Signal Matching Wavelet for Ultrasonic Flaw Detection in High

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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:,                 (SMW).         Different      from     those       methods       of, ,          and                                                            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
                                                                             )    ∑∑ d
                                                                                  j =1   k
                                                                                             j ,k   ⋅ψ j , k (t ) + ∑ aLmax , kφLmax , k (t ),
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
                                                                      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 ,
                                                                                              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                         (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
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

                                                                         Transducer                 T/R

                                Flaw              Signal processing
                               detector              unit (WT)                        echo signal

                                                          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
                          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
                                                                                       = ∑∫
                                                                            E j , ρτ                              s (t − ρτ )ψ j , k (t )dt ,   (8)
subsection, we describe explicitly the selection of these                              k
                                                                                              j ,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.
                    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
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 + i ) z − n , i= 0,1,, 2Q − 1 ,       (11)
considering tb = 2− jc ts in (6), we express Ri as
                            = 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.
                                                                                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
                                                                                  SNR of the echo signal in the whole signal length (or in the
                              Energy distribution (%)
                                                                                  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
                                                                                         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
   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
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
                             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
 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
  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 .
  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

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
                            ∞                                                                                ∞
                                                                   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

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

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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
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                                                                                                                         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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