Denoising ultrasonic pulse-echo signal using two-dimensional
analytic wavelet thresholding
Mohammad R. Hoseini, Ming J. Zuo, and Xiaodong Wang
Department of Mechanical Engineering, University of Alberta, Canada
Abstract
When an ultrasonic angle-beam pulse-echo setup is used, two kinds of noise are present in the
received signal: 1) wedge noise, and 2) random noise. In this study, we propose a method for
removing both random and wedge noises using a two-dimensional stationary wavelet
transform (2D SWT). To improve the performance of the 2D SWT, we employ analytic
wavelet thresholding. This requires calculating the analytic signal for a 2D signal and
extracting the envelope of the signal, but no regular generalization of analytic signals to multi
dimensions is available. We prove that under some certain conditions, the envelope of the
ultrasonic B-scan can be extracted exactly by introducing a reference vector. The
performance of the proposed denoising method is assessed using simulated and experimental
data. The experimental results show that our proposed denoising method is able to reveal the
echoes from a surface breaking crack as small as 0.1 mm.
Index Terms— Ultrasonic Testing, B-scan, Analytic Wavelet Thresholding, Stationary
Wavelet Transform.

The corresponding author: Ming J Zuo, ming.zuo@ualberta.ca
1
1.
INTRODUCTION
Ultrasonic diagnostic methods are widely used for non-destructive evaluations of engineering
materials. Ultrasound is usually produced by a piezoelectric crystal excited by a high voltage
pulse. The ultrasound waves are then transferred into the material being tested via a couplant;
they reflect off any discontinuity within the material, such as a defect. A portion of the reflected
ultrasound travels to the receiver producing voltage [1]. Either the transmitter or another
transducer can be used as the receiver. The first is called a pulse-echo configuration and the latter
is called a pitch-catch configuration [2]. Each single set of data collected from a fixed probe
position is called an A-scan. A position sensor can be used to measure probe movement and then
create a cross-sectional image of the object under examination [3]. This image is called a B-scan.
In this paper, a denoising method is proposed for enhancing the pulse-echo’s B-scan image. In
pulse-echo setups two types of noise are present in the signal: wedge noise and random noise.
There are some methods for dealing with removing wedge noise. Mosey et al. [3] recommended
calculating the mean of A-scans, iteratively subtracting the mean from all A-scans to remove the
background of the B-scan image. Although this method is conceptually simple, it can’t remove
the wedge noise effectively when there are small variations in it or the signal-to-noise ratio
(SNR) is low due to high level of random noise. Recently, Cygan et al. [4] used wavelet
transform to remove the wedge echo. They decomposed the B-scan image using wavelet
transform, then suppressed the wavelet coefficients (which include wedge noise) and
reconstructed the signal. Because wedge noise normally appears in the approximation and
vertical detail coefficients, removing the approximation coefficient may cause major distortion in
the reconstructed signal.
2
In this paper, we employ SWT to decompose the B-scan image. As mentioned above wedge
noise appears in the approximation and vertical detail coefficients, whereas crack echoes appear
mainly in the horizontal and diagonal detail coefficients. To remove wedge noise, the average of
all A-scans in the approximation coefficient is subtracted from each A-scan. This step is repeated
for the vertical detail. This way, removal of the wedge noise does not affect the crack echo signal
as much as does the method proposed by Mosey et al. [3]; this is because crack echoes appear
mainly in the horizontal and diagonal detail coefficients which remain unchanged during this
step. Because, the approximation coefficient contains the crack echo as well, our method does not
have the drawback of the method proposed by Cygan et al [4] in which the approximation
coefficient is totally removed. In order to remove random noise, we employ the SWT combined
with analytic wavelet thresholding, as proposed by Olhede and Walden [5]. In addition to its
good denoising performance, SWT allows better localization of the flaw than does discrete
wavelet transform (DWT) [6]. Applying analytic thresholding increases the denoising
performance of SWT even more [5]. Our contribution here is generalizing this method to two
dimensions.
The remainder of this paper is organized as follows. The experimental setup is described in
Section 2. Section 3 presents a model for pulse-echo B-scan imaging. Section 4 introduces
wavelet transform and investigates the analytical wavelet thresholding for a 2D signal. Section 5
explains the denoising method we propose. Section 6 presents a comparison between our
denoising method and two competing methods using simulated and experimental signals. Finally,
conclusions are given in Section 7.
3
2.
EXPERIMENTAL SETUP
The experimental setup, as shown in Figure 1, utilizes the pulse echo arrangement. R/D Tech
Omniscan UT module is used for data acquisition. The transducer is a Krautkramer, Benchmark
Series miniature angle beam transducer, with a center frequency of 2.25MHz, a bandwidth of 1.5
MHz, and an element with a diameter of 0.5”. The Lucite wedge used in this experiment has a
standard refracted shear wave angle of 45º in steel.
Figure 1
Nine 4140-steel specimens are cut perpendicular to the surface by electrical discharge machining
(EDM) to different depths ranging from 0.1mm to 3.0mm. The thickness of all specimens is 16
mm. The transducer is moved linearly along the specimen to generate a B-scan image. The
positioning system is programmed to move the transducer a distance of 30 mm with a step size of
0.25 mm. The starting position of the transducer has the probe beam index at a distance of 30 mm
from the EDM slot. At every step, 2048 data points are collected at a sampling frequency of 100
MHz. All experiments are repeated three times.
3.
MODELING ULTRASONIC PULSE-ECHO SIGNALS
Figure 2 illustrates the pulse-echo setup. As it can be seen in Figure 2, the received signal
consists of three echoes:
1. At the interface of the wedge and the specimen, a portion of the wave reflects back to the
wedge and is received by the transducer. This echo is called the wedge echo or wedge
noise [7].
2. The portion of the wave that is incident on the crack and is reflected back to the
transducer is commonly referred to as the corner trap echo.
4
3. When the ultrasonic wave impinges on the crack tip, it backscatters in a circular form [7].
Some of these scattered waves make their way back to the transducer.
Figure 2
Figure 3 shows a B-scan image for a specimen with a 3mm crack. The horizontal axis depicts
time and the vertical axis depicts the distance from beam index to slot. Two distinct patterns are
observed in the B-scan image: vertical lines and inclined lines. Vertical lines are produced by
echoes whose time of flight does not change by moving the transducer. These vertical lines are
caused by wedge noise. Though using absorbent material in the wedge mitigates the effect of
wedge noise, it still corrupts the signal, especially when crack echo is relatively weak. For
example, the crack echo is barely visible for a specimen with a 0.1mm crack as shown in Figure
4. For other crack lengths, SNR increases as the crack length increases. Other than that B-scan
images are similar to each other.
Figure 3
Figure 4
Figure 5(a) illustrates the travel paths of two ultrasonic rays through the wedge and the specimen.
We define the central ray as the one reflecting from the corner of the slot and it is represented by
a solid line. The other ray (we will call it the dash ray) is represented by a dashed line. It can be


shown that BC  CD  DF  BC  CH  DF  l s and AB  GF  l w , where ls is the travel
distance of the central ray in the specimen and lw is its travel distance in the wedge. Thus, the
total travel distance of the dash ray is equal to the total travel distance of the central ray. Figure
5(b) shows the travel paths of the central rays when the transducer is at two different positions.
From Figure 5(b), we can calculate the difference between the travel distances of these two
central rays as
5
l  2x sin  w
(1)
where x, l, w are respectively the horizontal distance from the transducer to the slot, the distance
the wave travels to the slot and back to the transducer, and the angle of incidence at the wedgespecimen interface on the wedge side. Substituting l  v w in Equation (1) gives,
 2 sin  w

x
vw
(2)
in which  , vw are the time of arrival and wave velocity in the wedge, respectively. Alternatively
from Snell’s law, one may rewrite Equation (2) as
 2 sin 

x
vs
(3)
where vs is the wave velocity in the specimen and  is the wave angle upon entering a specimen
with a normal surface. As this shows, the time of flight is a linear function of the distance
between transducer and slot. All parameters in Equation (3) are known before performing the
test. For our experimental setup, the speed of the refracted shear wave in the specimen has been
obtained using assisted calibration of Omniscan as 3229 m/s. Evaluating Equation (3) gives,
 2 sin   2 sin 45
.


 10 3  0.438 s
mm
x
vs
3229
(4)
Figure 5
Figure 6 gives the time of flight for the corner trap and tip diffracted echoes at different
transducer positions. From the experimental data, it can be observed for both cases that the slope
of the fitted line is in good agreement with the theoretical values in Equation (4); in both cases
there is less than 0.5% deviation from theory.
6
Figure 6
3.1
Mathematical model for ultrasonic echo
An A-scan signal is often modeled by superimposing multiple echoes [8-11]. Every backscattered
echo can be modeled in turn as,
g t   T t   cos2f c t      
(5)
in which g t  is the echo, T is a band-limited signal, t is time, τ is time of arrival (also called time
of flight), fc is the transducer’s center frequency, and is phaseEquation (5) gives the crack
echo signal for an A-scan. In a B-scan image, amplitude and time of flight are functions of
distance, x, hence
g  x, t   e x, t  cos2f c t    x      X  x T t   x  cos2f c t    x    
(6)
where X g and e are band-limited functions. As shown in Equation (3), the time of flight of the
crack echo is a linear function of the distance, i.e.   x   ax  b .
Gaussian echo is widely used to model the ultrasonic echo [11]. For this model T(t) is
approximated by a Gaussian function. Also, for the data collected in our experiment, we have
observed that the amplitude can be well approximated using a Gaussian function. In other words,
function X(x) can be also approximated by a Gaussian function for our experimental data. For
example, Figure 7 shows the amplitude of the corner trap echo as a function of the distance
between transducer and slot for a set of experimental data. Thus, we will use the following
equation to generate a simulated crack echo in Section 6.1:

 

g  x, t    exp   x  x  x0  exp   t t    x  cos 2f c t    x    
2
7
2
(7)
where x andtare the spatial and temporal bandwidth factors respectively, β is amplitude,
and x0 is the position where maximum amplitude occurs.
Figure 7
3.2
Minimum temporal and spatial resolution
To avoid aliasing, the Nyquist frequency, which is half of the sampling frequency, should be
higher than the maximum frequency of the sampled signal. Therefore, the sampling frequency
should be high enough such that,
G  f x , f t   0 if f x  f s ,x 2 or f t  f s ,t 2
(8)
where G is the Fourier transform (FT) of g(x,t), fs,t is the temporal sampling frequency, fs,x is the
spatial sampling frequency which is defined as the reciprocal of the scanning step size, ft is the
temporal frequency, and fx is the spatial frequency. The FT of the envelope, e(x,t), is equal to
E  f x , f t   T  f t X  f x  aft  exp  2if t b 
(9)
in which functions E, T , and X are the FT of functions e, T, and X, respectively. One can obtain
the FT of the echo signal in terms of E as
G f x , ft  
1
1
E  f x  af c , f t  f c  exp i   2f c b   E  f x  af c , f t  f c  exp  i   2f c b  .
2
2
(10)
Equation (10) implies that the cosine term in Equation (6) shifts the envelope to the 2nd and 4th
quadrants. For example, Figure 8(a) depicts the 2D FT of the ultrasonic signal shown in Figure 3.
It can be seen that the crack echoes in the frequency domain appear in the 2nd and 4th quadrants.
Figure 8
Inserting G  f x , f t  from Equation (8) into Equation (10) gives,
8
E  f x , f t   0 if f x  f s , x 2  af c or f t  f s ,t 2  f c .
(11)
Practically, it is reasonable to assume that E  f x , f t   0 in Equation (11) if the power of the
component at (fx , ft) is smaller than the maximum power by M dB or more. Thus, for a Gaussian
echo we have,
  f t 2 
E ( f x , ft ) T  ft 
  10M / 20 .

 exp 

E (0,0)
T 0 
 t 
(12)
After some algebraic manipulation we obtain,
min  f s ,t   2 f c 
2

M t ln 10
.
20
(13)
The frequency bandwidth of the transducer is 1.5 MHz, hence  t  16 MHz  . Letting M=20 dB
2
gives a minimum sampling frequency of 8.4 MHz which is well below the sampling frequency of
100 MHz used in our experiment.
Let X(x) also be a Gaussian function, then for the minimum spatial step size, we have,

 f t 2  f x  aft 2  
E ( f x , f t ) T  f t  X  f x  aft 
2

   10 M / 20 .

 exp   



E (0,0)
T 0  X 0 


x
 t


(14)
For a given fs,x, the condition expressed in Equation (14) should hold for all values of ft. This can
be achieved by maximizing E ( f x , f t ) E (0,0) with respect to ft. Letting partial derivative of
E ( f x , f t ) E (0,0) with respect to ft to be zero results in,
   2 f x2 
E ( f x , ft )
  10 M / 20 .
 exp  2
E (0,0)
a



t
x 

Finally, after some algebraic manipulation we obtain
9
(15)
min  f s , x   2af c 

2

M ln 10 2
a t   x .
20

(16)
The value of x is approximated from Figure 7 for the specimen with a 3mm slot. This value is
not the same for different slot lengths. We have performed curve fitting using the experimental
from specimens with slot lengths from 0.1 mm to 3 mm and have found the maximum value of x
to be 0.04 mm-2. From Equation (4), a=0.438 μs/mm and let M=20 dB again. Substituting these
parameter values in Equation (16) gives a minimum spatial sampling frequency of 3.7 mm-1 or a
step size of 0.27 mm; this is slightly larger than the 0.25 mm step size used in our experiment.
4.
WAVELET DENOISING
Wavelet transform is one of the most powerful transforms for image processing applications such
as feature extraction, denoising, and compression [12]. Its primary advantage is that it is localized
in both the spatial and frequency domains [13]. Wavelet transform also offers more flexibility in
selecting the basis than do other transforms such as the Fourier transform; however, it is
important to select a basis which matches the signal [14]. The discrete wavelet transform (DWT)
can be performed using the pyramidal algorithm proposed by Mallat [15]. The pyramidal
algorithm decomposes a signal, fm, at level m by convolving it with a low-pass filter, h (scaling
filter), to form an approximation signal, fm+1, at level m+1, and a high-pass filter, g (wavelet
filter), to form a detail signal, f m 1 , at level m+1.
f m 1 n    h2n  k  f m k 
k
f m 1 n    g 2n  k  f m k 
k
10
.
(17)
The resulting signals are subsequently sampled at every other point in order to avoid redundancy.
Although DWT is very efficient from the computational point of view, it is not translation
invariant. Translating the original signal leads to different wavelet coefficients [16]. This
degrades the quality of denoising [5]. To overcome this drawback, SWT was proposed [17]. The
only difference between SWT and DWT is that when calculating SWT the signal is not
decimated, instead, at each level the filters are up-sampled, i.e.
f m 1 n    hm n  k  f m k 
k
f m 1 n    g m n  k  f m k 
(18)
k
in which the scaling filter is defined recursively as:
hm k 2 , k even
hm 1 k   hm k   2  
0 , k odd
(19)
where h0 k   hk , and  2 denotes upsampling by a factor of 2. The wavelet filter, g m k  , can
be defined similarly. Figures 9 (a) and (b) illustrate how DWT and SWT are performed,
respectively.
The 2D wavelet transform can be formulated to be separable so it can be computed by extending
the 1D pyramidal algorithm transform. In other words, 1D wavelet decomposition is performed
first in one direction, then in the other direction. Thus, the 2D SWT of a 2D function, fm[n1,n2],
is calculated as:
11
f m 1 n     hmH n1  k1 hmV n 2  k 2  f m k1 , k 2 
f
H
m 1
f
V
m 1
k1
k2
k1
k2
n     hmH n1  k1 g mV n 2  k 2  f m k1 , k 2 
(20)
n     g n1  k1 h n 2  k 2  f m k1 , k 2 
H
m
k1
k2
k1
k2
V
m
f mD1 n     g mH n1  k1 g mV n 2  k 2  f m k1 , k 2 
where superscripts H, V, and D stand for horizontal, vertical, and diagonal coefficients,
respectively. The selection of horizontal and vertical filters is discussed in Section 5. Figure 9 (c)
illustrates how 2D SWT is performed using the pyramidal algorithm. As illustrated, a 2D signal
is decomposed into an approximation signal and three detail signals. Because each detail signal is
associated with the detail of the signal in one direction, they are called horizontal, vertical, and
diagonal detail coefficients. Table 1 gives the frequency content of each wavelet coefficient. For
example, the horizontal detail coefficient, which is passed through a scaling filter in horizontal
direction and through a wavelet filter in vertical direction, contains the horizontal low frequency
and vertical high frequency content of the signal.
Figure 9
Table 1. Frequency content of each wavelet coefficient
Horizontal
Vertical
Low Frequency
High Frequency
Low Frequency
Approximation Coefficient
Horizontal Detail Coefficient
High Frequency
Vertical Detail Coefficient
Diagonal Detail Coefficient
12
4.1
Wavelet thresholding
Wavelet thresholding combines simplicity and efficiency and exhibits certain asymptotic
optimality properties [18]. It has become a standard technique used extensively for signal
denoising. The wavelet thresholding algorithm consists of the following steps:
1) Perform the wavelet transform: In this work, we use the 2D SWT to decompose an
ultrasonic B-scan image. SWT often produces a lower estimation risk than does DWT,
thanks to its redundancy [12].
2) Estimate a threshold: There are many methods available for estimating the threshold.
Coifman & Donoho [19] found that universal thresholding when used in conjunction with
SWT produces a smaller number of noise spikes compared to Stein’s unbiased risk
thresholding. Minimax often results in a better SNR, though usually the visual
appearance of denoised images is inferior to that of images denoised using universal
thresholding. The universal threshold for a signal contaminated by white Gaussian noise
(WGN) is given by:  n 2 log n , where  n is the standard deviation of the noise and n is
the number of coefficients. If  n is unknown, it can be estimated using the robust median
absolute deviation of the finest scale,
 MAD 

median f 1,D1 , f1,D2 ,..., f 1,Dn

0.6745
(21)
where f 1,Di is the i’th coefficient of the diagonal coefficient at the first level.
3) Apply a shrinkage rule to the wavelet coefficients: The shrinkage rule defines how we
apply the threshold. Coifman & Donoho [19] found that combining the hard shrinkage
rule and SWT gave both good visual characteristics and good quantitative characteristics;
13
hence, we have adopted the hard shrinkage rule throughout. The so called hard shrinkage
rule sets all coefficients smaller than the threshold to zero and keeps all others unchanged.
4) Perform the inverse transform of the wavelet coefficients.
In spite of the appeal of SWT denoising, at least one of its aspects is unattractive; the
thresholding of an SWT coefficient which is oscillatory about zero degrades the denoising
efficiency. To overcome this problem, Olhede and Walden [5] proposed analytic wavelet
thresholding. They calculated the analytic signal (See Section 4.2 for the definition of the analytic
signal) and then applied wavelet thresholding to the amplitude of the signal rather than its
magnitude. In other words, they suppressed the effect of the phase. In setting the threshold value,
they proved that the amplitude of WGN asymptotically follows chi distribution and adapted the
universal threshold value,  n 2 log n log n  , proposed by Sardy [20] for chi-distributed white
noise. To apply analytic thresholding to a 2D signal we need to calculate the analytic signal, but a
regular generalization of the analytic signal is not available in multiple dimensions [21]. In the
following section, we address this problem.
4.2
Analytic signal of two dimensional signals
The analytic signal plays an important role in signal processing because its magnitude and
angular argument are frequently used to demodulate signals and define the instantaneous
amplitude and phase of real-valued signals. This can be achieved using Hilbert transform under
certain conditions stated in Bedrosian’s theorem. According to Bedrosian’s theorem, if the
envelope and carrier are non-overlapping band-limited signals, the amplitude of the analytic
signal is identical to the envelope [22]. The analytic signal, a(t), arising from a 1D signal, x(t), is
defined as:
14
a t   x t   iHx t 
(22)
where H denotes the Hilbert transform. a t  and x t  are defined to be the amplitude and
magnitude of the signal, respectively. Hilbert transform is an integral transform defined by:
H x t  
1

x  d
  t 
.
(23)

Analytic signals may be better understood in the frequency domain. If X() is the Fourier
transform of the real signal, x(t), then
A   1  sign   X  
(24)
where A() is the Fourier transform of the analytic signal and the sign function is defined as:
1

sign   0
 1

if   0
if   0 .
(25)
if   0
In other words, the negative frequency components are suppressed and the positive frequency
components are doubled.
In order to generalize the concept of the analytic signal to multiple dimensions, the positive and
negative frequencies in multiple dimensions should be defined. A straightforward way to define
positive and negative frequencies is to introduce a reference vector in the frequency domain.
Introducing a reference vector, n̂ , a frequency coordinate, u, is labeled positive if u T nˆ  0 and
negative if u T nˆ  0 [21]. Despite the argument put forward by Bülow and Sommer [23] that this
method is one-dimensional in nature, we will show it is suited for our purpose.
15
The envelope of the signal is best recovered in the direction of the reference vector and worst
recovered in the direction n̂  which is perpendicular to the reference vector [23]; however, there
is no general rule for selecting the direction n̂ . Figure 10 demonstrates a properly and an
improperly selected reference vectors. For a properly selected reference vector, high frequency
and low frequency components are projected to high frequency and low frequency regions of the
reference vector, respectively. In addition, the projection of the high-frequency carrier on the
reference vector does not overlap with the projection of the low-frequency envelope; hence,
according to Bedrosian’s theorem, the low frequency envelope can be extracted using the Hilbert
transform (see appendix A for the proof for 2D signals). In fact, the reference vector not only
defines the negative and positive frequencies but also defines the concept of high frequency and
low frequency in multiple dimensions. To determine the reference vector, it is often enough to
know roughly where the HF and LF components are located in the frequency plane.
Figure 10
Figure 11 depicts the 2D FT of the product of the envelope and carrier. As illustrated, selecting
any reference vector among n̂ , û 1 and û 2 and suppressing the negative frequency components
defined by these reference vectors leads to the same result. If no prior knowledge about the HF
and LF components is available, the reference vector can be chosen in the direction of the
principal axis of the second moment of the area with minimum moment (compare Figures 10 (a)
and (b)). Principal axes form a coordinate system in which the second moments of area assume
their maxima and minima.
Figure 11
16
As discussed earlier, crack echo frequency content appears in the 2nd and 4th quadrants. It is easy,
therefore, to introduce a reference vector in order to remove either the 2nd or the 4th quadrant and
keep the other one. Figure 8 shows an example of an experimental signal and its envelope
spectrum. To calculate the envelope spectrum of this experimental signal, the y-axis is chosen as
the reference vector.
5.
PROPOSED DENOISING METHOD
This section presents the proposed denoising method. To illustrate, Figure 4 gives an example of
an experimental signal to which we apply our denoising method. Figure 12 illustrates the
flowchart of the denoising method proposed for removing both random noise and wedge noise
from the B-Scan image.
Figure 12
Firstly, the signal is decomposed using the 2D SWT. The horizontal wavelet filter is selected to
best match the crack echo. Such a filter transforms signal energy to a few wavelet coefficients; in
other words wavelet coefficients provide a sparse representation of the signal. The selection is
made among three orthogonal wavelet families of Daubechies (dbN), symlet (symN), and coiflet
(coifN), based on the Shannon entropy criterion [24].
n
Shannon entropy of a vector, vi , i  1, 2,..., n , is S    pi log e pi where p i  vi2
i 1
n
v .
2
i
i 1
Shannon entropy, a measure of the randomness in a system, is an excellent indication of energy
concentration [25]. The higher the energy concentration of a signal, the lower the Shannon
entropy. We are looking for a mother wavelet that transforms the signal energy to only a few
coefficients; therefore, minimizing the Shannon entropy gives the best mother wavelet and level
17
of decomposition. An A-scan signal is decomposed using dbN (N=1, 2… 15), symN (N=1, 2…
15), and coifN (N=1, 2… 5) wavelets; this done, its entropy is calculated. Figure 13 depicts the
entropy of wavelet coefficients for different wavelets. The minimum entropy belongs to sym9
and the best level of decomposition for all wavelets is three.
Figure 13
We chose the Haar mother wavelet as the vertical filter. It is employed to separate wedge noise
from the crack echo as much as possible. Haar wavelet has one vanishing moment. In other
words, the approximation coefficient contains the locally constant portion of the signal, and the
detail coefficients contain the varying portion. Because the wedge echo does not change when
transducer is moved, it appears mainly in the approximation and/or vertical coefficients
depending on its horizontal frequency; in contrast the time of flight of the crack echo changes as
the transducer’s position does, hence the crack echo appears in the horizontal and/or diagonal
coefficients.
Figure 14 shows the SWT decomposition of the experimental signal. As illustrated, the
approximation coefficient contains most of the energy; however, the crack echo is mainly
captured by the horizontal detail. Cygan et al. [4] used the detail coefficients to detect the crack
echoes. Nonetheless, some of the flaw signal’s energy is captured by the approximation signal so
some information will be lost if we discard it from our analysis. Moreover, the filter causes a
delay which affects the size-estimation. The image can be decomposed to higher levels to better
separate the wedge and flaw signals, as well as to achieve better denoising.
Figure 14
18
The next step is to remove wedge noise from the approximation and vertical coefficients. In order
to remove this noise, the average value of all A-scans in the approximation/vertical is subtracted
from each A-scan in the approximation/vertical coefficient. Figure 15 depicts the approximation
coefficient after wedge noise has been removed. One can observe the crack echo clearly in this
image.
Figure 15
Finally, every coefficient is denoised and reconstructed using the method explained in Section 4.
The Sym9 and Haar mother wavelets are used as horizontal and vertical filters, respectively, for
three-level SWT decomposition of the signal. Universal thresholding is used with a threshold
value of  n 2 log n log n  . Then, the analytic signal of all coefficients (including the
approximation) is calculated and they are hard thresholded.
Figure 16 shows the denoised coefficients. Compared to Figure 14, the reduction in wedge noise
and random noise is evident. Finally, Figure 17 shows the denoised B-scan image. It can be
observed that the noise has been greatly reduced. Yet, some artifacts can be observed in the left
portion of Figure 17. These artifacts are caused by echoes reflected by irregularities on the near
surface. Since these echoes arrive earlier than any other echoes reflected by defects inside the
material, they do not distort the echoes reflected by defects. To assess the performance of the
proposed method quantitatively and compare it to other methods, in the following section,
simulated B-scan images are generated with different levels of noise and are denoised using the
proposed denoising method.
Figure 16
Figure 17
19
6.
EVALUATING THE PERFORMANCE OF THE PROPOSED METHOD
6.1
Generating a simulated B-scan
We set up a simulated B-scan as b=c+w+r, consisting of three terms: 1) c: crack echo 2) w:
wedge noise 3) r: random noise. The parameters of each term are given as follows.

Crack echo
The crack echo is generated using Equation (7). The simulated signal is intended to be similar to
the experimental signal, thus the parameters of Equation (7) are estimated from the experimental
signal. The parameters of function X(x) in Equation (7) are given in Figure 7. The arrival time of
the echo is given in Figure 6. The center frequency of the transducer is 2.25MHz. The parameters
of function T(t) is obtained by fitting a Gaussian signal to an envelope of an experimental A-scan
and finally the amplitude is set to 1 for convenience. By substituting these parameters, Equation
(7) yields

 

g t , x   exp  0.0256 x  14.39 exp  5.67t    cos4.5 t   
2
where   0.4378x  7.486

2
.
(26)
Wedge noise
Similarly, the Gaussian echo model is used for generating wedge noise. The parameters of this
Gaussian echo are obtained by curve fitting of an experimental signal. Unlike the crack echo, the
amplitude and the time of flight of the wedge echo do not change by time. Nonetheless, we
introduce small random changes to amplitude and time of flight in order to model variations in
wedge echo caused by contact conditions which can be observed in the experimental data.
20


g t , x    exp  3.3t    cos4.5 t   
2
(27)
where:   Aw 1  0.1 * rnd  ,   14.4  0.02 * rnd , rnd is a uniformly distributed random
number between -1 and 1, and Aw is chosen to achieve desired noise level.

Random noise
To incorporate the effect of random noise, a Gaussian distributed white noise with different
powers is added to the signal.
6.2

Performance evaluation
Simulated signal
The signal-to-noise ratio of the denoised B-scan image is calculated to indicate the performance
of each method. SNR is defined as follows:
 Psignal 

SNR  10 log

 Pnoise 
(28)
in which P is the power of a 2D signal defined as P  x    xij2 . Here, the signal is the crack
i
j
echo and “noise” is the sum of the random and wedge noises.
Figure 18 demonstrates the amount of improvement in SNR achieved by the proposed method
compared to two other methods for different noise levels. In each plot the wedge noise amplitude
is fixed and the random noise level is varied. The wedge noise amplitude, Aw, is set to 0.5 and 1
which are equivalent to signal-to-wedge noise ratio of -0.5 dB and -6.5 dB, respectively. Signalto-wedge noise ratio is defined as 10 logPsignal Pwedge noise  . The first method, ‘swt’, is similar to the
21
proposed method but its thresholding is based on the magnitude not the amplitude of the signal.
The second method, ‘dwt’, is similar to ‘swt’ but it employs DWT instead of SWT.
Figure 18
It can be seen that when the random noise level is low, all methods perform similarly. This shows
their ability to remove wedge noise is almost the same. Indeed, if there is no random noise
present in the signal, the proposed method and swt are exactly the same with regard to wedge
noise removal. For a given wedge noise amplitude, as the random noise level increases the
performance of the proposed method remains almost constant, increasing the SNR by 23 dB. This
came from the observation that the improvement in SNR from our method (solid line) is always
about 23 dB for SNR values up to -16 dB. In contrast, the performance of the other two methods
drops; especially that of dwt. This shows that swt outperforms dwt in removing random noise,
and analytic wavelet thresholding further increases denoising performance.
It can be seen that dwt’s improvement of SNR increases as the noise level increases. The
resulting B-scan images show dwt removes much of the signal when the random noise level is
high. For instance, Figure 19 (a) shows a simulated B-scan image with an SNR of -22 dB and
Aw=1. Figures 19 (b), (c), and (d) show the B-scan denoised by the proposed method, swt, and
dwt, respectively. In all cases, wedge noise is perfectly removed and little random noise is left,
but the proposed method retains the crack echo better than the two other methods do.
Figure 19
Denoised signals finally will be used for detecting, locating and sizing flaws. One important
feature which is often extracted from ultrasonic data is time of flight. It can be used to locate and
size flaws such as cracks. To further compare the proposed method with the swt and dwt
22
methods, we estimate the time of flight of the crack echo from A-scans. First, we exclude any Ascan with an amplitude less than 10% of the maximum amplitude. Time of flight is then
estimated by finding the time at which maximum amplitude occurs in each A-scan. Next, the
estimated time of arrival error is calculated and averaged for different levels of noise (Figure 20).
Clearly, it can be seen that the proposed method outperforms other methods, especially at low
SNRs.
Figure 20

Experimental data
In order to evaluate the performance of the proposed method using experimental data, time of
flight is estimated and compared with a theoretical value, for different slot lengths. The
theoretical time of flight is   ax  b where a is given by Equation (4), and constant b is
calibrated using experimental data. From Figure 6, b’s value is 7.486 s.
Figure 21 shows the error for estimated time of arrival using experimental signals from
specimens with different crack lengths. As it can be seen in Figure 21, for cracks smaller than
1mm, arrival times can be best estimated from images denoised by the proposed method. This
can be attributed to better denoising performance of the proposed method. For cracks larger than
1mm, the arrival times obtained from images denoised by the proposed method and ‘swt’ have
better accuracy than those obtained from images denoised by ‘dwt’. This is because SWT better
localize flaws than does DWT.
Figure 21
23
7.
CONCLUSIONS
In this paper, a method has been proposed for noise removal from ultrasonic pulse-echo B-scan
images. The proposed method decomposes the B-scan image using 2D SWT and removes wedge
noise present in the B-scan from the approximation and vertical detail coefficients. The only
assumption on which our method of noise removal is based is that changes in wedge noise are
negligible as a transducer moves along a specimen. Thus, the proposed method can be applied
without limitation to B-scan images unless conditions related to the transducer-specimen
interface change dramatically causing large variations in ultrasonic reflection at the interface. In
removing random noise, we have used analytic wavelet thresholding to better perform the
denoising. Generalizing the analytic signal concept to two dimensions has been studied in
relation to ultrasonic B-scans. The performance of the proposed method has been evaluated by
applying it to simulated and experimental signals; this has shown that analytic wavelet
thresholding can enhance noise removal. It has also been observed that SWT performs better than
wavelet transform. Using the proposed method we have been able to detect echoes from a crack
as small as 0.1 mm. In the future, we will investigate on the application of the proposed denoising
method to other kinds of flaws such as voids and surface corrosion.
8.
ACKNOWLEDGEMENT
This research was supported by the Natural Sciences and Engineering Research Council of
Canada (NSERC). Reviewers and the Editor comments would be very much appreciated.
24
REFERENCES
[1] Drury JC, Ultrasonics Part 4, Transducers for generating and detecting sound waves, Insight - NonDestructive Testing and Condition Monitoring 2005; 47(2): 98-100.
[2] Shull PJ. Nondestructive evaluation: theory, techniques, and applications. M. Dekker Inc.; 2002.
[3] Mosey S, Charlton PC, Wells I. Iterative method for background noise removal in ultrasonic B-scan
images, Insight - Non-Destructive Testing and Condition Monitoring 2006, 48(11): 677-81.
[4] Cygan H, Aknin P, Simard P, Girardi L. B-Scan ultrasonic image analysis for internal rail defect
detection, World Congress on Railway Research (WCRR’03), Edinburgh, 2003.
[5] Olhede SC, Walden AT. Analytic wavelet thresholding, Biometrika 2004, 91: 955-73.
[6] Chen Y, Ma H. Signal De-noising in Ultrasonic Testing Based on Stationary Wavelet Transform, WRI
Global Congress on Intelligent Systems 2009, 2: 474-478.
[7] Golan S, Adler L, Cook KV, Nanstad RK, Bolland TK. Ultrasonic Diffraction Technique for
Characterization of Fatigue Cracks, J NONDESTRUCT EVAL 1980, 1(1): 11-9.
[8] Chen CH. Ultrasonic and Advanced Methods for Nondestructive Testing and Material
Characterization, World Scientific Publishing; 2007.
[9] Zhang G, Hou C, Wang Y, Zhang S. Optimal frequency-to-bandwidth ratio of wavelet in ultrasonic
non-destructive evaluation, Ultrasonics 2001, 39: 13–7.
25
[10] Liao TW, Ni J. An automated radiographic NDT system for weld inspection: Part I- Weld extraction,
NDT & E Int 1996, 29(3): 157-62.
[11] Demirli R, Saniie J. Model-Based Estimation of Ultrasonic Echoes Part I: Analysis and Algorithms,
IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 2001, 48(3): 787-802.
[12] Mallat SG. A wavelet tour of signal processing, 2nd ed. Academic press;1999.
[13] Wang J, Huang HK. Handbook of Medical Image Processing and Analysis: Three-Dimensional
Image Compression with Wavelet Transforms, 2nd ed. Academic Press; 2009.
[14] Ruiz-Reyes N, Vera-Candeas P, Curpian-Alonso J, Mata-Campos R, Cuevas-Martinez JC. New
matching pursuit-based algorithm for SNR improvement in ultrasonic NDT, NDT & E Int 2005, 38(6):
453-8.
[15] Mallat SG. A Theory for Multiresolution Signal Decomposition: The Wavelet Representation, IEEE
T PATTERN ANAL 1989, 11(7): 674-93.
[16] Gyaourova A, Kamath R, Fodor IK. Undecimated Wavelet Transforms for Image Denoising,
Lawrence Livermore National Laboratory technical report UCRL-ID-150931, 2002.
[17] Lang M, Guo H, Odegard JE, Burrus CS, Wells RO. Noise reduction using an undecimated discrete
wavelet transform, IEEE Signal Proc Let 1996, 3(1): 10-2 .
[18] Donoho DL, Johnstone IM. Ideal spatial adaptation by wavelet shrinkage, Biometrika 1994, 81: 425–
55.
26
[19] Coifman RR, Donoho DL. Translation-invariant denoising, In Antoniadis A, Oppenheim G, editors.
Wavelets and Statistics, Lecture Notes in Statistics, New York: Springer-Verlag; 1995, p. 125–50.
[20] Sardy S. Minimax Threshold for Denoising Complex Signals with Waveshrink, IEEE Trans Signal
Process 2000, 48 (4): 1023-1028.
[21] Granlund GH, Knutsson H. Signal processing for computer vision, Kluwer Academic Publishers;
1995.
[22] Bedrosian E. A product theorem for Hilbert transforms, Proceedings of the IEEE 1963, 51(5): 868-9.
[23] Bülow T, Sommer G, A Novel Approach to the 2D Analytic Signal, Lecture Notes In Computer
Science, Proc of the 8th Int Conf on Comp Anal of Images and Pat 1999, 1689: 25-32.
[24] Coifman RR, Wickerhauser MV, Entropy-based Algorithms for best basis selection, IEEE Trans. on
Inf. Theory 1992, 38(2): 713-718.
[25] Shannon CE, The Mathematical Theory of Communication (Reprinted), M D Computing 1997,
14(4): 306–317.
APPENDIX A
Theorem: Let f(x) and g(x) be generally complex functions of 2D real variable x. If
27
a) Fourier transform, F(u), of f(x) vanishes for u1  a and the Fourier transform, G(u), of g(x)
vanishes for u1  a , where a is an arbitrary positive constant,
or
b) f(x) and g(x) are analytic signals with respect to a reference vector u1, i.e., both F(u) and G(u)
vanish for u1  0 .
then the directional Hilbert transform (HT) of the product of f(x) and g(x) is given by,
H 1  f x g x   f x  H 1 g x 
(A-1)
in which the directional Hilbert transform of a function f(x) with respect to a reference vector u1
is defined as
FT H 1  f x   i sgn u1 F u 
(A-2)
or
H 1  f x  
1
i sgn u F u e
2  
2
1
U
iu . x
dU
(A-3)
in which FT denotes the Fourier transform, dU  du1du 2 , and the integration is performed over
the interval of   ,   for two variables of u1 and u2.
Proof: The Fourier transform of the product of two functions is equal to the convolution of their
Fourier transform, thus
f x g x  
1
F u G v e
2   
4
V U
Now from Equations (A-2) and (A-4) we have
28
i u  v . x
dUdV .
(A-4)
H 1  f x g x  
1
2 4 V U
i sgn u1  v1 F u G v e i u  v .x dUdV .
(A-5)
But, either
a) Functions f and g are band-limited signals, hence the product F(u)G(v) is non-vanishing when
u1  a and v1  a .
or
b) Functions f and g are analytic, thus F(u)G(v) is non-vanishing when u1  0 and v1  0 .
In either case, for the non-vanishing region sgn u1  v1   sgn v1  , hence Equation (A-5) yields
H 1  f x g x  
1
2 4 V U
i sgn v1 F u G v e i u  v .x dUdV .
(A-6)
Integrating over U gives,
H 1  f x g x  
1
2 
2
f x  i sgn v1 G v e iv .x dV .
(A-7)
V
Finally from the definition of the directional HT, Equation (A-7) becomes
H1  f x g x   f x  H1 g x  .
(A-8)
Proof ends.
In this proof, for simplicity and without loss of generality we let the frequency axis, u1, be the
reference vector. This theorem implies that the directional HT of the product of a low frequency
envelope and a high frequency carrier is equal to the envelope multiplied by the directional HT of
the carrier. Note that high frequency and low frequency are defined with respect to the reference
vector. Likewise the analytic signal can be defined as amplitude of the analytic signal of this
product which is defined as Au   1  sgn u1 F u  or ax   f x   iH 1  f x  . Now, let
29
f x   ex  cos ω.x  where e(x) is the low frequency envelope which takes only positive values.
According to the above theorem, the directional HT and the analytical signal of f(x) are:
H 1  f x   H 1 ex  cos ω.x   ex  H 1 cos ω.x    ex sin ω.x 
ax   ex  cosω.x   i ex sin ω.x   ex e iω.x .
Clearly, the amplitude of the analytic signal obtained from the directional HT is equal to the
envelope signal.
Figure 1- The experimental setup for ultrasonic testing of a specimen with an EDM slot
Figure 2- Ultrasound travel path in the wedge and the specimen
30
(A-9)
(A-10)
Transducer Position (mm)
Wedge
echoes
25
Corner trap
echo
20
Tip
diffracted
echo
15
10
5
0
5
10
Time ( s)
15
20
Transducer Position (mm)
Figure 3- B-Scan image of a specimen with a 3mm slot
25
20
15
10
5
0
5
10
Time ( s)
15
Figure 4- B-scan image of a specimen with a 0.1mm slot
31
20
(a)
l/2
(b)
lw/2
A
l/2
w
Wedge
G
x
B
Specimen
F
Central ray
w
C
ls/2
D
x
H
Figure 5- (a) Travel paths of two ultrasonic rays through the wedge and specimen, (b) Travel path of central
ray at two different transducer positions
14
20
a
b
y = 0.4366*x + 6.245
 s)
y = 0.4378*x + 7.486
2
R =0.9999
16
Arrival time (
Arrival time (
 s)
18
14
12
12
2
R =0.9980
10
8
10
8
6
5
10
15
20
25
4
6
8
10
12
14
distance (mm)
distance (mm)
Figure 6- Time of flight for (a) the corner trap echo and (b) the tip diffracted echo
Figure 7- Amplitude of the corner trap echo
32
16
2
1.5
a
b
spatial frequency (mm-1)
spatial frequency (mm-1)
1.5
1
0.5
0
-0.5
-1
-1.5
-5
0
frequency (MHz)
5
1
0.5
0
-0.5
-1
-1.5
-3
-2
-1
0
1
frequency (MHz)
2
3
Figure 8- a) 2D Fourier transform of the signal, shown in Figure 3, after removing the wedge noise, and b) its
envelope spectrum
(a)
h
2
fm+1
g
2
f’m+1
fm
 2 Down-sampling by factor 2
hm
fm+1
gm
f’m+1
fm
(b)
H
m
hmV
f m 1
g mV
f mH1
hmV
f mV1
g mV
f mD1
h
(c)
fm
g mH
Figure 9- Decomposition of a signal, fm, at level m into approximation and detail signals at level m+1, (a)
wavelet decomposition, 1D signal, (b) SWT decomposition, 1D signal, (c) SWT decomposition, 2D signal.
33
(a)
(b)
Figure 10- Illustration of high frequency (HF) carrier, low frequency (LF) envelope, reference vector and
positive and negative frequency regions in two dimensions; Reference vector, n̂ , is selected properly in (a) so
that the HF and LF components are projected on HF and LF regions of axis n̂ respectively, while in (b) HF is
projected on the LF region.
Figure 11- 2D frequency spectrum of the product of envelope and carrier; as illustrated selecting any
reference vector among n̂ , û 1 and û 2 and suppressing the negative frequency components leads to the same
result.
34
Decompose the B-Scan image using stationary wavelet
Remove wedge noise from the approximation and vertical coefficients
For every coefficient find the reference vector
Calculate the envelope for each coefficient
Estimate the threshold
Apply hard threshoding to all coefficients
Reconstruct the B-scan image from the denoised coefficeints
Figure 12- Flowchart of the proposed denoising method
Figure 13- The Shannon entropy of wavelet coefficients of an A-scan signal for different mother wavelets
35
A
H
V
D
Figure 14- Stationary wavelet decomposition of signal shown in Figure 4 , A: Approximation, H: Horizontal,
V: Vertical, D: Diagonal. The x-axis of all wavelet coefficients is time ranging from 0 to 20.48 s and y-axis is
the transducer distance to the slot ranging from 0 to 30 mm.
Position (mm)
25
20
15
10
5
0
5
10
Time (  s)
15
20
Figure 15- Approximation coefficient after removing the wedge noise
36
Figure 16- Denoised stationary wavelet coefficients
Figure 17- Denoised B-scan image
37
Figure 18- Improvement in SNR obtained by the proposed method, ‘swt’, and ‘dwt’.
a
b
c
d
Figure 19- a) Simulated B-scan image with SNR of -18dB denoised by b) proposed method, c) swt, and d) dwt
38
Figure 20- Error of the estimated arrival time using simulated signal denoised by proposed method, ‘swt’, and
‘dwt’.
Figure 21- Error of the estimated arrival time using experimental signal denoised by proposed method, ‘swt’,
and ‘dwt’.
39