A Comparative Study of Shearlet, Wavelet, Laplacian Pyramid, Curvelet, and Contourlet Transform to Defect Detection

Journal of Soft Computing in Civil Engineering 7-2 (2023) 1-42
How to cite this article: Vafaie S, Salajegheh E. A comparative study of shearlet, wavelet, laplacian pyramid, curvelet, and
contourlet transform to defect detection. J Soft Comput Civ Eng 2023;7(2): 1–42. https://doi.org/10.22115/scce.2023.356475.1505
2588-2872/ © 2023 The Authors. Published by Pouyan Press.
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Journal of Soft Computing in Civil Engineering
Journal homepage: www.jsoftcivil.com
A Comparative Study of Shearlet, Wavelet, Laplacian Pyramid,
Curvelet, and Contourlet Transform to Defect Detection
Sepideh Vafaie1*
, Eysa Salajegheh2
1. Ph.D. Student, Faculty of Earth and Environmental Studies, Montclair State University, United States
2. Professor, Faculty of civil Engineering, Shahid Bahonar University of Kerman, Kerman, Iran
Corresponding author: vafaies1@montclair.edu
https://doi.org/10.22115/SCCE.2023.356475.1505
ARTICLE INFO ABSTRACT
Article history:
Received: 16 August 2022
Revised: 04 January 2023
Accepted: 31 January 2023
This study presents a new approach based on shearlet
transform for the first time to detect damages, and compare it
with the wavelet, Laplacian pyramid, curvelet, and contourlet
transforms to specify different types of defects in plate
structures. Wavelet and Laplacian pyramid transforms have
inferior performance to detect flaws with different multi-
directions, such as curves, because of their basic element
form, expressing the need for more efficient transforms.
Therefore, some transforms, including curvelet and
contourlet, have been evaluated so far for improving the
performance of traditional transforms. Although these
transforms have overcome the deficiencies of previous
methods, they have a weakness in detecting several
imperfections with various shapes in plate structures —one
of the essential requirements that each transform should
possess. In this study, we have used the shearlet transform
that is used for the first time to detect identification and
overcome all previous transform dysfunctionalities. In this
regard, these transforms were applied to a four-fixed
supported square plate with various defects. The obtained
results revealed that the shearlet transform has the premier
capability to demonstrate all kinds of damages compared to
the other transforms, namely wavelet, Laplacian pyramid,
curvelet, and contourlet. Also, the shearlet transform can be
considered as an excellent and operational approach to
demonstrate different forms of defects. Furthermore, the
performance and correctness of the transforms have been
verified via the experiment.
Keywords:
Wavelet transform;
Laplacian pyramid transform;
Curvelet transform;
Contourlet transform;
Shearlet transform;
Damage detection.

2 S. Vafaie, E. Salajegheh/ Journal of Soft Computing in Civil Engineering 7-2 (2023) 1-42
1. Introduction
Non-destructive methods in civil engineering (e.g., wavelet transform) have received significant
attention in recent decades [1]. In the past few years, the application of wavelet transforms for
health monitoring of structures (SHM) and detecting damages have been investigated by many
researchers. Ovanesova and Suarez [1] used wavelet transform to find defects in frame
structures, showing that this method merely needs response data of the defective structure. Kim
and Melhem [2] offered the wavelet analysis method for defect identification. They first
presented the theory of wavelet transform and then used it in the detection of cracks in a beam
and mechanical gear. Yun et al. [3] proposed a technique based on analyzing the wavelet signal
of the smart wireless sensor for the identification of the decentralized defect. They verified this
proposed method with experimental tests. Bagheri et al. [4] expressed the ability of a two-
dimensional discrete wavelet transform to identify damage of plates using modal data. Also, they
used experimental data to validate the proposed technique. Cao et al. [5] suggested the utilization
of the Teager energy operator along with wavelet transform for beam damage recognition in
noisy conditions. They applied this method on several analytical cases to show the competence
of their proposed technique. Yang and Nagarajaiah [6] suggested the blind damage detection by
analyzing independent component via wavelet transform. Moreover, examples of the seismic-
excited structures are stated to indicate the ability of the developed method. Ulriksen et al. [7]
developed a new technique based on wavelet transform and modal analysis to identify defects of
wind turbine blades. Shahsavari et al. [8] presented the mode shape analysis with wavelet
transform to detect defects of beams. Wang et al. [9] introduced a new form of wavelet transform
based on residual force vector for fault recognition of underground tunnel structures. They
introduced a novel damage index, which can be used as an efficient defect detection indicator.
Zhu et al. [10] proposed an approach for crack recognition using continuous wavelet transform
through introducing a new index for defect discernment. Jahangir et al. [11] presented an
approach based on wavelet analysis to identify damage to RC beams. Fakharian and Naderpour
[12] utilized two various methods including wavelet packet transform and peak picking to assess
the quantification of defect severity. Naderpour et al. [13] presented shear strength prediction
using three different approaches including ANN, GMDH-NN, and GEP. They showed all of the
methods are capable of predicting properly. Ghanizadeh et al. [14] used evolutionary polynomial
regression to develop a prediction model for collapse settlement and stress release coefficients.
Naderpour et al. [15] utilized a new approach based on the data handling group method to the
estimation of the moment capacity of ferrocement members. Bagheri and Kourehli [16]
presented a wavelet analysis based-method to defect identification. Kourehli [17] used wavelet
transform to structural health monitoring of steel frames. Ghannadi and Kourehli [18] used a
slim mold algorithm to damage detection. Ghannadi and Kourehli [19] suggested a new method
based on a moth-flame algorithm to defect identification. Also, there are some useful methods
including wavelet transform and optimization that have been used recently for damage
demonstration [20–23]. Another transform that has been evaluated in this investigation is the
Laplacian pyramid transform. Burt and Adelson [24] suggested the new method based on the

S. Vafaie, E. Salajegheh/ Journal of Soft Computing in Civil Engineering 7-2 (2023) 1-42 3
Laplacian pyramid, well suited for image analysis and compression. Also, Do and Vetterli [25]
used the Laplacian pyramid to create a new transform, named pyramid directional filter bank.
They demonstrated that once the Laplacian pyramid is applied on a signal, two parts are
produced, including approximation and detail. Although wavelets have been widely utilized in
damage detection, they have offered a poor performance on representing objects with highly
anisotropic elements. Accordingly, other transforms have been introduced to overcome the
weakness of traditional approaches. In what follows, some of these transforms are expressed.
Candes et al. [26] presented the fast discrete curvelet transform, i.e., the second generation of
curvelet transform. Bagheri et al. [27] used the curvelet transform for recognition of vibration-
based defects of plate structures, demonstrating its excellent ability to show line features.
Nicknam et al. [28] propounded curvelet transform via wrapping procedure for fault discernment
in two-dimensional structures. They used both numerical and experimental data to display the
superior performance of this technique. Another transform that has been developed to improve
the traditional multiscale representation is the contourlet transform. Do and Vetterli [29]
proposed a new two-dimensional image representation named the contourlet transform. Po and
Do [30] exhibited that contourlets are composed of basics oriented at various directions, which
enables this transform to show smooth contours of natural images effectively. Vafaie and
Salajegheh [31] compared wavelet and contourlet transforms for the identification of vibration-
based damage of plate structures, showing the superiority of contourlets over wavelets in the
detection of curved cracks in the plate structures. Jahangir et al. [32] proposed using contourlet
transform to damage localization and assessment of severity. Although these transforms have
been successful in overcoming the weakness of the previous techniques, they could not detect
damages with various shapes excellently. Thus, the shearlet transform has been presented as an
efficient technique in this study. Lim [33] suggested the discrete shearlet transform, utilized to
provide efficient multiscale directional representation. Xu et al. [34] applied the shearlet
transform for the surface defects classification of metals. They expressed that since various
damages have information in several directions and on different scales, another transform
superior to wavelet should be used, i.e., the shearlet transform.
2. Research significance
This investigation mainly aims to present the effectiveness of a shearlet transform-based
approach which is used for the first time to defect detection, and compared to the wavelet,
Laplacian pyramid, curvelet, and contourlet transforms, to detect imperfections with various
shapes in plate structures. Thus, eight examples with multiple damages have been discussed to
check the transform performance. The rest of this paper is organized as follows. In section 2, the
overview of the algorithm of wavelet, Laplacian pyramid, curvelet, contourlet, and shearlet
transforms is presented. Then, the process of defect identification with these five transforms is
expressed, and a damage index is also introduced to show flaws. In section 3, eight numerical
examples are addressed. Performance evaluation of the transforms through an experimental
model is carried out in section 4. Finally, section 5 represents the concluding remarks.

3. Methods
Defect detection with wavelet, Laplacian pyramid, curvelet, contourlet, and shearlet transforms
includes the following steps:
 Model a plate
 Obtain the structural response of the plates
 De-noise the structural response of the plates
 Apply the transforms
 Determine the damage index (DI)
 Plot the damage index
These steps are presented comprehensively as below:
3.1. Model a plate
In this part, the plate with and without damage is simulated. A defect in the plate can be
modeled as the reduction of cross-sectional area, material properties, stiffness, and so forth. In
this research, the flaw is simulated as the reduction of Young’s modulus in the damaged plate.
As can be seen in Eq. (1), Young’s modulus for damaged plate is denoted by 𝐸𝑑𝑎𝑚𝑎𝑔𝑒𝑑, where 𝑑
stands for the intensity of defect, and 𝐸𝑢𝑛𝑑𝑎𝑚𝑎𝑔𝑒𝑑 is Young’s modulus of the undamaged plate. It
is worth mentioning that d could be located between 0 and 1 (in our examples 0.05, 0.1, and 0.2
was considered). In addition, d is equal to 0 and 1 meaning undamaged and fully damaged states,
respectively.
𝐸𝑑𝑎𝑚𝑎𝑔𝑒𝑑 = 𝐸𝑢𝑛𝑑𝑎𝑚𝑎𝑔𝑒𝑑(1 − 𝑑) (1)
3.2. Obtain the structural response of the plates
After modeling a plate, the structural response of the plate is required for fault identification. In
this investigation, the plate mode shape was used to obtain the displacement of nodes in the
fundamental mode shape of the plate as the structural response needed for the procedure of
damage identification. It is worth noting that the finite element method (FEM) was implemented
to specify the plate mode shape. The equation of free vibration is defined as:
𝑀𝑢̈ (𝑡) + 𝐾𝑢(𝑡) = 𝑓(𝑡) (2)
Where 𝑀, 𝐾, and 𝑓 are the mass, stiffness matrices, and force vector, respectively. Also, 𝑢̈ and 𝑢
are acceleration and displacement, respectively. According to the harmonic motion, the natural
frequencies and the modes of vibration are gained as
𝐾 − 𝜔𝑖
2
𝑀)𝜑𝑖 = 0, 𝑖 = 1,2, … , 𝑛𝑚 (3)
Where 𝜔𝑖 is the natural frequency, 𝜑𝑖 is the 𝑖th vibration mode shape vector, and 𝑛𝑚 is the
structural modes number [35].

3.3. De-noise the structural response of the plates
Since measured data of the structures contain noise in a real experiment, de-noising plays an
essential role in the process of defect recognition. However, the noise amount is ambiguous, and,
in turn, it is likely to delete useful information containing noise in the process of de-noising the
measured information with low noise. Therefore, de-noising is one of the significant sections in
the defect detection process. In this study, de-noising was performed as follows:
1. The transform was applied to the signal (the displacement in the plate fundamental mode
shape) to obtain the transform coefficients.
2. Set the coefficient threshold.
3. De-noised coefficients were utilized to reconstruct the de-noised signal.
It should be noted that the white Gaussian noise has been added to the signal randomly.
𝑍𝑖~𝒩(0, 𝑁)
𝑌𝑖 = 𝑋𝑖 + 𝑍𝑖
(4)
Where 𝑍𝑖 is drawn from a normal distribution with a mean value of zero and the variance N, and
it adds the noise randomly to our signal which is 𝑋𝑖. In addition, we have compared two various
methods based on wavelet sym4 and db1 to denoise the noisy signal, and we have used sym4
according to the superior performance of sym4 [36].
3.4. Apply the transforms
In this section:
a) An overview of the wavelet, Laplacian pyramid, curvelet, contourlet, and shearlet
transform is given.
b) Then, the structural response of the plates —considered as the displacement of the
nodes in the first mode shape of the plates in this study— is utilized as a signal for
these transforms to acquire the transform detail coefficients.
c) Finally, these detail coefficients are used in the next stage in the damage index for
defect detection.
In all formulas, the following notations are used:
 𝑗 = decomposition level of the transform
 𝑙 =number of directions for displaying detail coefficients, 𝑙 = 0,1, … , 𝐿
 low-pass of signal=approximation part; high-pass of signal= detail part
3.4.1. Wavelet transform
Two parts are produced by applying two-dimensional wavelet to a signal for 𝑗 = 𝐽;
approximation (𝑐𝐴𝐽) and three detail coefficients
(𝑐𝐷𝐽
ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙,𝑙=1
, 𝑐𝐷𝐽
𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙,𝑙=2
, 𝑐𝐷𝐽
𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙,𝑙=3
, 𝑙 = 1,2,3 (𝐿 = 3)) (Fig. 1).

Fig. 1. The scheme of wavelet transform for damage detection with an example for j=1.
Also, as choosing the proper wavelet is very important due to its significant impact on damage
identification, in this study, three wavelets – Haar, Symlet, and Discrete Meyer – are used to
compare their performance and choose the one having the superior efficiency in detecting a flaw.
Haar wavelet: The Haar wavelet is a rescaled function sequence (square-shaped) that form a
wavelet family. The Haar wavelet mother and its scaling function—𝜓(𝑥), 𝜙(𝑥)—are determined
as follows [35]:
𝜓(𝑥) = {
1 0 ≤ 𝑥 < 0.5
−1 0.5 ≤ 𝑥 < 1
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
(5)
𝜙(𝑥) = {
1 0 ≤ 𝑥 ≤ 1
(6)
Symlet wavelet: The family of Symlet wavelets is a modified version of Daubechies wavelets
with increased symmetry [37]. The mother wavelet and scaling function are represented in Fig.
2.
(a) Mother function (b) Scaling function
Fig. 2. Symlet wavelet function [38].
Discrete Meyer wavelet: The Meyer wavelet is infinitely differentiable with infinite support;
defined in the frequency domain in terms of function 𝜈 as [39]:
Approximation
coefficient
Detail
coefficients
These are used
in Damage
index
Apply wavelet
𝒄𝑫𝟏
𝒉,𝒍=𝟏
𝒄𝑫𝟏
𝒗,𝒍=𝟐
𝒄𝑫𝟏
𝒅,𝒍=𝟑

𝜓(ω) ≔
{
1
√2𝜋
sin (
𝜋
2
𝜈(
3|𝜔|
2𝜋
− 1)) 𝑒𝑗𝜔/2
𝑖𝑓
2𝜋
3
< |𝜔| <
4𝜋
3
1
√2𝜋
cos (
𝜋
2
𝜈(
3|𝜔|
4𝜋
− 1)) 𝑒𝑗𝜔/2
𝑖𝑓
4𝜋
3
< |𝜔| <
8𝜋
3
(7)
ν(x) ≔ {
0 𝑖𝑓 𝑥 < 0
𝑥 𝑖𝑓 0 < 𝑥 < 1
1 𝑖𝑓𝑥 > 1
(8)
In this investigation, all three above wavelets are evaluated in the second numerical example.
3.4.2. Laplacian pyramid and Contourlet transform
Contourlet transform is a multiresolution transform with basic functions 𝜓𝑎,𝑗,𝑙
𝑛
. Thus, the
contourlet transform of a signal is determined as:
𝐶𝑐𝑜𝑛𝑡(𝑎, 𝑗, 𝑙, 𝑛) = 〈𝑠𝑖𝑔𝑛𝑎𝑙, 𝜓𝑎,𝑗,𝑙
𝑛
〉 (9)
Where 𝐶𝑐𝑜𝑛𝑡(𝑎, 𝑗, 𝑙, 𝑛) is the inner product of signal with the basic functions (𝜓𝑎,𝑗,𝑙
𝑛
); 𝑎, 𝑎 >
0 scale; 𝑛 = decomposition levels number of the directional filter bank. Also, 𝐶𝑐𝑜𝑛𝑡(𝑎, 𝑗, 𝑙, 𝑛)
determines coefficients of the contourlet transform, which includes high and low-frequency
parts, see [25] for more information. In this study, we considered an effective discrete contourlet
transform scheme based on a Laplacian pyramid combined with proper directional filter banks,
which was proposed in [29], (section a=Laplacian pyramid transform, section(a+b)= contourlet
transform).
Section a) First, Laplacian pyramid transform is applied to a signal. After the Laplacian
pyramid stage, the output is 𝐽 high-pass signal 𝑐𝐷𝑗
𝑙=1
, 𝑗 = 1,2, … , 𝐽; 𝐿 = 1) (in the fine
to coarse order) and a low-pass signal 𝑐𝐴𝐽.
Therefore, by applying Laplacian pyramid on a signal for 𝑗 = 𝐽, two parts are produced;
approximation (𝑐𝐴𝐽) and detail coefficient (𝑐𝐷𝐽
𝑙=1
, 𝐿 = 1). In this study, as indicated in Fig. 3,
Laplacian pyramid transform is applied on the structural response of the intact and imperfect
plate, which is the node displacement in the plate initial mode shape, to generate the detail
coefficient. In the next step, the detail coefficient (𝑐𝐷𝐽
𝑙=1
, 𝐿 = 1) is used as 𝑐𝐷𝑙
𝑑𝑎𝑚𝑎𝑔𝑒𝑑 (detail
coefficients of the damaged plate) and 𝑐𝐷𝑙
𝑢𝑛𝑑𝑎𝑚𝑎𝑔𝑒𝑑 (detail coefficients of the undamaged plate)
in section 3.5. Damage index, Eq. 15. Fig. 3 illustrates the Laplacian pyramid transform
decomposition for 𝑗=1.

Fig. 3. The scheme of Laplacian pyramid transform for damage detection with an example for j=1.
Section b) Second, a directional filter bank is applied on 𝑐𝐷𝑗
𝑙=1
, 𝐿 = 1 (high-pass signal).
Each high-pass signal 𝑐𝐷𝑗
𝑙=1
, 𝑗 = 1,2, … , 𝐽; 𝐿 = 1 is further decomposed by a directional
filter bank into 𝑙 high-pass directional signal (𝑐𝐷𝑗
𝑙
, 𝑙 = 1,2, … , 𝐿), (𝑐𝐴𝐽 approximation
coefficient, and 𝑐𝐷𝑗
𝑙
, detail coefficients) [29].
Therefore, by applying contourlet transform to a signal for 𝑗 = 𝐽, two parts are produced;
approximation (𝑐𝐴𝐽 ) and detail coefficient (𝑐𝐷𝐽
𝑙
), 𝑙 = 1,2, … , 𝐿 (contourlet can show details in
various directions). In this research, as demonstrated in Fig. 4, the contourlet transform is applied
to the structural response of flawless and imperfect plate, the node displacement in the
fundamental mode shape, to produce the detail coefficients. In the next step, the detail
coefficients (𝑐𝐷𝐽
𝑙
, 𝑙 = 1,2, … , 𝐿 ) are utilized as 𝑐𝐷𝑙
𝑑𝑎𝑚𝑎𝑔𝑒𝑑 (detail coefficients of the damaged
plate) and 𝑐𝐷𝑙
𝑢𝑛𝑑𝑎𝑚𝑎𝑔𝑒𝑑(detail coefficients of the undamaged plate) in section 3.5. Damage
index, Eq. 15. Fig. 4 indicates the decomposition of contourlet transform when 𝑗=1, 𝐿=4.
Fig. 4. The scheme of contourlet transform for damage detection with an example for j=1, L=4.
Also, selecting the appropriate Laplacian filter is very important in contourlet and Laplacian
pyramid transforms for damage identification. Accordingly, 9/7 wavelet filter bank and PKVA
filter (filters from the ladder structure) are utilized in this study in the second example.
Approximation
coefficient
Detail
coefficients
These are used
in Damage
index
Apply
contourlet
Approximation
coefficient
The detail
coefficient
This is used in
Damage index
Apply
Laplacian
pyramid
𝒄𝑫𝟏
𝒍=𝟏
𝒄𝑫𝟏
𝒍=𝟏 𝒄𝑫𝟏
𝒍=𝟐
𝒄𝑫𝟏
𝒍=𝟑
𝒄𝑫𝟏
𝒍=𝟒

3.4.3. Curvelet transform
Curvelets consists of the two forms of block ridgelet transform and curvelet transform on the
Fourier domain. In block ridgelet transform, signals are segmented into blocks, and ridgelet is
performed on the blocks. In the curvelet transform, frequency partitioning is performed on the
frequency domain. Therefore, the curvelet transform of a signal with basic functions 𝜓𝑎,𝑗,𝑙
𝑝
is
defined as:
𝐶𝑐𝑢𝑟𝑣(𝑎, 𝑗, 𝑙, 𝑝) = 〈𝑠𝑖𝑔𝑛𝑎𝑙, 𝜓𝑎,𝑗,𝑙
𝑝
〉 (10)
Where 𝐶𝑐𝑢𝑟𝑣(𝑎, 𝑗, 𝑙, 𝑝) is the inner product of 𝑠𝑖𝑔𝑛𝑎𝑙 with the basic functions (𝜓𝑎,𝑗,𝑙
𝑝
); 𝑎, 𝑎 >
0 scale; 𝑝 = (𝑝1, 𝑝2)𝜖𝑍2
the sequence of translation parameters. Also,
𝐶𝑐𝑢𝑟𝑣(𝑎, 𝑗, 𝑙, 𝑝) determines the coefficients of the curvelet transform, including high and low-
frequency parts, see ([[40]; [26]) for more information. In this research, digital curvelet
transform is considered as the following:
𝑆𝑖𝑔𝑛𝑎𝑙 = 𝑐𝐴𝐽 + ∑ 𝑐𝐷𝑗
𝑙
𝐽
𝑗=1 , 𝑙 = 1,2, … , 𝐿 (11)
Where 𝑐𝐴𝐽 is a low pass version of the signal (approximation part), and 𝑐𝐷𝑗
𝑙
, 𝑙 =
1,2, … , 𝐿 represents details of the signal, see [26] for more information. Therefore, by applying
curvelet transform on a signal for 𝑗 = 𝐽, two parts of approximation (𝑐𝐴𝐽 ) and detail coefficient
(𝑐𝐷𝐽
𝑙
), 𝑙 = 1,2, … , 𝐿 (curvelet can exhibit details in several directions) are produced. In this
investigation, as exhibited in Fig. 5, the curvelet transform is applied to the structural response of
intact and imperfect plate to generate the detail coefficients. In the next step, the detail
coefficients (𝑐𝐷𝐽
𝑙
, 𝑙 = 1,2, … , 𝐿 ) are used as 𝑐𝐷𝑙
𝑑𝑎𝑚𝑎𝑔𝑒𝑑 (detail coefficients of the damaged
plate) and 𝑐𝐷𝑙
𝑢𝑛𝑑𝑎𝑚𝑎𝑔𝑒𝑑(detail coefficients of the undamaged plate) in section 3.5. Damage
index, Eq. 15. Fig. 5 demonstrates the curvelet transform decomposition when 𝑗=1, 𝐿=4.
Fig. 5. The scheme of curvelet transform for damage detection with an example for j=1, L=4.
3.4.4. Shearlet transform
The shearlet transform is a multiresolution transform with basic functions 𝜓𝑎,𝑏,𝑠 defined as:
𝜓𝑎,𝑏,𝑠(𝑥) = 𝑎
−3
4 𝜓(𝐴𝑎
−1
𝑆𝑠
−1(𝑥 − 𝑏)) (12)
Approximation
coefficient
Detail
coefficients
These are
used in.
Damage
index
Apply
curvelet
𝒄𝑫𝟏
𝒍=𝟏 𝒄𝑫𝟏
𝒍=𝟐
𝒄𝑫𝟏
𝒍=𝟑 𝒄𝑫𝟏
𝒍=𝟒

𝐴𝑎 = ( 𝑎 0
0 √𝑎
), 𝑆𝑠 = ( 1 𝑠
0 1
) (13)
Where 𝑎, 𝑎 > 0 is scale; 𝑏𝜖𝑅2
is position; 𝑠𝜖𝑅 is the slope in the frequency domain; 𝐴𝑎 is
parabolic scaling matrix; and 𝑆𝑠 is the shear matrix. Therefore, the shearlet transform of a signal
is defined as:
𝐶𝑠ℎ𝑒𝑎𝑟(𝑎, 𝑏, 𝑠) = 〈𝑠𝑖𝑔𝑛𝑎𝑙, 𝜓𝑎,𝑏,𝑠 〉 (14)
Where 𝐶𝑠ℎ𝑒𝑎𝑟(𝑎, 𝑏, 𝑠) is the inner product of signal and the basic functions (𝜓𝑎,𝑏,𝑠). Also,
𝐶𝑠ℎ𝑒𝑎𝑟(𝑎, 𝑏, 𝑠) defines the coefficients of the shearlet transform, which includes high and low-
frequency parts ([41]; [42]). In this study, a discrete shearlet transform was considered, which
included the Laplacian pyramid and shearing filters [34]. First, the Laplacian pyramid transform
was applied to a signal where the outputs consisted of 𝐽 high-pass signal 𝑐𝐷𝑗
𝑙=1
, 𝑗 =
1,2, … , 𝐽; 𝐿 = 1 and low-pass signal 𝑐𝐴𝐽.Second, proper shearing filters were applied to
𝑐𝐷𝑗
𝑙=1
, 𝐿 = 1, (high-pass signal) to generate 𝑙 high-pass directional signal (𝑐𝐷𝑗
𝑙
, 𝑙 = 1,2, … , 𝐿 );
(𝑐𝐴𝐽 defines approximation coefficient, and 𝑐𝐷𝑗
𝑙
defines detail coefficients in 𝑙 directions).
Therefore, by applying shearlet transform to a signal for 𝑗 = 𝐽, two parts are produced;
approximation (𝑐𝐴𝐽 ) and detail coefficient (𝑐𝐷𝐽
𝑙
), 𝑙 = 1,2, … , 𝐿 (shearlets are capable of
demonstrating details in various directions). In this study, as demonstrated in Fig. 6, the shearlet
transform is applied to the structural response of damaged and undamaged plates, i.e., the node
displacement in the initial mode shape of the plates, to produce the detail coefficients. In the next
step, the detail coefficients (𝑐𝐷𝐽
𝑙
, 𝑙 = 1,2, … , 𝐿 ) were used as 𝑐𝐷𝑙
𝑑𝑎𝑚𝑎𝑔𝑒𝑑 (detail coefficients of
the damaged plate) and 𝑐𝐷𝑙
𝑢𝑛𝑑𝑎𝑚𝑎𝑔𝑒𝑑 (detail coefficients of the undamaged plate) in section 3.5.
Damage index, Eq. 15. Fig. 6 displays shearlet transform decomposition when 𝑗=1, 𝐿=6.
Fig. 6. The scheme of shearlet transform for damage detection with an example for j=1, L=6.
Detail
coefficients
These are used
as in Damage
index
Apply
shearlet
Approximation
coefficient
𝒄𝑫𝟏
𝒍=𝟏
𝒄𝑫𝟏
𝒍=𝟐
𝒄𝑫𝟏
𝒍=𝟑
𝒄𝑫𝟏
𝒍=𝟒 𝒄𝑫𝟏
𝒍=𝟓 𝒄𝑫𝟏
𝒍=𝟔

3.5. Determine the damage index (DI)
The specification of the appropriate connection, named as damage index, is a central part of the
flaw identification process [14]. Accordingly, after obtaining detail coefficients of the damage
and undamaged plates, a proper relationship was defined between both of them that led to the
efficient demonstration of the defect place. The damage index is defined as follows:
𝐷𝐼 = √
1
𝐿
∑ (𝑐𝐷𝑑𝑎𝑚𝑎𝑔𝑒𝑑
𝑙
−𝑐𝐷𝑢𝑛𝑑𝑎𝑚𝑎𝑔𝑒𝑑
𝑙
)2
𝐿
𝑙=1
1
𝐿
∑ (𝑐𝐷𝑢𝑛𝑑𝑎𝑚𝑎𝑔𝑒𝑑
𝑙
)2
𝐿
𝑙=1
(15)
𝑐𝐷𝑑𝑎𝑚𝑎𝑔𝑒𝑑
𝑙
, and 𝑐𝐷𝑢𝑛𝑑𝑎𝑚𝑎𝑔𝑒𝑑
𝑙
determine the detail coefficients of the transforms for the damaged
and undamaged plates, respectively.
3.6. Plot the damage index
Finally, the defect location was obtained for each transform by plotting the damage index. In
other words, the defect location was determined as the maximum value of the damage index in
all transforms. As indicated in Fig. 7, the fault detection procedure was identical for all
transforms.
Fig. 7. The general procedure of damage detection using the structural response for the horizontal linear
defect example.
4. Results
4.1. Numerical results
This research aims to present a new approach via shearlet transform compared with the other
transforms; the wavelet transforms, Laplacian pyramid transform, curvelet transform, and
contourlet transform; to find defect types in plate structures. To this end, the ability of all
Apply wavelet transform and plot
damage index
The damaged
plate
Damage index in 2D Damage index in 3D
Damage location
The undamaged
plate

transforms to detect cracks of plate structures was evaluated with eight numerical examples. In
all cases, a square plate with a thickness of 10 𝑐𝑚 was considered with four fixed boundary
conditions. Properties of plate material included Young’s modulus of 𝐸 = 20 𝐺𝑃𝑎 , the mass
density 𝜌 = 2500 𝑘𝑔/𝑚3
, and Poisson’s ratio of 𝜐 = 0.2. It is worth mentioning that for
simulating different geometry defects in the ABAQUS software, after modeling a plate, the
defect with different shapes was considered in the plate as a new section with various materials.
Then, all the plate could have meshed as one unit. The next step includes gathering results of
ABAQUS such as nodal coordinates (x and y), and displacement in the y direction for the
damaged and undamaged plates. In MATLAB software; these four vectors are considered input
variables. In this step, the scatter interpolate method has been used to make a mesh grid in the x
and y directions.
4.1.1. The plate with a point defect
The first example contains a plate with a width of 400 𝑐𝑚 and a length of 400 𝑐𝑚, including a
square point damage with 5 𝑐𝑚 length and width, located as shown in Fig. 8.
(a) The geometry of the plate with point defect
(b) Wavelet
200 cm
195
cm

(c) Laplacian pyramid
(d) Curvelet
(e) Contourlet
(f) Shearlet
Fig. 8. Point damage detection.

4.1.2. The plate with a horizontal linear defect
The second example includes a plate with a width of 400 𝑐𝑚 and a length of 400 𝑐𝑚, containing
a horizontal linear defect with 60 𝑐𝑚 length and 5 𝑐𝑚 width, which is located as illustrated in
Fig. 9.
(a) The geometry of the plate with horizontal linear defect
(b) Haar wavelet
(c) Symlet wavelet
200 cm
200
cm

(d) Discrete Meyer wavelet
(e) Laplacian pyramid (9/7 filter)
(f) Laplacian pyramid (PKVA filter)
(g) Curvelet

(h) Contourlet (9/7 filter)
(i) Contourlet (PKVA filter)
(j) Shearlet
Fig. 9. Horizontal linear damage detection.
4.1.3. The plate with a diagonal linear defect
The third example consists of a fixed support plate with a width of 400 𝑐𝑚 and a length
of 400 𝑐𝑚, including a diagonal linear damage with 60 𝑐𝑚 length, 5 𝑐𝑚 width, and the angle of
45° located as in Fig. 10.

(a) The geometry of the plate with a diagonal linear defect
(b) Discrete Meyer wavelet
(c) Laplacian pyramid (9/7 filter)
300
cm
300 cm

(d) Curvelet
(e) Contourlet (9/7 filter)
(f) Shearlet
Fig. 10. Diagonal linear damage detection.

4.1.4. The plate with an arc defect
The fourth example comprises a fixed support plate with a width of 400 𝑐𝑚 and a length
of 400 𝑐𝑚, containing a curved defect with 66 𝑐𝑚 length and 1𝑐𝑚 width, and coordinates of the
arc center are as shown in Fig. 11.
(a) The geometry of the plate with arc defect
313
cm
65 cm

(d) Curvelet
(f) Shearlet
Fig. 11. Arc damage detection.

150 cm
150
cm
33 cm
33
cm
22cm
28
cm
4.1.5. The plate with diverse linear defects
The fifth example includes a fixed support plate with a width of 300 𝑐𝑚, length of 300 𝑐𝑚, and
three linear damages, including horizontal, vertical, and diagonal. The size of the defects is as
follows: 60 𝑐𝑚 length and 5 𝑐𝑚 width; the diagonal linear defect with the angle of 30°. The
location of these defects is depicted in Fig. 12.
(a) The geometry of the plate with diverse linear defects
Damage A:
Damage B :
Damage C:
A
B
C

(d) Curvelet

22cm
150cm
150cm
22cm
43cm
22cm
(f) Shearlet
Fig. 12. Diverse linear damages detection.
4.1.6. The square plate with arc defects
The sixth example includes a fixed support plate with a width of 300 𝑐𝑚, length of 300 𝑐𝑚, and
three curved defects. The sizes of these defects are as follows: curve damage 𝐴 with a length of
31 𝑐𝑚 and width of 4 𝑐𝑚; curve damage 𝐵 with the length of 50 𝑐𝑚 and width of 1 𝑐𝑚; curve
damage 𝐶 with the length of 50 𝑐𝑚 and width of 4 𝑐𝑚. The locations of these defects are shown
in Fig. 13.
(a) The geometry of the plate with arc defects
A
B
C
Damage A:
Damage B:
Damage C:

(d) Curvelet

(f) Shearlet
Fig. 13. Arc damage detection.
4.1.7. The plate with five diverse defects
The seventh example includes a plate with a width of 300 𝑐𝑚 and a length of 300 𝑐𝑚, which
contained five defects. The linear defects (horizontal, vertical, and diagonal) have a length of
50 𝑐𝑚 and width of 5 𝑐𝑚, and the diagonal linear defect with an angle of 30°. Also, the curve
damage has a length of 48 𝑐𝑚 and width of 2 𝑐𝑚, and dimensions of the square defect are 5 𝑐𝑚.
The locations of these defects are indicated in Fig. 14.

60cm
22cm
65cm
65cm
38cm
10cm
20cm
22cm
150cm
140cm
(a) The geometry of the plate with five diverse defects.
(c) Laplacian Pyramid (9/7 filter)
A
B
C
D
E
Damage A:
Damage B :
Damage C:
Damage D:
Damage E:

(d) Curvelet
(f) Shearlet
Fig. 14. Five diverse defects detection.

37cm
48cm
58cm
55cm200cm
35cm
170cm
60cm
200cm
170cm 37cm
190cm
4.1.8. The plate with six diverse defects
The eighth example includes a plate with a width of 400 𝑐𝑚 and a length of 400 𝑐𝑚 containing
six defects. The linear defects (horizontal, vertical, and diagonal) have a length of 60 𝑐𝑚 and a
width of 5 𝑐𝑚, and the diagonal linear defect with the angle of 45°. Also, the curve damage has a
length of 50 𝑐𝑚 and width of 5 𝑐𝑚, and square defect dimensions are 7 𝑐𝑚. The locations of
these defects are displayed in Fig. 15.
(a) The geometry of the plate with six diverse defects
A
B
C
D
E F
Damage A:
Damage B :
Damage C:
Damage D:
Damage E:
Damage F:

(c) Laplacian Pyramid (9/7 filter)
(d) Curvelet
(f) Shearlet
Fig. 15. Six diverse defects detection.

4.2. Experimental validation
In this section, the capabilities of the transforms (wavelet, Laplacian pyramid, contourlet,
curvelet, and shearlet) were evaluated using the vibration response data of a steel plate proposed
by Rucka and Wilde [43]. It is worth mentioning that the data from Rucka and Wilde's paper is
not publicly available. Also, the experimental data includes the undamaged state exclusively;
hence, the damaged state was attained numerically. In addition, using experimental data has
some problems such as lack of all measured DOFs and incomplete modal data; see these
references for more information [44–46]. Fig. 16 depicts the steel plate with the length of 𝐿 =
560 𝑚𝑚, width of 𝐵 = 480 𝑚𝑚, and height of 𝐻 = 2 𝑚𝑚. The plate Young’s modulus,
Poisson’s ratio, and mass density were 𝐸 = 192 GPa, 𝜐 = 0.25, and 𝜌 = 7430 𝑘𝑔/𝑚3
,
respectively. The plate included a rectangular defect with 𝐿𝑑 = 80𝑚𝑚, 𝐵𝑑 = 80𝑚𝑚, and 𝐻𝑑 =
0.5 𝑚𝑚. The starting point of the damage was located at 𝑥 = 200𝑚𝑚 and 𝑦 = 200𝑚𝑚. The
transforms are applied to the fundamental mode shape to identify faults. Also, data de-noising
was carried out based on the process in section of de-noise the structural response of the plates.
(a) Experimental set-up [43]
(b) Wavelet

(c) Laplacian pyramid
(d) Curvelet
(e) Contourlet
(f) Shearlet transform
Fig. 16. Single experimental defect detection.

5. Discussion
In the following, the results of numerical and experimental examples are presented
comprehensively (Fig. 17 demonstrates the overall summary of the numerical examples):
 The experimental results have displayed that the maximum values of damage index are
located approximately at the place of the defect in all the transforms. Accordingly, the
transforms—wavelet, Laplacian pyramid, curvelet, contourlet, and shearlet transform—
can be adopted as applicable procedures to detect defects in all studies.
 To choose the proper wavelet based on the wavelet impact on fault demonstration, the
ability of various wavelets— including Haar, Symlet, and Discrete Meyer wavelet – is
evaluated in the second example. As the shape of the horizontal linear defect is more
vivid in the Discrete Meyer wavelet demonstration. Hence, damage presentation in all
examples is investigated via this wavelet.
 Since selecting the proper Laplacian filter has a considerable impression on the results of
the Laplacian pyramid and contourlet transform, both filters 9/7 and PKVA, have been
evaluated in this study to choose the qualified filter for the defect demonstration. As the
9/7 filter has better performance in comparison with the PKVA filter, it is considered in
all examples to defect identification.
 As can be seen in Fig. 17, when there is a single defect in the plate, shearlet transform
demonstrates excellent performance compared with the wavelet and Laplacian pyramid
transform; however, the shearlet has a relatively similar function to the curvelet and
contourlet transform. Therefore, it can be concluded that all these three transforms are
useable tools to detect single damages in the plate structures.
 Based on the Fig. 17, wavelet, Laplacian pyramid, curvelet, and contourlet transform
have poor ability to detect multiple damages with different shapes in plate structures.
Nonetheless, the shearlet transform has shown the perfect ability to exhibit several
damages on the plate.
 In conclusion, based on the numerical and experimental results, it could be concluded that
the shearlet transform is a practical and efficient transform to identify all kinds of
damages in the plate structures.

First to the Fourth Example
Damage
Point
Horizontal linear
Diagonal linear Curve
Laplacian
pyramid
Wavelet
Curvelet
Contourlet
Shearlet
Performance
of
the
transforms
in
damage
detection
(First
to
the
fourth
example)
1) Wavelet: poor
2) Laplacia: poor
3) Curvelet: excellent
4) Contourle: excellent
5) Shearlet: excellent
1) Wavelet: poor
2) Laplacian: poor
3) Curvelet: excellent
4) Contourlet: excellent
1) Wavelet: poor
2) Laplacian: poor
3) Curvelet: good
4) Contourlet: good
1) Wavelet: poor
2) Laplacian: poor
3) Curvelet: good
4) Contourlet: good

Fifth Example
Damage
Laplacian
pyramid
Wavelet
Curvelet
Contourlet
Shearlet
Sixth Example
Damage
Laplacian
pyramid
Wavelet
Curvelet

Contourlet
Shearlet
Seventh Example
Damage
Laplacian
pyramid
Wavelet
Curvelet
Contourlet
Shearlet

Eighth Example
Damage
Laplacian
pyramid
Not shown the
damage
Wavelet
Not shown the
damage
Curvelet
Not shown the
damage
Contourlet
Not shown the
damage
Shearlet
Not shown the
damage
Performance
of
the
transforms
in
damage
detection
(Fifth
to
the
eighth
example)
1) Wavelet: poor
2) Laplacian: poor
3) Curvelet: poor
4) Contourlet: poor
1) Wavelet: poor
2) Laplacian: poor
3) Curvelet: poor
4) Contourlet: poor
1) Wavelet: poor
2) Laplacian: poor
3) Curvelet: poor
4) Contourlet: poor
1) Wavelet: poor
2) Laplacian: poor
3) Curvelet: poor
4) Contourlet: poor
Fig. 17. The overall summary of the numerical examples.

Appear as Usually occur in
Show
There is the question of why the shearlet transform has superior efficiency in comparison with
other transforms? To answer this question, let’s consider two steps:
• First step: Comparison of shearlet with Laplacian pyramid and wavelet transform.
This section is summarized in three stages.
1. The capability to show more details causes better defect detection.
2. Shearlet has a better function to show details compared to Laplacian and wavelet
transform.
3. Therefore, Shearlet has superior performance to exhibit defects.
These stages have been explained in detail as follows:
Fig. 18. The relationship between defects and detail part.
Consider Fig. 18; this Fig presents this fact that defects appear in discontinuities of signals.
Discontinuities occur in the detail part. Therefore, to detect defects, we should find the detail part
of a signal. Accordingly, each transform with the superior capability to illustrate details has
better efficiency to display defects. Now the question is which of wavelet, Laplacian pyramid,
and the shearlet transform can demonstrate the detail part more effectively. To find the answer
see Fig. 19.
Fig. 19. The capability to show details.
As indicated in Fig. 19, the wavelet transform can demonstrate details in three directions
(horizontal, vertical, and diagonal), which are three matrices. Also, the Laplacian pyramid can
give the details in one matrix exclusively. Nonetheless, the shearlet transform can indicate detail
coefficients in several diverse directions. Hence, shearlet demonstrates more details, providing
superior performance to demonstrate damages.
Defects Discontinuities in
signal
Detail part of
signal
Apply Laplacian
Apply Wavelet
Apply Shearlet
Signal
Detail part is produced in one matrix
(no direction)
Detail parts are produced in three
matrices (three directions)
Detail parts are produced in various

• Second step: Comparison of shearlet with Curvelet and contourlet transform.
These three transforms can show details in various directions. It means they have the same
ability to show details. Thus, there is a question of why shearlet has presented better performance
to damage identification. The answer is the shape of basic elements, see Fig. 20.
(a) curvelet (b) contourlet (c) shearlet
Fig. 20. The basic elements.
As shown in Fig. 20, each transform has elements with elongated shapes oriented in different
directions; however, their basic forms are different. The shape of the basic elements of the
shearlet transform is more proper to show defects; therefore, it offers a superior capability to
detect damage.
6. Conclusions
This research proposed a new shearlet transform-based approach and compared it with four
transforms, including wavelet, Laplacian pyramid, curvelet, and contourlet, to detect several
types of damages, such as point, linear, and curve, in the plate structure. To assess, the
performance of various wavelets; Haar, Symlet, and Discrete Meyer wavelets were applied, and
the Discrete Meyer wavelet was considered the superior one. Also, 9/7 and PKVA filters were
evaluated to find the best ability of contourlet transform to detect damages, and the 9/7 filter was
considered the better one. Based on what was mentioned, the best performance of each transform
was used to compare the results. According to numerical simulation results, wavelets and
Laplacian Pyramid could not demonstrate curve defects perfectly. In addition, shearlet, curvelet,
and contourlet transform have similar, excellent performance in the plate with single damage.
However, only the shearlet demonstrates perfect performance to identify multiple defects in the
plate structure. It is worth mentioning that signal-based methods have some limitations such as
the inability to determine the severity of the damage. Hence, based on the results, the shearlet
transform overcome the weakness of the wavelet transform. Also, it offered superior
performance to the other multiresolution transforms (Laplacian Pyramid, curvelet, and
contourlet) to detect damages with different shapes. In addition, the correctness of the shearlet
transform was validated by the experimental example. Thus, the shearlet transform can be
employed as an efficient and practicable tool for the detection of all types of damages.

Acknowledgments
The author would like to thank Professor Rucka and Wilde for providing the experimental data,
and Professor Forgoston for his constructive feedback on the paper.
Funding
This research received no external funding.
Conflicts of interest
The authors declare no conflict of interest.
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