bad localization result in the elimination of confusion between the clean echo property and overlap with the noise. In this case, the flaw echo and the noise in WT domain. is difficult to be separated from the noise. Different from Fig. 1, Considering the time-frequency location property of a we show a non-overlap energy distribution in Fig. 2 [6], in wavelet basis function, we directly design the transmitted which the energy distribution of the flaw echo is localized and signal to be a wavelet basis function (here, we focus on the far from that of the noise. In this case, the flaw echo can be orthogonal wavelet basis). Since a flaw echo s (t − τ i ) is a easily separated from the noise. By comparing these two cases delayed form of the transmitted signal as shown in (2), the flaw shown in Fig. 1 and Fig. 2, we derive two problems for the echo will also possess a localized distribution in WT domain. optimal energy match as follows: Then the transmitted signal s (t ) can be expressed by the (i) Concentration: the flaw echo signal should have a localized energy distribution. following form (ii) Separation: the energy distribution area of the flaw echo s (t ) = W (t ) , (5) should be far from that of the noise. where W (t ) is a mother wavelet function. When a wavelet B. Optimal Energy Match function is selected to be an actual transmitted signal, the pulse duration (that is, support length) of the signal must be As mentioned above, the two problems, concentration and determined. Considering that the support length is unstable to separation, are crucial for optimal energy match. In this different wavelet functions at the same decomposition scale, subsection, we propose two concentration rules and one > IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL< 4 Decision block material Transducer T/R Flaw Signal processing Receiver detector unit (WT) echo signal flaws Fig. 3. A diagram of the SMW-based UFD system. separation rule to solve the two problems. We firstly present concentrated energy distribution around scale jc . these three rules and then give their explanations. Concentration Rule 2 is given with the consideration of Concentration Rule 1: The flaw echo s (t − τ i ) has a wavelet characteristics and its proof is listed detailedly in Appendix for the complicated derivation. The explanation of concentrated energy distribution around a scale jc , if the Separation Rule is straightforward from the fact: 1) the following constraint is satisfied Gaussian noise has localized energy at lower scales and its tb = 2− jc ts , jc ∈{1, 2,} , (6) energy decreases quickly with the increasing scale j, 2) the flaw echo has concentrated energy distribution around scale jc . where jc is defined as the central scale where the major energy As a result, the concentration and separation are achieved by of a flaw echo s (t − τ i ) locates in WT domain. the three rules. According to these rules, the transmitted signal s (t ) and the wavelet bases {ψ j , k (t )} can be properly designed Concentration Rule 2: The flaw echo s (t − τ i ) decays fast, (or selected) to achieve the optimal match between the flaw if the conditions in the following two cases are satisfied. echo and the wavelet basis. (i) In the case that the scale j decreases with j < jc , the Notably, there is a conflict between the high vanishing transmitted signal s (t ) has a high Lipschitz regularity, and moment and the short support length in the construction of {ψ } wavelet. Thus, during constructing practically the transmitted the selected wavelet bases j ,k (t ), j , k ∈ have high signal and wavelet basis, we should carefully select the vanishing moment and short support length. parameters to solve the conflict. In the next section, we will (ii) In the case that the scale j increases with j > jc , the apply the proposed SMW to the practical UFD and give the selection of parameters. transmitted signal s (t ) has short support length. Separation Rule: The energy distribution of a flaw echo s (t − τ i ) is far from that of background noise, if the central IV. DESIGN OF PROPOSED SMW FOR UFD scale jc is selected large enough. In this section, we apply the proposed SMW to UFD. First of Next, the explanations of these rules are listed as follows. all, we give the framework of the SMW-based UFD system. As for Concentration Rule 1, considering the spectrum of Then we analyze the selection of parameters used in the UFD system. Finally, a low-rate DWT structure is presented to wavelet, the transmitted signal s (t ) as well as the facilitate the implementation of actual system. corresponding flaw echo s (t − τ i ) have a band-limited energy 1 A. UFD Based on the Proposed SMW distribution in region ± [π -δ1 , π +δ 2 ] , where both δ1 and 2 ts Fig. 3 shows the SMW-based UFD system. Unlike the δ 2 describe the fluctuation of the spectrum region where the traditional UFD system in which the design of transmitted signal is independent of the selection of wavelet basis, the major energy localizes and their values are small. Also, a proposed system adds a decision block to control the design of wavelet basis {ψ j , k (t )} at the jc -th scale has a band-limited transmitted signal and the selection of wavelet basis, aiming to 1 π -δ1 2π +δ 2 achieve the optimal match between flaw echo and wavelet basis. energy distribution in ± [ , jc ] . Provided the The echo signal obtained by a receiver is processed by the tb 2 jc 2 WT-based signal processing unit, which is determined by a constraint in (6) is satisfied, the wavelet basis function selected wavelet basis according to the decision block. ψ jc ,k (t ) has the similar band limitation with the transmitted Consequently, the flaw echo is preserved and the background signal s (t ) . Thus the flaw echo s (t − τ i ) will exhibit a better noise is suppressed effectively. Finally, the position and size of > IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL< 5 the flaws can be determined via a flaw detector. {ψ j , k (t ) = ψ j +∆j , k (t ), j , k , ∆j ∈ , their generated wavelet } In the proposed system, the decision block is crucial for the noise suppression and flaw detection. Next, we give the coefficients d j , k and d j , k satisfy the following relation decision procedure as follows: d j , k = d j +∆j , k , ∆j ∈ . (7) Step 1: Determine the wavelet type of the transmitted signal and select the wavelet basis according to Concentration Rules. The relation (7) indicates that the wavelet coefficients of a Step 2: Set the parameter of transmitted signal ts and the flaw echo have the same energy distribution for the same system parameters including central scale jc , threshold scales wavelet base with different mother wavelets, whereas the Llow and Lhigh , theoretical sampling rate Ri , and practical central scale containing the major energy of the input signal is different. So we assume the central scale jc to be fixed in the sampling rate R shown in Sec. IV-B and Sec. IV-C. study of the energy distribution of flaw echo. Step 3: Calculate the excitation signal e(t ) by the method Due to the orthogonal and shift property of {ψ j , k (t )} , the shown in [20], [21] and generate the transmitted signal to detect a material. energy distribution, denoted by E j , of the wavelet coefficients Step 4: Receive the echo signal and sample it at the rate R . of s (t − τ i ) in a j-th scale is a periodic function of the variable The sampled signal is decomposed by a low-rate DWT structure shown in Sec. IV-C, generating the wavelet τ i with a period 2 j tb . So it is enough to study the distribution coefficients. By using the pruning technique associated with of E j in one period but not the whole wide range of τ i . Then the parameters in Step 2, the useful wavelet coefficients are we denote the delay time τ i in one period by ρτ and express preserved and then the denoised echo signal is reconstructed. E j as a function of ρτ as follows The above procedure includes several parameters related to the transmitted signal and the UFD system. In the following ∞ 2 = ∑∫ ∑d 2 = E j , ρτ s (t − ρτ )ψ j , k (t )dt , (8) subsection, we describe explicitly the selection of these k j ,k k −∞ parameters for enhancing the match capability. j ∈{1, 2,}, ρτ ∈ [0, 2 j tb ) B. Selection of Parameters Experiments in Sec. V-A illustrate such a fact: if Concentration Rules are satisfied properly, the energy In this subsection, we discuss the selection of parameters, distribution E ( j , ρτ ) is mainly localized at three scales: which is closely related to the performance of the proposed SMW for UFD. The parameters: the central scale jc and the jc − 1, jc , jc + 1 . Based on this fact, we can select the threshold scales Llow and Lhigh determine the performance of = parameters properly Llow jc − 1 , and Lhigh jc + 1 , which = guarantees the major energy of flaw echo preserved. noise suppression. The parameter ts and the sampling rate Ri are related to the system characteristics. Determination of Central Scale jc The central scale jc influences the separation degree of flaw Determination of Parameters Llow and Lhigh in ‘Pruning’ echo from noise and the sampling rate of the practical system The pruning technique is a simple and useful WT-based according to Separation Rule and (6). To determine a proper jc , method for noise suppression in flaw detection [14]. In this the energy distribution of the white Gaussian noise should be paper we employ the pruning to separate the clean echo from quantitatively analyzed in WT domain. We calculate the noise. In the pruning technique, the input signal is supposed to energy distribution of the Gaussian noise in several typical have concentrated energy at the scales {Llow, Llow + 1,, Lhigh } wavelets by percentage and show the results in Table 1. ( 0 < Llow ≤ Lhigh ≤ Lmax ) apart from the noise in WT domain, Table 1 shows that the white Gaussian noise has the similar where Llow and Lhigh are respectively the minimal and energy distribution for different wavelets. Specifically, over 96% energy of white Gaussian noise is localized at the first 5 maximal threshold scales. So the recovered signal can be scales, over 98% energy at the first 6 scales and over 99% obtained only by the reconstruction of the coefficients in the energy at the first 7 scales. Under high background noise, the preserved scales from Llow to Lhigh . wavelet coefficients of at least the first 5 scales should be When Concentration Rule 1 is satisfied, the major energy of discarded in order to obtain a good result of noise suppression. flaw echo s (t − τ i ) will be localized around scale jc . It can be So the central scale jc should be chosen larger than 6, which is found, for two wavelet-transform with their wavelet base a lower bound of jc . A practical jc should be determined by {ψ j , k (t )} and {ψ j , k (t )} satisfying both the system cost and the noise intensity. > IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL< 6 TABLE 1. TABLE 2. STATISTIC ENERGY DISTRIBUTION OF GAUSSIAN NOISE IN WT DOMAIN FOR RANGES OF P , t s AND Ri FOR EACH SELECTED jc . SEVERAL TYPICAL WAVELETS. Ri (MHz) Energy distribution (%) P (mm) ts ( µ s ) Selected jc = 7 jc = 8 jc = 9 bavelet [0.6, [0.38, [39.38, [78.77, [157.54, basis first 4 first 5 first 6 first 7 first 8 5.2] 3.26] 341.33] 682.67] 1365.33] scales scales scales scales scales Haar 93.46 97.07 98.93 99.42 99.88 practical requirement, the detection precision P should db4 93.46 96.63 98.21 99.20 99.75 belong to the region [0.6,5.2] mm when the steel sample has db10 93.78 96.83 98.36 99.30 99.72 several centimeters or decimeters in thickness [5]. Here we set ζ = 1 and calculate the parameter ts and Ri for the central dmey 94.02 96.92 98.48 99.25 99.69 scale jc being 7, 8 and 9, respectively. Table 2 shows the sym6 93.04 96.46 98.43 99.22 99.62 ranges of the parameter ts and the sampling rate Ri for each sym12 93.55 96.81 98.56 99.17 99.73 selected jc . Table 2 shows that the sampling rate Ri varies higher with increasing jc . For jc = 9 , the upper bound of Ri even Determination of Parameter t s and Sampling Rate Ri reaches GHz order of magnitude, which results in the high Both the parameter ts and the system sampling rate Ri are hardware cost. In fact a larger jc is needed to suppress the related to the required detection precision P which is the noise with high-intensity. In the following subsection, we will minimal distance between two flaws distinguished. Here we give an equivalent DWT structure to reduce the sampling rate will derive their relations. without degrading noise suppression performance. Let D be the distance propagated by an ultrasound in the time C. Low-rate DWT Structure Based on Polyphase interval ts in a certain metal material. The relation between D Decomposition and ts can be expressed as In order to reduce the sampling rate of the DWT structure, an = 2= 2ζ P / v , ts D/v (9) equivalent DWT structure is constructed based on polyphase decomposition [22]. As shown in Sec. IV-A, the wavelet where v is the velocity of ultrasound propagating in the metal coefficients only at the scales { jc − 1, jc , jc + 1} are preserved medium, the distance D and the detection precision P are to reconstruct the signal. All these wavelet coefficients at the related though a calibration parameter ζ ∈ (0,1] . Usually scales above can be obtained from the approximation different wavelet corresponds to a different ζ . coefficients {aQ , k , k ∈ , 1 ≤ Q ≤ jc − 2 , so we only need } From (9), it can be seen that the parameter ts is proportional study the equivalent DWT structure at the first Q scales shown to P . It indicates that a high detection precision can be in Fig. 4. In Fig. 4, the matrix EQ ( z ) is a linear module with achieved by setting a small ts which indicates a short support 2Q -input and 1-output. Each element of EQ ( z ) can be length of transmitted signal. As is well known, the sampling rate Ri of the system is expressed as inversely proportional to the sampling interval tb . And EQ ( i ) ( z )= ∑ h '(2 n Q n + i ) z − n , i= 0,1,, 2Q − 1 , (11) considering tb = 2− jc ts in (6), we express Ri as where = 1= 2 jc / ts . Ri / tb h '(n) =h(n) ∗ (h(n) ↑ 2 ∗ (h(n) ↑ 22 ) ∗ ∗ (h(n) ↑ 2Q −1 ) ) Combining (9), we can deduce the sampling rate Ri by and h( n) is the scale filter in DWT. And symbols ‘ ↑ ’ and ‘ ↓ ’ denote the up-sampling and down-sampling of a digital Ri = 2 jc −1 v / (ζ P ) . (10) signal, respectively. Here, we give a practical example to visually show the In Fig.4, we assume the input a0,k is the high-rate sampling numerical range of t s and Ri . We assume that the UFD is result of an analog echo signal. It can be seen that the output applied to the flaw detection in steel sample, where the signal aQ , k is obtained by down-sampling and linearly ultrasound velocity is v = 3200 m / s . According to the > IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL< 7 samplers. Therefore, we should trade off Q and the hardware costs in the design of a practical detection system. V. EXPERIMENTAL RESULTS In this section, we give the experiments from two aspects, energy match and noises suppression, to illustrate the performance of the SMW. Fig. 4. The polyphase decomposition of the DWT structure at the first Q scales. A. Performance of Optimal Energy Match of SMW The performance of the optimal energy match can be determined by two problems: concentration and separation. Since the separation is only determined by the central scale and the central scale can be easily adjusted to perform the separation, we only focus on the concentration of energy. Here, we give three experiments to illustrate the ability of concentration of our proposed method. The first and the second are given to illustrate the energy distribution in the aspect of the wavelet basis and the transmitted signal, respectively. The third experiment is given to show that the echo energy will be dispersed when the Concentration Rule is not satisfied. Experiment 1: We test the influence of wavelet basis on the energy distribution of flaw echo. Here, the wavelet bases are Fig. 5. Block diagram of the low-rate equivalent DWT structure at the first Q scales. sym3, sym5, sym8, sym13, sym18, and sym25, respectively, the transmitted signal is selected to be sym3, and the central scale jc = 8 . The continuous variable ρτ in (8) is discretized transforming of the delayed versions of a0,k . We find there are into 256 samples, thereby obtaining 256 delay values. Fig. 6 two sampling operations when the analog echo signal is (a)~(f) shows the energy distribution E ( j , ρτ ) of the flaw transformed in DWT. In order to reduce the sampling rate for echo for each selected wavelet basis. In these figures, the analogy echo signal, we could merge these two sampling horizontal-axis is the scale j varying from 1 to 11 and the procedure. Based on this consideration, we design a low-rate vertical-axis is the delay τ varying from 0 to 255. Obviously, equivalent DWT structure, shown in Fig. 5. the major energy of the flaw echo concentrated at the 7th, 8th In Fig. 5, the paralleled channel number is M = 2Q . Each and 9th scales. A/D sampler works at a low sampling rate R = Ri / M at To illustrate the energy distribution of flaw echo further, the different sampling time in this structure. The sampling time of average values of E ( j , ρτ ) of all delays at each scale are each A/D sampler is controlled by a high-rate clock and its calculated and shown in Table 3. It can be seen from Table 3, sampling delay corresponding to the first A/D sampler is with the vanishing moment of wavelets increasing, the energy marked in Fig. 5. The A/D samplers are followed by the digital of flaw echo at the 7th, 8th and 9th scales tends to increase time calibration module and the linear module EQ ( z ) . Since gradually. This result illustrates that the concentration of the structure in Fig. 5 is only the equivalence of the DWT energy distribution corresponds to Concentration Rules. But structure at the first Q scales, the output aQ , k should be when we select sym8 to sym25 as the wavelet bases, the increasing trend of energy becomes obviously slow. The reason decomposed consecutively from the scale Q + 1 to jc + 1 , in is that such wavelet bases usually have longer support length, order to obtain the wavelet coefficients at scales which is not suitable for the energy concentration. So the { jc − 1, jc , jc + 1} . balance of the vanishing moment and the support length for selecting the wavelet basis is important in a practical system. With the low-rate equivalent DWT structure, the sampling rate of the detection system can be significantly reduced to Ri / 2Q , which shows its superiority in practical applications. The higher Q is, the lower the sampling rate. However, the higher Q is, the more the parallel channels M and the A/D > IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL< 8 (a) sym3 (b) sym5 (c) sym8 (d) sym13 (e) sym18 (f) sym25 Fig. 6. The energy distribution E ( j , ρτ ) with each selected wavelet basis. TABLE 3. AVERAGE ENERGY OF E ( j , ρτ ) WITH EACH WAVELET BASIS. Experiment 2: This experiment tests the effect of the transmitted signal on the energy distribution of flaw echo. Here, Average energy (%) the transmitted signal is sym3, sym5, sym8, sym10 and sym13, Wavelet basis respectively, and the wavelet basis is selected to be sym8. And j≤5 j=6 j=7~9 j >9 the central scale jc is selected to 8. From the results shown in sym3 0.49 2.63 96.32 0.55 Table 4, we can conclude that the transmitted signal with a higher Lipschitz regularity results in a more concentrated sym5 0.39 1.66 97.78 0.16 energy distribution of flaw echo, which conforms to Concentration Rule 2. sym8 0.33 1.40 98.25 0.01 Experiment 3: We further verify the influence of sym13 0.32 1.38 98.27 0.01 Concentration Rule 1 on energy distribution of flaw echo. Here the transmitted signal is selected to be sym3 wavelet, and the sym18 0.29 1.56 98.12 0.00 wavelet basis is sym8. Parameters ts and tb are selected as sym25 0.21 1.49 98.29 0.00 follows: tb= a ⋅ 2− j ' ts , a ∈ [1, 2] , j ' ∈ + , TABLE 4. AVERAGE ENERGY OF E ( j , ρτ ) AT THE 7~9-TH SCALES OF FIVE where j ' represents the scale including the major energy of TRANSMITTED SIGNALS. flaw echo and j ' = 8 . Only when a = 1 or 2 in the above equation, the constraint of Concentration Rule 1: tb = 2− jc ts is Transmitted signal sym3 sym5 sym8 sym10 sym13 satisfied. Then we test the average distribution of E ( j , ρτ ) for Average energy all the ρτ with the parameter a being 1.0, 1.2, 1.4, 1.6, 1.8, 98.25 99.57 99.92 99.96 99.97 (%) 2.0, respectively. As Table 5 shows, in the case of 1 < a < 2 , the energy of flaw echo is dispersed to four scales (from the 6-th to 9-th scale), which decreases the performance of noise > IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL< 9 (a) Clean echo signal (b) Noisy echo signal with jc = 8 (c) Noisy echo signal with jc = 10 (d) Reconstructed echo signal with jc = 8 (e) Reconstructed echo signal with jc = 10 Fig. 7. Noise suppression for flaw echo detection. TABLE 5. follows. The input-SNR: SNRI (or SNRI ( i ) ) is defined as the AVERAGE ENERGY AT VARIOUS SCALE j AND PARAMETER A. SNR of the echo signal in the whole signal length (or in the Energy distribution (%) a support region of the i-the flaw echo). The output-SNR: SNRO j≤5 j=6 j=7 j =8 j =9 j >9 (or SNRO ( i ) ) is defined as the SNR of the reconstructed echo 1.0 0.33 1.40 22.37 64.13 11.75 0.01 signal after noise suppression in the whole signal length (or in the support region of the i-the flaw echo). 1.2 0.59 5.60 31.05 53.65 9.32 0.01 Firstly, a concrete example of the noise suppression is given. Here, we select sym5 and sym13 as the transmitted signal and 1.4 0.84 9.72 38.23 42.11 7.05 0.01 the wavelet basis, respectively. The clean echo signal r (t ) 1.6 1.17 14.90 47.40 33.04 4.15 0.00 includes five flaw echoes with various energy intensities, which are labeled from 1 to 5 in Fig. 7(a). The noisy echo signal is 1.8 1.47 19.22 55.82 22.23 2.30 0.00 generated by adding the Gaussian white noise n(t ) to the 2.0 1.73 22.37 64.13 11.75 0.01 0.00 clean echo signal. Then the performance of the proposed method on noise suppression is tested in two cases below. Case 1: the noisy echo signal (including the flaw echoes 1~5) suppression. While the energy distribution is concentrated at SNRI = −9.18 dB and the 5th flow echo three scales for a = 1 and a = 2 . SNRI (5) = −15.03 dB (as Fig. 7(b) shows). From the above three experiments, if the transmitted signal and the wavelet basis conform to Concentration Rules, the Case 2: the noisy echo signal (including the flaw echoes 1~5) energy distribution of the echo signal is concentrated and the SNRI = −15.17dB and the 5th flow echo optimal energy match is achieved. SNRI (5) = −20.31 dB (as Fig. 7(c) shows). B. Performance of Noise Suppression by SMW Considering the intensity of noise, we choose the central In this subsection, we test the performance of noise scale jc for the two cases as 8 and 10, respectively. Fig. 7(d) suppression and the robustness of the proposed method. Before and (e) show the reconstructed echo signals by the SMW showing experimental results, some definitions are given as method. It can be found that even the flaw echoes are almost (in > IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL< 10 TABLE 6. AVERAGE SNR IMPROVEMENT FOR SEVERAL TRANSMITTED SIGNALS AND WAVELET BASE. Wavelet basis Transmitted signal sym10 sym13 sym15 sym17 sym20 sym4 21.82 22.02 21.98 21.87 21.83 sym5 22.01 22.07 21.96 21.85 21.99 sym6 21.92 21.96 21.86 21.95 21.86 sym7 22.01 21.95 21.96 22.04 22.07 Fig. 8. Curves of output-SNR against input-SNR for jc = 8, 9, 10 . sym8 22.09 22.04 22.00 22.01 22.07 the first case) or totally (in the second case) buried in the noise, all the flaw echoes can be detected successfully. By calculation, the SNR of the output echo signal are: Case 1: proposed method. SNRO = 9.39 dB , SNRO (5) = 4.90 dB (as Fig. 7(d) shows) As the experimental results show, the proposed SMW is effective in the noise suppression and the flaw detection. and Case 2: SNRO = 9.58dB , SNRO (5) = 5.18dB (as Fig. 7(e) Compared with [6], the proposed method can improve the flaw shows), respectively. The above experiment shows the SMW echo detection ability about 5 dB in high background noise. method can efficiently detect the flaw in high background noise. In the follows, we further verify the performance of noise VI. CONCLUSIONS suppression in various noise levels. Here, the input-SNR is select in the interval [-25, 5] dB. The echo signal includes one In this paper, we propose the SMW method for UFD in high background noise. SMW overcomes the shortcoming of flaw echo. We select three central scales jc being 8, 9 and 10, traditional methods which is difficult to generate the optimal respectively. To obtain the stable statistical result, we test 100 match due to the constraints of wavelet. In SMW, the times for each selected central scale and in each test the delay of transmitted signal is designed to be a wavelet function to obtain flaw echo is randomly set. The approximate linear relation of a localized energy distribution of flaw echoes in WT domain. the output-SNR against the input-SNR is depicted in Fig. 8. Then three rules for optimal energy matching are proposed by The average SNR improvements reach 19.40 dB, 21.96 dB and analyzing the energy distribution of the echo signal in WT 24.47dB for the three cases of jc , respectively. domain. Furthermore, the scheme for choosing parameters is As pointed in [6], the flaw echo can be detected when the put forward in applying the proposed SMW to the actual UFD. output-SNR is higher than 4 dB. As Fig. 8 shows, for the In addition, a low-rate equivalent DWT structure based on central scales jc = 8, 9, 1 0, the flaw echo can be detected polyphase decomposition is developed, which reduces the with input-SNR above -15.40 dB, -17.96 dB and -20.47 dB, hardware cost and farther facilitates the practical application. respectively. So, the SMW-based method is effective in the The sufficient experiments are provided in two aspects: the improvement on noise suppression. validation on optimal match from energy distribution of flaw On the other hand, we test the robustness of the proposed echo in WT domain and the performance of noise suppression. method on the SNR improvement for various transmitted The experimental results show that SMW can efficiently detect signal and wavelet basis, both of which satisfy Concentration the flaw under high background noise even for the input-SNR low to -20 dB. Rules. Let the central scale jc = 9 , and the input-SNR set in the interval [-20, 0] dB. The transmitted signal is selected as ACKNOWLEDGEMENT sym4, sym5, sym6, sym7 and sym8, respectively. And the wavelet bases are chosen as sym10, sym13, sym15, sym17 and The authors would like to thank the reviewers for their sym20, respectively. Table 6 shows the average SNR valuable comments to improve the quality of the paper. And improvement of various combination of the transmitted signal they wish to show great appreciation to Prof. Xuemei Xie for with the wavelet basis. Each case in Table 6 is tested for 400 her help and suggestions during the revision of the paper. times. As Table 6 shows, the minimal SNR improvement is 21.82 dB and the maximal SNR improvement is 22.09 dB. Therefore, this experiment verifies the robustness of the > IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL< 11 APPENDIX And the smaller the length of R j , k , the larger α j , k . Given an Proof of Concentration Rule 2 arbitrary pair ( j1 , k1 ) satisfying j1 < jc and 2 j1 k1 ∈ Rτ , the following relation is deduced according to Lemma 1. To facilitate the proof, we first cite a lemma which describes the decay property of a signal f in WT domain. ∀j < j1 , 2 j k ∈ R j1 , k1 (A3) Lemma 1 [4]: If a mother wavelet ψ (t ) has a vanishing ∞ j β j1 ,k1 +1/ 2 ∫ s = (t − τ )ψ j , k (t )dt ≤ C2 (2 ) + d r , C2 ∈ moment K , K ∈ + , and there is a function f ∈ L2 ( j ,k −∞ ) which is uniformly Lipschitz α < K over [ξ1 , ξ 2 ] , then there where β j1 , k1 = min{α j1 , k1 , Kψ } . Equation (A3) shows that, if exists A > 0 such that the vanishing moment Kψ is large enough, when the scale α +1/ 2 ∀(u , s ) ∈ [ξ1 , ξ 2 ] × , Wf (u , s ) ≤ As + , (A1) parameter j decreases, the wavelet coefficients has the decay with the degree higher than α 0 + 1 / 2 . where u is the shift parameter, s is the scale parameter, and Wf (u , s ) is the wavelet coefficients corresponding to the WT Support length: of f on the basis 1 ψ( t −u ), (u , s ) ∈ [ξ1 , ξ 2 ] . { } Given two wavelet bases, one is ψ j , k (t ) with vanishing s s moment Kψ and support region R j , k for the function The proof of Concentration Rule 2 can be addressed in the following two cases. Case I: scale j < jc and Case II: scale ψ j ,k (t ) , and the other is {ψ j ,k (t )} with the same vanishing ˆ j > jc . ˆ moment Kψ but different support region R j , k for function Case I: scale j < jc . ψ j ,k (t ) . Let the lengths of these two support regions satisfy ˆ We give the proof from three aspects: Lipschitz regularity, R j , k < R j , k and R j , k ⊂ R j , k . The wavelet coefficients of ˆ ˆ vanishing moment and support length. For easing the deduction, the support region of a flaw echo s (t − τ ) is s (t − τ ) on {ψ j , k (t )} and {ψ j , k (t )} are denoted by d r, k and ˆ j denoted by Rτ and its length is denoted by | Rτ | . ˆ d jr, k , respectively. Then for an arbitrary pair ( j2 , k2 ) Lipschitz regularity: satisfying j2 < jc and 2 j2 k2 ∈ Rτ , we have R j2 , k2 ⊂ R j2 , k2 . ˆ Let the transmitted signal s (t ) be uniformly Lipschitz α 0 Let s (t − τ ) be uniformly Lipschitz α j2 , k2 in the region over [ −∞, ∞] , then a flaw echo s (t − τ ) , as a delayed version of s (t ) , is also Lipschitz α 0 . Applying the Lemma above to R j2 , k2 and γ j2 , k2 in the region R j2 , k2 respectively. Then we ˆ DWT, if a basis {ψ j , k (t ), j , k ∈ has a vanishing moment } can get γ j2 , k2 ≥ α j2 , k2 . Let χ j2 , k2 = min{γ j2 , k2 , Kψ } , we have K > α 0 , the following relation is satisfied ∀j < j2 , 2 j k ∈ R j2 , k2 ˆ (A4) ∞ ∞ ∫ j α 0 +1/ 2 ∀j , k ∈ d = s (t − τ )ψ j , k (t )dt ≤ C1 (2 ) j χ j2 ,k2 +1/ 2 ∫ s =(t − τ i )ψ j , k (t )dt ≤ C3 (2 ) r + , j ,k −∞ , (A2) ˆ d r j ,k ˆ , C3 ∈ −∞ + C1 ∈ Considering the arbitrariness of ( j2 , k2 ) , we conclude that It can be found from (A2) that for a basis {ψ j , k (t )} with for a fixed vanishing moment, the decay of wavelet coefficients vanishing moment K > α 0 , when the scale j decrease, the is faster if the support length of selected wavelet is smaller. Combining these three relations (A2)-(A4), the wavelet coefficients d r, k decays with the degree of α 0 + 1 / 2 . j Concentration Rule 2 for Case I is concluded. Thus, if the transmitted signal s (t ) have a large uniformly Case II: scale j > jc . Lipschitz regularity α 0 , when scale j decreases ( j → 0 ), the Since the flaw echo s (t − τ ) is selected as a wavelet wavelet coefficients has a fast decay. Vanishing moment: { function from the wavelet basis W j , k (t − τ ), j , k ∈ , it } satisfies the dilation and shift relations. Thus we express the Let ψ j , k (t ) have a vanishing moment Kψ , Kψ ∈ + and coefficients d r, k as its support region be R j , k . Let s (t − τ ) be uniformly Lipschitz j α j ,k in R j ,k . The relation α j ,k ≥ α 0 can be easily obtained. > IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL< 12 [14] S. Legendre, D. Massicotte, J. Goyette, and T. K. Bose, ∞ = ∫ W (t − τ )ψ j , k (t )dt “Wavelet-transform-based method of analysis for lamb-wave ultrasonic NDE d r, k j signals,” IEEE Trans. Instrum. Meas., vol. 49, no. 3, pp. 524-530, 2000. −∞ [15] R. J. Liou, K. C. Kao, C. Y. Yeh, and M. S. Chen, “Flaw detection and sizing ∞ 1 u − 2− j τ i t / 2j − k ∫ + 2 j k ψ (u )du of ultrasonic images using wavelet transform and SAFT,” in Proc. IEEE Int. Conf. u =W −j Intelligent Signal Process. Commun. Systems, 2004, pp. 106-110. −∞ −j 2 2 [16] E. Oruklu and J. Saniie, “Ultrasonic flaw detection using discrete wavelet W− j , −2 j k ( u − 2− j τ i )ψ (u )du. ∞ transform for NDE applications,” in Proc. IEEE Int. Ultrason. Symp., 2004, pp. ∫ (A5) 1054-1057. −∞ [17] A. H. Tewfik, D. Sinha, and P. Jorgensen, “On the optimal choice of a wavelet for signal representation,” IEEE Trans. Inform. Theory, vol. 38, no. 2, pp. The equation (A5) shows a distinct fact, not only can the 747-765, 1992. coefficient d r, k be thought to be the projection of the echo j [18] J. O. Chapa and R. M. Rao, “Algorithms for designing wavelets to match a specified signal,” IEEE Trans. Signal Process., vol. 48, no. 12, pp. 3395-3406, signal W (t − τ ) on basis ψ j , k (t ) , but it can be thought to be 2000. [19] A. Gupta, S. D. Joshi, and S. Prasad, “A new approach for estimation of the projection of ψ (t ) on the basis W− j , −2 j k t − 2− j τ i .( ) statistically matched wavelet,” IEEE Trans. Signal Process., vol. 53, no. 5, pp. 1778-1793, 2005. [20] R. Hamilton and G. Hayward, “The modelling and design of controllable Therefore, the analysis procedure and the conclusion for Case composite transducers,” in Proc. IEEE Int. Ultrason. Symp., 1990, pp. 393-396. II are similar with that for Case I, differing only on the suitable [21] G. Hayward and J. E. Lewis, “A theoretical approach for inverse filter design signal being s (t ) instead of ψ j , k (t ) . While a higher in ultrasonic applications,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 36, no. 3, pp. 356-364, 1989. vanishing moment of a function usually causes longer support [22] P. P. Vaidyanathan, Multirate systems and filter banks. Englewood Cliffs, NJ: Prentice-Hall, 1993. length, which will reduce the detection precise when it is selected as a transmitted signal, so the requirement of high vanishing moment is not included in the Concentration Rule 2 Guangming Shi (M’07–SM’10) was born in for Case II. Based on the analysis above, the Concentration Jiangxi, China, in 1965. He received the B.S. Rule 2 for Case II is concluded. degree in electronic engineering in 1985 and the M.S. degree in automation in 1988, Ph. D. degree in intelligent signal processing in 2002, all from Xidian University, Shaanxi, China, and now he is a professor at the same university. His REFERENCES research interests include ultrasonic signal [1] M. G. Gustafsson, “Nonlinear clutter suppression using split spectrum detection, wavelet application, compressed processing and optimal detection,” IEEE Trans. Ultrason., Ferroelect., Freq. sensing theory and its application on UWB, Contr., vol. 43, no. 1, pp. 109-124, 1996. signal sampling and processing theory, image [2] Q. Tian and N. M. Bilgutay, “Statistical analysis of split spectrum processing and video compression, and optimization. for multiple target detection,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 45, no. 1, pp. 251-256, 1998. [3] C. Cudel, M. Grevillot, J. J. Meyer, L. Simonin, and S. Jacquey, “Detecting echoes of different spectral characteristics in absorbing media by variable moving bandwidth SSP minimization,” in Proc. IEEE Int. Ultrason. Symp., 2002, pp. Xuyang Chen (S’07) was born in Hebei, 785-788. China, in 1980. He received the B.S. degree in [4] S. Mallat, A wavelet tour of signal processing. New York: Academic, 1998. electronic engineering from Xidian University, [5] A. Abbate, J. Koay, J. Frankel, S. C. Schroeder and P. Das, “Application of Shaanxi, China, in 2005. He is currently wavelet transform signal processor to ultrasound,” in Proc. IEEE Int. Conf. pursuing the M.S. and Ph.D. degree at Xidian Ultrasonics Symposium, Nov. 1994, pp. 1147-1152. University. His research interests include [6] A. Abbate, J. Koay, J. Frankel, S. C. Schroeder, and P. Das, “Signal detection wavelet analysis, pattern recognition and and noise suppression using a wavelet transform signal processor: application to optimization theorem. ultrasonic flaw detection,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 44, no. 1, pp. 14-26, 1997. [7] L. Bechou, D. Dallet, Y. Danto, P. Daponte, Y. Ousten, and S. Rapuano, “An improved method for automatic detection and location of defects in electronic components using scanning ultrasonic microscopy,” IEEE Trans. Instrum. Meas., vol. 52, no. 1, pp. 135-142, 2003. [8] S. P. Song and P. W. Que, “Wavelet based noise suppression technique and its Xiaoxia Song received her B.S. degree in application to ultrasonic flaw detection,” Ultrasonics, vol. 44, no. 2, pp. 188-193, computer science in 1998 from Yanshan 2006. University, Hebei, China. And she earned her [9] D. L. Donoho, “De-noising by soft-thresholding”, IEEE Trans. Information M.S. degree in computer software and theory in Theory, vol. 41, no. 3, pp. 613-627, 1995 2005 from Guangxi Normal University, [10] E. Pardo, J. L. San Emeterio, M. A. Rodriguez, and A. Ramos, “Noise Guangxi, China. Now she is pursuing her Ph.D. reduction in ultrasonic NDT using undecimated wavelet transforms,” Ultrasonics, degree in intelligent information processing at vol. 44, supp. 1, pp. e1063-e1067, 2006. Xidian University, Shaanxi, China. From 1998, [11] B. Molavi, A. Sadr, and H. A. Noubari, “Design of optimum wavelet for noise she was with a teacher in Shanxi Datong suppression and its application to ultrasonic echo delay estimation”, IEEE Signal University, Shanxi, China, and she is an Process. Commun. Conf., 2007, pp. 209-212. associate professor at the same University. Her [12] S. F. Qi, C. Zhao, and Y. Yang, “Research on ultrasonic detection of seabed current research interests include compressed oil pipeline based on wavelet packet de-noising”, in Proc. IEEE Int. Conf. sensing theory, image processing, and Wireless Commun., Networking and Mobile Computing, 2009, pp. 1-4. optimization calculation. [13] K. Kaya, N. M. Bilgutay, and R. Murthy, “Flaw detection in stainless steel samples using wavelet decomposition,” in Proc. IEEE Int. Ultrason. Symp., 1994, pp. 1271-1274.

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