1. Introduction
Laser ultrasonic technology, with its advantages of convenience and flexibility, high detection frequency, and spatial resolution, as one of the more commonly used major technical methods for non-destructive detection, can be used to detect metal grain size [
1], defects [
2], welding quality [
3], stress [
4], etc. Laser ultrasonic technology allows a laser to propagate throughout the material and extract internal information, such as changes in ultrasonic velocity and the frequency-related attenuation coefficient, as well as changes in the metal microstructure [
5,
6,
7]. To achieve strictly non-destructive results, the thermoelastic mechanism needs to be utilized when the laser emits ultrasonic but at the same time, this method exhibits the disadvantage of low energy and a weak signal. To solve this problem, the laser phased array principle [
8] can be used to convert the point-source light into a ring-shaped source [
9] through a cone lens to achieve enhanced signal energy under the thermoelastic mechanism. Even so, the laser ultrasonic signal attenuates with an increasing propagation distance, and the signal is often drowned out by the noise. Therefore, it is particularly important to denoise the ultrasonic laser signal and accurately extract valid information.
Empirical mode decomposition (EMD), which has been widely used in signal processing, is a time-frequency analysis method proposed by Huang et al. [
10] for non-stationary signal analysis with an adaptive nature,. However, the local characteristics of the conventional EMD may produce mode mixing. To improve this problem, Wu et al. [
11] proposed a new method: ensemble EMD (EEMD), which uses white noise to superimpose the original signal so that the signal is continuous at different scales. This method can reduce the mode mixing, but the signal after denoising contains residual noise and the decomposition completion is poor.
In 2014, Dragomiretskiy et al. [
12] proposed the variational mode decomposition (VMD) algorithm, which effectively eliminated the problem of EMD through extreme value envelope decomposition and decomposed the original signal into signal components with different frequencies and magnitudes. VMD can determine the number of mode decompositions for a given sequence according to the actual situation, and can adaptively match the optimal center frequency and finite bandwidth of each mode in the subsequent search and solution process to achieve the effective separation and frequency domain division of the intrinsic mode components of the signal. Due to the above advantages, VMD is widely used in signal processing methods involving bearing fault diagnosis, bioscience, earthquake monitoring, etc. Two key parameters in the VMD decomposition process need to be set: the number of mode decomposition layers
K and the penalty factor
α. However, artificially defined parameters can lead to many problems, including frequency mixing and incomplete noise separation. Several scholars have investigated various methods to obtain optimal parameters. Long et al. [
13] utilized particle swarm optimization (PSO) to find the optimal parameters of the VMD for the removal of various interferences from the UHF PD signals. Hua et al. [
14] used the grasshopper optimization algorithm (GOA) to obtain the best combination of parameters for VMD by decomposing the Lidar signal and thus achieving the denoised signal. Qi et al. [
15] used the gray wolf optimization algorithm to obtain the best combination of parameters for VMD, and parallel EMD decomposes the signal to achieve the denoising of the Lidar signal.
In addition to the optimization method of optimal parameters, the signal denoising process based on VMD decomposition requires that the associated mode functions of the decomposition be reconstructed. When VMD is applied to noisy signals, the physical meaning of the modes—as pure signal, pure noise, or both—needs to be determined [
16]. Li et al. [
17] realized that the identification of noisy band-limited intrinsic mode functions (BLIMF) by using correlation-based thresholding will extract the modes and simulate the signal. However, the first mode still has a strong correlation with the noisy signal, low signal-to-noise scenario, leading to a poorly performing method. Liu et al. [
18] developed a simple criterion based on detrended fluctuation analysis (DFA) to select correlated modes. This method can measure the long-range dependence of non-stationary time series, but it ignores the relationship with the simulated signal and lacks adaptability. Qi et al. [
15] chose the first local maximum of the Pearson correlation coefficient method as the dividing line in EMD denoising. Before this maximum, the noise is the main part of the IMFs, and the rest is the main part of the signal. Xu et al. [
19] calculated the correlation with the original function sequentially by gradually increasing the accumulated mode components and selected the mode component accumulated when the threshold is reached for the first time as the denoised signal. Although this method can obtain reasonable results, the threshold of the approximate entropy needs to be preset.
Given the above problems, a novel laser ultrasonic signal denoising method based on parameter-optimal VMD combined with the Hausdorff distance (HD) is proposed in this paper. First, the number of decomposition modes and the quadratic penalty factor are optimized using the whale optimization algorithm (WOA). Then, the optimal parameters are inputted to decompose the signal into modes; finally, the HD is used to select the relevant mode functions and reconstruct the signal to obtain the denoised signal. The proposed method is validated by simulating signals with five different signal-to-noise ratios and experimental laser ultrasonic signals for real lasers. The results show that the proposed WOA-VMD method achieves better performance compared with the other three denoising methods, effectively preserving the useful details while at the same time, denoising the original signal.
The main aim of this paper is to use sample entropy as the optimal fitness function of the whale optimization algorithm to obtain optimization parameters that can effectively perform variational pattern decomposition and avoid mode mixing or spurious components in order to achieve better denoising performance.
It can effectively prevent the weak signal from being annihilated by environmental noise or electromagnetic interference, facilitate the accurate extraction of the transverse echo signal of the laser ultrasound, and improve the accuracy for the subsequent research on the microstructure of metals.
The structure of this paper is as follows.
Section 2 briefly introduces the principles of VMD, WOA, WOA-VMD, and HD, as well as the flow of the algorithm proposed in this paper. In
Section 3, we analyze the denoising performance of WOA-VMD based on the processing results of simulated noisy signals and compare WOA-VMD with some commonly used denoising methods. In
Section 4, we select an experimental signal to further analyze the denoising process and compare the performance between WOA-VMD and EMD.
Section 5 is the conclusions of this paper.
2. Methods
In this section, we will study the laser ultrasonic signal denoising method based on the WOA for VMD. First, the basic principles of the VMD algorithm and the whale optimization algorithm are briefly introduced; second, the specific VMD based on WOA is illustrated when the combined fitness function is the sample entropy; then, the judgment of the relevant modes using HD is introduced; finally, the specific process of the algorithm proposed in this paper is shown.
2.1. The Principle of VMD
VMD treats the signal to be analyzed as a linear superposition of several mode components. Each band-limited intrinsic mode function (BLIMF) is defined as an amplitude-modulated frequency signal with the following expression [
12]:
where
t is time;
uk(
t) is the
kth BLIMF;
Ak(
t) is the instantaneous amplitude; and
φ(
t) is the signal phase.
The variational modes are constructed by calculating the one-sided spectrum of each BLIMF component using the Hilbert transform, and then estimating the central frequency [
12]:
where
k = 1,2,…,
K; σ(t) is the unit pulse function;
ωk is the center frequency;
f(
t) is the input signal; ⊗ indicates the convolution operation;
∂t denotes the partial derivative operation; and
j is an imaginary number.
The constrained variational problem is transformed into an unconstrained variational problem by introducing penalty factors and the Lagrange multiplier operator to obtain the extended Lagrange expression [
12].
where α is the bandwidth parameter.
The center frequency and bandwidth of each component are continuously updated during the solution process until the iteration-stopping condition is satisfied [
12].
where:
is the expression after the kth BLIMF update and φ is the discriminative accuracy, generally taken as 10
−6.
At the termination of the iteration, the signal frequency domain characteristics have been adaptively separated and then converted to the time domain by an inverse Fourier transform.
2.2. VMD Parameter Optimization Based on WOA
According to the VMD theory, the layers of decomposition
K have a great influence on the decomposition results, while the penalty factor
α is related to the decomposition accuracy [
12]. Therefore, the determination of these two parameters is the key to improving the performance of VMD. The whale optimization algorithm (WOA) [
20,
21] is a new population intelligence optimization algorithm proposed by Mirjalili et al. of Griffith University, Australia, in 2016, which has the advantages of simple operation, few parameters, and a high ability to jump out of the local optimum.
2.2.1. WOA
- 1.
Encircling prey
The search range of the whale is the global solution space, and the location of the prey needs to be determined first in order to surround it. Since the location of the optimal design in the search velocity is not known, the WOA algorithm assumes that the current best candidate solution is the target prey, or close to the optimal solution. After defining the best search agent, the other search agents will try to update their positions toward the best search agent. This behavior is represented by the following Equations (5) and (6) [
20].
where
t is the current number of iterations;
and
are the coefficient vectors,
denotes the best whale position vector so far, and
denotes the current whale position vector.
The vectors
and
can be obtained from the following calculation [
20]:
where the value of
decreases linearly from 2 to 0, and
is a random vector in [0,1].
- 2.
Hunting behavior
According to the hunting behavior of the humpback whale, it swims toward its prey with a spiral motion, so the mathematical model of the hunting behavior is as follows [
20]:
where
indicates the distance between the whale and its prey,
denotes the best position vector so far,
b is the constant of the logarithmic spiral shape, and
l is a random number in [−1, 1].
The humpback whale swims around its prey within a shrinking envelope while following a spiral path. To simulate this simultaneous behavior, we assume a 50% probability to choose either the shrinking envelope mechanism or the spiral model to update the whale’s position during optimization. The mathematical model is as follows [
20]:
The algorithm is set when , and the whale attacks the prey.
- 3.
Search the prey
The mathematical model is as follows [
20]:
where
is the randomly selected whale position vector, the algorithm is set to randomly select a searching individual. When
, updating the position of other whales based on the randomly selected whale position, the whales are forced to deviate from the prey and thus find a more suitable prey, which can enhance the exploration capability of the algorithm, enabling the WOA algorithm to perform a global search.
2.2.2. WOA-VMD
Sample Entropy [
22] (SampEn) is a measurement of time series complexity performed by measuring the magnitude of the probability of generating new patterns in the signal; the greater the probability of generating new patterns, the greater the complexity of the sequence. The lower the value of sample entropy, the higher the sequence self-similarity is; the higher the value of sample entropy, the more complex the sample sequence is. When WOA is used to search for the optimal parameters of the VMD algorithm, the smallest sample entropy of the mode component is used as the fitness function, and the steps are as follows.
Step 1: Input the signal, set the parameter ranges of K and α in the VMD algorithm, and initialize the parameters in the WOA model, including population size, the maximum number of iterations, spatial dimension, and initial population individuals.
Step 2: Perform the VMD decomposition of the signal and calculate the fitness of each individual in the initial population. When the sample entropy is the smallest, the corresponding parameter is optimal.
Step 3: When , the whale position corresponding to the minimum ranking entropy is selected as the target value for local exploitation, and then Equation (10) is selected to update the position of individual whales according to the magnitude of the p-value. When , the position of one whale is randomly selected to update the position of individual whales according to Equation (12), preserving the optimal fitness and the corresponding parameter combination.
Step 4: Keep the updated whale population position as the initial population for the new round and iterate through the cycle until the set maximum number of iterations is reached.
Step 5: Output the parameter combinations corresponding to the optimal whale individuals.
2.3. Hausdorff Distance Identification of Correlated Modes Reconstructs the Signal
After the VMD, the properties of the extracted modes must be determined, and the relevant modes need to be selected, which is a prerequisite for signal reconstruction. The VMD algorithm decomposes a signal from low to high frequencies. It is generally considered that the high-frequency area is the noise mode (uncorrelated mode), and the low-frequency area represents the pure signal mode (correlated mode).
The signal
x(
t) is decomposed by VMD to obtain a series of mode functions
uk,
k = 1, 2, …,
K. The reconstructed signals are:
where
kth denotes the index of the reconstructed area.
The probability density function (PDF) can reflect the difference between signal distributions. In this paper, we use HD as a similarity metric to distinguish between relevant and irrelevant patterns.
HD is a nonlinear operator that measures the similarity between two sets or two geometric figures [
23]. The HD between two point sets
P and
Q is defined as follows [
16]:
HD is sensitive to outliers and can react to the sharpness and narrowness of the PDF of the VMD pattern. To identify the relevant patterns,
x(
t) is decomposed into BLIMFs using VMD. The similarity index
L is defined as follows [
16].
The correlation pattern can be determined by the difference between two adjacent PDF distances. The larger the difference, the greater the change in similarity. The maximum difference value is defined as follows.
Assuming that the maximum difference is generated between BLIMF
m and BLIMF
m+1, the estimated signal
y(
t) can be obtained as follows.
2.4. Flow of the Proposed Algorithm
The flow of the proposed algorithm is shown in
Figure 1.
First, the optimal parameters K and α are obtained by using the whale optimization algorithm, with the minimum sample entropy as the best fitness function.
Secondly, the optimal parameters are used as the input of VMD to decompose the signal.
Finally, using the Hausdorff distance, the relevant mode functions are selected. The relevant mode functions are accumulated to obtain the final denoised signal.
5. Conclusions
In this paper, a novel denoising method, called the WOA-VMD, is proposed to deal with the laser ultrasonic signal denoising problem.
(1) The proposed method utilizes the WOA to optimize the VMD parameters, including the number of decomposition modes K and penalty factor α. Then, relevant modes are selected using the HD between the decomposed modes and the original signal. The denoised signal is obtained by accumulating the correlation modes.
(2) The feasibility and effectiveness of this proposed method are verified via simulations. The simulation results of the ultrasonic signals with different SNRin have shown that the proposed method can effectively denoise the noisy signal and outperform other denoising methods.
(3) Finally, a laser ultrasonic experiment is carried out. The results prove that the proposed method could reduce the Gaussian white noise and high-frequency noise, preserve the useful signal, and increase the SNRout for laser ultrasonic signals. This helps to extract the transverse signal echoes of laser ultrasound and improves the accuracy of subsequent studies of the microstructure of metals.