Ultrasonic Detection of Aliased Signal Separation Based on Adaptive Feature Dictionary and K–SVD Algorithm for Protective Coatings of Assembled Steel Structure

Abstract

When using ultrasound to detect the thickness of protective coatings on assembled steel structures, the coatings are extremely thin, which can cause echo signals to overlap and impair the detection accuracy. Therefore, the study of the separation of the superimposed signals is essential for the precise measurement of the thickness of thinner coatings. A method for signal time domain feature extraction based on an adaptive feature dictionary and K–SVD is investigated. First, the wavelet transform, which is sensitive to singular signal values, is used to identify the extreme values of the signal and use them as the new signal to be processed. Then, the feature signal extracted by wavelet transform is transformed into Hankel matrix form, and the initial feature dictionary is constructed by period segmentation and random extraction. The optimized feature dictionary is subsequently obtained by enhancing the K–SVD algorithm. Finally, the time domain signal is reconstructed using the optimized feature dictionary. Simulations and experiments demonstrate that the method is more accurate in separating mixed signals and extracting signal time domain feature information than the conventional wavelet transform and Gabor dictionary-based MP algorithm, and that it is more advantageous in detecting the thickness of protective coatings.

1. Introduction

Due to their obvious advantages such as light weight, high strength, and rapid construction speed, assembled steel structures are widely used around the world. However, compared to reinforced concrete structures, assembled steel structures have poor fire resistance and corrosion resistance, limiting their application perspective. To improve the fireproof and anti-corrosion performance of the assembled steel structure, the surface is first coated with anti-corrosion coating with strong adhesion to prevent corrosion of the steel structure by the external environment [1,2]; secondly, the fireproof coating with better fireproof performance, convenient construction, moderate cost, and short construction period is applied on top of the anti-corrosion coating to form a secondary protection for the assembled steel structure [3]. In general, the thickness of the protective layer of assembled steel structures guarantees their service life in harsh environments; therefore, it is essential to measure the surface coating thickness of the assembled steel structure.

Piezoelectric ultrasonic inspection [4] is distinguished by its high resolution, simple operation, and resistance to external environmental interference; its application in water immersion ultrasonic pulse echo technology has wide application prospects in coating inspection [5]. For the conventional ultrasonic signature scanning technique [6], the vertical incidence of ultrasound reduces the effect of waveform conversion on the echo signal, and the received reflected echo can be directly analyzed to obtain the necessary time-domain information. In the process of assembled steel structure coating inspection, the coating thickness is extremely thin, resulting in the mixing of the reflected echoes from the interface of fireproof coating and anticorrosive coating, which makes it difficult to identify the characteristic information of each coating and impossible to precisely determine the coating thickness, as depicted in Figure 1. In this paper, we use an algorithm to separate the mixed echoes of each coating and accurately obtain the time domain information of each coating for precise thickness measurement.

Research for ultrasonic detection of coating thickness has been the focus of domestic and international research. Malikov et al. [7] used Morlet continuous wavelet transform to define the time and instantaneous center frequency information of the received waveform, while the phase information at the local peak was obtained by Hilbert transform. The obtained instantaneous center frequency ratio, continuous wavelet transform amplitude ratio, and phase difference were used as inputs to the deep neural network, and the trained network system was able to identify 100% of the coating debonding. Kim et al. [8] proposed the use of fast Fourier transform combined with guided wave technique for the coating debonding problem of protecting ships, which can effectively distinguish the debonding condition of coatings in water. By investigating the variation pattern of the sound pressure reflection coefficient at the bonding interface, we explored the characteristic values that characterize the variation pattern. Luoming Sun et al. [9] applied the variation law to the C-scan imaging of coating debonding and enhanced the quantitative accuracy of the detection. Lin Li et al. [10] used wavelet transform to analyze the thickness measurement signal of ceramic coatings 250~350 $\mathsf{\mu }\mathrm{m}$ thick, which was consistent with the metallographic inspection results. Zhang Yu et al. [11] improved the detection signal-to-noise ratio by employing wave packet decomposition for high-frequency ultrasonic sweep signals of iron-based amorphous coatings. Through scanning electron microscopy, their results revealed a relative error of about 3.2%. It can be seen that domestic research on coating thickness focuses primarily on single-layer coatings with a thickness between 0.2 and 0.3 mm; for multi-layer and thinner coating thickness detection, the detection sensitivity of the above method is yet to be verified.

Currently, coating applications are becoming more and more widespread, and there are various methods for detecting coatings, such as high-frequency wave packet decomposition techniques [12], scanning imaging [13], intelligent algorithmic feature recognition [14], acoustic decay properties [15,16], time-frequency joint analysis [17], multiple data fusion methods [18], and time series parameter identification [19]. However, the effectiveness of these methods for processing coating detection signals with severe degree of aliasing is yet to be verified.

In recent years, sparse decomposition methods [20,21] have demonstrated remarkable advantages in signal aliasing problems. In the sparse decomposition method, the prototype atoms that match the original signal well are first screened. The prototype atoms are then expanded to form an overcomplete dictionary library by either parameter expansion [22] or dictionary learning [23], and finally, the original signal is characterized by a sparse decomposition algorithm to obtain the cleanest linear representation. Wang Fan et al. [24] used the Orthogonal Matching Pursuit (OMP) algorithm combined with the K Singular Value Decomposition (K–SVD) algorithm to update the dictionary, reconstructed the optical think stemmed laminar scan image by weighted average and exponential transform, and evaluated the noise reduction effect by mean square error, peak signal-to-noise ratio, structural similarity, and edge retention index. The proposed algorithm was more capable of reducing image scatter noise. Cheng Gu et al. [25] studied the effect of Morlet wavelets as multi-class atomic parameters of time-frequency atoms on the sensitivity of matched tracking algorithm, analyzed the accuracy of obtaining the initial values of atomic time-frequency parameters using Hilbert transform, and experimentally demonstrated that the improved algorithm has significantly better noise reduction than time-frequency analysis techniques such as generalized S-transform. Mor et al. [26] pioneered the Support Matching Pursuit (SMP) algorithm by improving the idea of combining the parameter expansion dictionary with the Orthogonal Matching Pursuit algorithm to successfully separate the mixed signals using the correspondence between ultrasonic echoes and time domain information, with significantly better results than the Matching Pursuit (MP) and Basis Pursuit (BP) algorithms. The aforementioned method achieved good results for the separation of partially mixed signals, but the overcomplete dictionary formed based on parameter expansion requires enormous memory, and algorithms such as MP must calculate the inner product of signal and residuals once in each iteration step, which requires an enormous number of operations. These limitations limit the practical application of sparse decomposition, and this is the key problem that will be addressed in this paper.

In light of the limitations of the sparse decomposition algorithm, the current focus of research is on optimizing the atomic search process for speedup using intelligent algorithms. Liu Xia et al. [27] used the particle swarm algorithm to improve the accuracy of the MP algorithm for fast decomposition reconstruction of seismic signals. The computational complexity of the particle swarm fast optimization MP algorithm is reduced by a factor of 69 compared to the original MP algorithm. Jicheng Liu et al. [28] used the asynchronous change learning factor to optimize the artificial bee colony algorithm, with improvements in both global and local search capabilities of the algorithm, and its application to the MP algorithm effectively improved the sparse decomposition efficiency. Applying it to the MP algorithm improves the efficiency of the sparse decomposition by a minimum of 67 times and obtains a good reconstructed signal. Elsayed et al. [29] combined dynamic multigroup and fast optimization ideas based on the quantum particle swarm algorithm for optimizing the optimization process of the MP algorithm, which improved the efficiency by about 90% compared to the quantum particle swarm-optimized MP algorithm and ensured that the signal-to-noise ratio of the image was reduced by less than 0.58 dB. On the other hand, the construction of a data-driven dictionary (i.e., dictionary learning) was studied, where the K–SVD algorithm was widely used as a classical dictionary learning algorithm [23]. Youxi Le et al. [30] used the K–SVD algorithm to reconstruct the signal based on the complete overall empirical modal decomposition noise reduction of the noise-containing signal. The experiment shows that the signal-to-noise ratio is improved to 16.65 dB after denoising at 0.88 dB, and the effect is significantly better than 8.11 dB after F-X domain denoising, 9.98 dB after wavelet soft threshold denoising, 7.94 dB after Complete Ensemble Empirical Mode Decomposition (CEEMD) frequency division denoising, and 13.97 dB after K–SVD sparse denoising. Ma L Y et al. [31] applied the K–SVD algorithm to the noise reduction study of synthetic aperture radar images, and the initial dictionary was obtained by orthogonal matching tracking. Then, the dictionary was updated by learning training, and its noise reduction effect was significantly better than the wavelet noise reduction algorithm and the discrete cosine dictionary-based sparse decomposition algorithm. Zhao et al. [32] converted two-dimensional sound signals into Hankel matrices and then applied the analytical K–SVD algorithm for noise reduction and feature extraction. However, the size of the original signal determines the dimensionality of the Hankel matrix constructed from the original signal, and if there are sufficient signal sampling points, the Hankel matrix will be redundant, which will place a significant load on the subsequent K–SVD algorithm. In addition, K–SVD algorithms typically employ fixed dictionaries, such as Fourier dictionaries, whose features are not guaranteed to match perfectly with the signal to be analyzed. As a result, the sparsest representation discovered by the algorithm during the iterative process might not be the optimal solution. Therefore, it is necessary to study the new initial signal matrix and the initial dictionary’s construction method. Based on this, Qin et al. [33] proposed to construct the initial matrix using period partitioning and cyclic shifting in order to obtain the initial dictionary after time-domain averaging of specific columns of the signal matrix and achieve the desired results in fault feature extraction experiments. However, it is ineffective to directly construct an initial dictionary using the original mixed signals for training and then separate the mixed signals; therefore, it is more effective to preprocess the original signals with feature extraction and then perform dictionary learning based on the extracted features in order to separate the mixed signals.

As stated previously, because the thickness of the protective coating of the assembled steel structure is thinner, the ultrasonic detection signal is more severely mixed, and it is difficult to obtain the thickness information of the coating by directly extracting the signal features through the K–SVD algorithm. Therefore, in this paper, we propose a method to separate ultrasonic mixed signals based on adaptive feature dictionary and improved K–SVD algorithm, aiming to improve the algorithm computing efficiency and the accuracy of the separation of mixed signals. The main objective of this paper is to study the adaptive learning dictionary that better fits the signal characteristics and can distinguish the characteristics of coating signals in the overlapped signals more effectively. Detailed simulation and experimental results show that the algorithm in this paper can identify and separate the overlapping detection signals of thin coatings of 0.1–0.2 mm, and the algorithm computational efficiency is also improved.

In subsequent sections of this paper, the proposed enhanced algorithm will be described in detail. In Section 2, the algorithm for the wavelet transform modulo maximal is described. The improved K–SVD algorithm based on the adaptive feature dictionary is described. In Section 3, the mixed ultrasonic detection signals are first obtained by simulation experiments, and then, the enhancement algorithm proposed in Section 2 is applied to process the signals to verify the rationality of the algorithm. Experiments were conducted in Section 4 on steel test blocks with protective coatings, and the experimental signals were processed. The effectiveness and advantages of the algorithm in this paper for coating ultrasonic inspection are verified. In Section 5, a summary of the study is given.

2. Theoretical Backgrounds

2.1. Wavelet Transform

Wavelet analysis is an effective method to identify the characteristics of non-stationary signals due to its ability to adaptively adjust the resolution in different frequency components and its multiscale and advective nature [34]. When applying wavelet basis functions to process and analyze an unknown signal, different wavelet bases lead to different processing results due to their unique properties. Therefore, when choosing a wavelet basis, the characteristics of the wavelet basis should be as similar as possible to the those of the signal to be processed.

The echo signal received by ultrasonic detection is a non-smooth signal, and the Mexh wavelet is the second-order derivative of the Gauss function, which provides good time-frequency resolution for signal analysis in its support length region and has symmetry and non-orthogonality, similar to the signal characteristics to be pre-processed in this paper. The expression of the Mexh wavelet function is shown in Equation (1), and its construction shape is shown in Figure 2.

$$\varphi \left(x\right)=\frac{2}{\sqrt{3}}{\pi }^{-1/4}\left(1-x^2\right)e^{-x^2/2}$$

(1)

The wavelet sequence is generated by stretching and translational transformation of the mother wavelet function, and the time domain expression is as follows:

$${\psi }_{a,b}\left(t\right)={\left|a\right|}^{-1/2}\psi \left(\frac{t-b}{a}\right)$$

(2)

where $a$ denotes the scale change, and $b$ denotes the displacement change. When the Mexh wavelet satisfies the constraints, the continuous wavelet transform can be applied to any function, as shown in Equation (3).

$$W_f\left(a,b\right)={\left|a\right|}^{-1/2}{\int }_{R}z\left(t\right)\psi \left(\frac{t-b}{a}\right)dt$$

(3)

If $\varphi \left(t\right)$ denotes the derivative of some smooth function $\theta \left(t\right)$ and the function $\theta \left(t\right)$ satisfies the conditions of Equation (4),

$${\int }_{-\infty }^{+\infty }\theta \left(t\right)dt=1$$

(4)

At this point, $\varphi \left(t\right)$ satisfies the constraint.

Define the wavelet transform expressions of $\varphi \left(t\right)$ and ${\varphi }^2\left(t\right)$, as shown in Equations (5) and (6).

$$\varphi \left(t\right)=d\theta \left(t\right)/dt$$

(5)

$${\varphi }^2\left(t\right)=d^2\theta \left(t\right)/d{t}^2$$

(6)

Let ${\theta }_m=\theta \left(t/s\right)/s$. Then, for a real function $z\left(t\right)\in L^2\left(R\right)$, its wavelet transform is expressed as shown in Equations (7) and (8).

$$W_m^{1}z\left(t\right)=W^{1}z\left(s,t\right)=z\ast {\varphi }_m\left(t\right)=z\ast \left(s\cdot \frac{d{\theta }_m\left(t\right)}{dt}\right)=\frac{sd\left(z\ast {\theta }_m\left(t\right)\right)}{dt}$$

(7)

$$W_m^{2}z\left(t\right)=W^{2}z\left(s,t\right)=z\ast {\varphi }_m^2\left(t\right)=z\ast \left(s^2\cdot \frac{d^2{\theta }_m\left(t\right)}{d{t}^2}\right)=s^2{d}^2\left(z\ast {\theta }_m\left(t\right)\right)/d{t}^2$$

(8)

It can be seen that $W_m^{1}f\left(t\right)$ and $W_m^{2}f\left(t\right)$ are related to the function after being smoothed by ${\theta }_m\left(t\right)$, and their values are positively related to the first-order and second-order derivatives of the function being smoothed. Therefore, the amplitude extreme points of $W_m^{1}z\left(t\right)$ are mapped to the set consisting of the mutation points of $z\left(t\right)$, while the zero points of $W_m^{2}z\left(t\right)$ are mapped to the set consisting of the inflection points with $z\ast \theta \left(t\right)$ [35]. Based on the above analysis, in this paper, we conduct our study on the basis of the family of Mexh wavelet functions and perform wavelet transform on the received signal $z\left(t\right)$. Moreover, the maximum value of wavelet transform mode $\left|W_{m}z\left(t\right)\right|$ is taken as the time domain eigenvalue of the reflected echo of the mixed coating interface.

2.2. K–SVD Algorithm

After the wavelet transforms modal maxima processing, the time domain characteristics of the original signal regarding the coating signal are amplified, with the initial effect of separating the mixed signal. However, the signal distortion and distortion characteristics caused by signal overlap are also highlighted, which are detrimental to the extraction of the time-domain information of the signal. An improved K–SVD algorithm based on the Hankel matrix construction and the initial feature dictionary is investigated for extracting the time domain information of the original signal more effectively.

Aharon et al. [23] proposed the K–SVD dictionary learning algorithm after an extended study of the K−means clustering algorithm, which is free from the limitations of parametric extended dictionaries and is able to update dictionary atoms adaptively. The algorithm is nestable with other decomposition algorithms, and the sparsity of the training dictionary is reduced continuously during dictionary optimization. Therefore, the atoms trained by the K–SVD algorithm are more tightly coupled to the original signal [36]. The ultimate objective of K–SVD is to solve the optimization problem shown in Equation (9).

$$\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\left|\left|Y-\mathit{DX}\right|\right|}_F^2\right\} s.t \forall i,{{\left|\right|x}_i\left|\right|}_0\le T_0$$

(9)

where $D$ represents the original overcomplete dictionary; $Y$ denotes the training sample; $X$ represents the sparse representation coefficient matrix; and $T_0$ is the threshold value for the number of non-zero component vectors in the sparse representation coefficient matrix.

The K–SVD algorithm comprises two phases: the encoding phase for determining the best sparse solution and the learning phase for determining the best matching dictionary. In the phase of sparse coding, it is assumed that

$$D\in R^{n\ast K},\mathrm{y}\in R^n, \mathrm{x}\in R^K$$

(10)

$$\mathrm{Y}={\left\{y_i\right\}}_{i=1}^N,\mathrm{X}={\left\{x_i\right\}}_{i=1}^N$$

(11)

where $D$ represents the original overcomplete dictionary; $y$ is the training signal; $x$ represents the sparse representation coefficient vector of the training signal; and $Y$ is the set of $N$ training signals. $X$ is the set of solution vectors of $Y$, and $N$ must be guaranteed to be much larger than $K$. In the sparse coding phase, the K–SVD algorithm needs to solve the optimization problem, as shown in Equation (12).

$$\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\left|\left|Y-\mathit{DX}\right|\right|}_F^2\right\} s.t \forall i,{{\left|\right|x}_i\left|\right|}_0\le T_0,\mathrm{i}=\mathrm{1,2},\cdots ,\mathrm{N}$$

(12)

The sparse representation coefficient vector corresponding to the original dictionary is obtained in the sparse encoding phase. Next is the lexicon learning phase, where the overcomplete dictionary library $D$ is trained iteratively. Let $d_k$ be the k-th column vector of the overcomplete dictionary library $D$ to be updated. Then, the sample vector is decomposed in the form, as shown in Equation (13).

$${‖Y-DX‖}_F^2={‖\left(Y-{\sum }_{j\ne k}{d}_j{x}_T^j\right)-d_k{x}_T^k‖}_F^2={‖E_k-d_k{x}_T^k‖}_F^2$$

(13)

where $x_T^k$ is the kth row vector in the coefficient matrix $X$, and $E_k$ represents the decomposition error matrix after removing $d_k$. To perform the singular value decomposition (SVD), the following definitions are introduced:

$$\begin{gathered} {\omega }_k=\left\{i|1\le i\le K,x_T^k\left(i\right)\ne 0\right\} x_R^k=x_T^k{\mathsf{\Omega }}_k \\ Y_k^R=Y{\mathsf{\Omega }}_k E_k^R=E_k{\mathsf{\Omega }}_k \end{gathered}$$

(14)

where ${\omega }_k$ represents the index set of atoms used for the sparse representation. ${\mathsf{\Omega }}_k$ is a matrix of size N*l(${\omega }_k$), where l(${\omega }_k$) is the length of ${\omega }_k$, whose elements are all 1 at $\left({\omega }_k\left(i\right),i\right)$ and 0 otherwise. $x_R^k,Y_k^R,{\mathrm{a}\mathrm{n}\mathrm{d} E}_k^R$ are the shrinkage results of $x_T^k,Y,\mathrm{a}\mathrm{n}\mathrm{d} E_k$ obtained by removing zero inputs, respectively. Equation (13) can therefore be rewritten as

$${‖E_k{\mathsf{\Omega }}_k-d_k{x}_T^k{\mathsf{\Omega }}_k‖}_F^2={‖E_k^R-d_k{x}_R^k‖}_F^2$$

(15)

SVD is performed on $E_k^R$ such that $E_k^R=\mathrm{U}\mathsf{\Delta }{V}^T$. At this point, the first column of $U$ represents the result of $d_k$ being optimally trained. In this way, each column of $D$ is optimally trained to the last column one by one to obtain the new dictionary. By repeating sparse coding and dictionary learning, the optimal dictionary of sample set Y and its corresponding coefficient matrix can be obtained.

2.3. Construction of Signal Matrix and Initial Feature Dictionary

The basic condition for the execution of K–SVD is to satisfy $N\gg K$. The dictionary for training optimization in K–SVD is also an over-complete dictionary, and it satisfies $K>n$. Thus, $N>n$, and it is guaranteed that the input signal matrix satisfies a condition whose number of columns should be as far as possible larger than the number of rows. While the ultrasonic detection signal of the coating is a two-dimensional signal about time and amplitude, and the number of rows of the signal is significantly greater than the number of columns, it cannot be separated directly using K–SVD. Therefore, the ultrasonic detection signal is first converted into the form of a Hankel matrix and then decomposed on this basis. After performing the wavelet transform, the signal with prominent features $x={\left[\begin{array}{cccc}{x}_1& x_2& \dots & x_n\end{array}\right]}^T$ is obtained, and the corresponding Hankel matrix is generated using the hankel function in the MATLAB function library, as shown in Equation (16).

$$x=\left[\begin{array}{c}\begin{array}{ccc}{x}_1& x_2\dots & x_n\\ x_2& x_3\dots & 0\\ ⋮& ⋮& ⋮\end{array}\\ \begin{array}{ccc}{x}_n& 0\dots & 0\end{array}\end{array}\right]$$

(16)

To execute the K–SVD algorithm to process the signal, a condition must be met, namely that the input signal matrix should have as many columns as possible that are significantly greater than the number of rows. Obviously, an excessive number of data sampling points will result in an excessively long signal length, and the generated initialized input matrix is also excessively large, reducing the computational efficiency. More seriously, for large matrices, singular value decomposition cannot be implemented. Moreover, for the signal matrix displayed previously, it is evident that the square matrix of order n does not satisfy the K–SVD operation requirements. Therefore, a new method of signal matrix construction is proposed. First, after selecting the frequency of the excitation signal, the number of sampling points per cycle is determined, and then, the number of sampling points of the received signal is added, segmenting the signal according to (17).

$$\mathrm{h}=n/T$$

(17)

where T represents the number of points in one cycle of the excitation signal; the number of points sampled in the best resolution is substituted when the number of points in the excitation signal is unknown; n represents the total number of points sampled in the received signal. By partitioning the signal Hankel matrix, it is possible to convert Equation (16) into the matrix depicted in Equation (18).

$$x_0=\left[\begin{array}{c}\begin{array}{ccc}{x}_1& x_{T+1}\dots & x_{hT+1}\\ x_2& x_{T+2}\dots & x_{hT+2}\\ ⋮& ⋮& ⋮\end{array}\\ \begin{array}{ccc}{x}_T& x_{2T}\dots & x_{\left(h+1\right)T}\end{array}\end{array}\right]\in R^{T\ast h}$$

(18)

The ultrasonic detection method is chosen for the detection of the coating thickness, and the coating is very thin. A higher detection frequency is used, and the excitation signal pulse width is kept as small as possible in order to improve the detection’s resolution. Therefore, after segmentation, the signal Hankel matrix satisfies the K–SVD requirement that the number of columns exceeds the number of rows. Meanwhile, the $x_0$ matrix formed by splitting the signal matrix according to the pulse width of the excitation signal on the basis of wavelet transform highlights the time-domain characteristics of the signal more and guarantees a better match between the output training dictionary of the K–SVD algorithm and the input signal.

During the execution of the K–SVD algorithm, the characteristics and properties of the initial dictionary will have an effect on the efficiency of algorithm training and the precision of sparse representation. At each step of the algorithm iteration, convergence to the global optimum is expected, and the nature of the dictionary determines the optimization accuracy of the algorithm; therefore, the properties of the selected initial feature dictionary determine the accuracy of the subsequent iterations of the optimization algorithm. It also ensures the accuracy, speed, and adaptiveness of transient feature extraction. A new method of constructing an adaptive initial feature dictionary is proposed based on the established $x_0$ signal matrix. The matrix form is shown in Equation (19).

$$D_1=\left[\begin{array}{ccc}{d}_1^{x_0}& d_2^{x_0}\cdots & d_q^{x_0}\end{array}\right]\in R^{T\ast h}$$

(19)

where $q$ denotes the number of columns of the feature dictionary. The initial feature dictionary with $q$ atoms is generated by randomly selecting $d_i^{x_0},i=\mathrm{1,2},\dots ,q$ column vectors from the signal matrix $x_0$. According to experience, the number of columns of the dictionary has little effect on the accuracy of the algorithm as long as the preconditions of the K–SVD algorithm are satisfied. In this study, the number of columns of the feature dictionary is set to $q=10\times h$, so that the highest degree of matching between $D_1$ and $x_0$ is achieved, and the time-domain characteristics of the signal can be better highlighted.

2.4. Signal Reconfiguration

After the K–SVD algorithm described in this paper has been applied, the learned dictionary $D_L$ and its corresponding coefficient matrix $A_L$ can be obtained. Since the effect of the waveform after mixing distortion remains in the wavelet transform and the construction of the feature dictionary, the dictionary $D_L$ can be further optimized by solving the problem shown in Equation (20).

$$\genfrac{}{}{0pt}{}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}{D_1}{‖D_0-D_L‖}_2^2+{\mu ‖D_0‖}_0$$

(20)

where $D_L$ denotes the dictionary to be optimized, and $D_0$ denotes the optimized dictionary. For the time domain characteristics of the coating detection signal, the time domain information is read through the wave crest; therefore, it is necessary to highlight the wave crest information while suppressing the non-correlated amplitude. For the signal, the degree of dispersion and concentration of the data is revealed by standard deviation. Therefore, an adaptive threshold condition is set for the dictionary $D_L$ by standard deviation, as shown in Equation (21).

$$D_0=max\left(std\left(D_L\right)\right)$$

(21)

where $std$ denotes the standard deviation of the dictionary $D_L$. The signal is reconstructed using the optimized dictionary, as shown in Equation (22).

$$X^{\prime }=D_0 \ast A_L=\left[\begin{array}{ccc}{X}_0^{\prime }& X_1^{\prime }\cdots & X_m^{\prime }\end{array}\right]$$

(22)

The submatrix $X_0^{\prime }$ in $X^{\prime }$ is reconstructed from the constructed initial matrix $x_0$, as shown in Equation (23).

$$x_0=\left[\begin{array}{c}\begin{array}{ccc}{x}_1^{\prime }& x_{T+1}^{\prime }\dots & x_{hT+1}^{\prime }\\ x_2^{\prime }& x_{T+2}^{\prime }\dots & x_{hT+2}^{\prime }\\ ⋮& ⋮& ⋮\end{array}\\ \begin{array}{ccc}{x}_T^{\prime }& x_{2T}^{\prime }\dots & x_{\left(h+1\right)T}^{\prime }\end{array}\end{array}\right]\in R^{T\ast h}$$

(23)

Subsequently, all columns of $X_0^{\prime }$ are combined to obtain the final coating detection time domain feature signal $x^{\prime }$, as shown in Equation (24).

$$x^{\prime }=\left[\begin{array}{ccc}{x}_1^{\prime }& x_2^{\prime }\dots & x_n^{\prime }\end{array}\right]$$

(24)

The signal is pre-processed by wavelet transform, and the proposed adaptive feature dictionary and improved K–SVD algorithm are applied to the pre-processed signal to obtain an ultrasonic coating detection signal that successfully separates the overlapped signal, and its explicit steps are described in Figure 3.

3. Simulation Analysis

3.1. Protective Coating Modeling

The simulation environment chosen for this study is the COMSOL multiphysics field simulation software. The simplified two-dimensional physical model created in COMSOL is shown in Figure 4.

From top to bottom in Figure 4: 1 mm thickness of water layer, 0.1 mm thickness of the fireproof coating, 0.1 mm thickness of the anti-corrosion coating, and 3 mm thickness of the steel. As shown in

Table 1. Acoustic physical parameters of each layer material.

Structural Layer	Poisson Ratio	$\mathrm{Elastic} \mathrm{Modulus} \left(GPa\right)$	Density $(\mathbf{k}\mathbf{g}/{\mathbf{m}}^3)$
water layer	/	/	1000
fireproof coating	0.45	0.99	800
anti-corrosion coating	0.28	30	2400
steel layer	0.3	200	7850

, the simulation model for each layer of the medium includes the physical parameters required for ultrasonic detection.

This study employs the high-frequency ultrasonic water immersion method to determine the thickness of protective coatings. In the simulation environment, the time domain display module of the physical field of pressure acoustics is selected, and the time domain waveform is solved by a transient solver. The ultrasonic wave and the propagation in the coating are also affected by the model’s boundary conditions [37]. The contact layer and excitation source boundary conditions are set as hard acoustic field boundaries. Then, the contact layer’s continuity is set, and the left and right boundaries of the coating and substrate layers are set as impedance boundaries.

The division of cells satisfies the requirement of Equation (25):

$$l_{max}={\lambda }_L/6=\frac{c_L}{6f_0}$$

(25)

where $l_{max}$ denotes the maximum cell size; ${\lambda }_L$ denotes the ultrasonic longitudinal wave wavelength; $c_L$ is the wave speed of the longitudinal wave in the medium; and $f_0$ is the excitation signal frequency.

The speed of ultrasound in different media is not the same. For this study, the longitudinal speed of sound in the media is calculated as follows:

$$C=\sqrt{\frac{E\left(1-\sigma \right)}{\rho \left(1+\sigma \right)\left(1-2\sigma \right)}}$$

(26)

where $E-$ elastic modulus; $\rho -$ medium density; $\sigma -$ poisson ratio. The theoretical value of ultrasonic velocity of $2167 \mathrm{m}/\mathrm{s}$ for fireproof coating and $3997 \mathrm{m}/\mathrm{s}$ for anticorrosive coating can be calculated using Equation (26). When the ultrasonic excitation frequency is 20 MHz, the maximum cell size of the grid in the water layer is set to 0.012 mm; the maximum cell size of the grid in the steel body layer is set to 0.049 mm; the maximum cell size of the grid in the fireproof coating is set to 0.018 mm; and the maximum cell size of the grid in the anti-corrosion coating is set to 0.033 mm according to the acoustic physical parameters of each layer material in Table 1.

The excitation signal is chosen to use a sinusoidal modulated signal with the mathematical expression shown in Equation (27) and the signal waveform shown in Figure 5.

$$h(t)=\sin \left(2\pi ft\right)\cdot \exp \left\{-{\left[\left(t-2T\right)/\left(T/2\right)\right]}^2\right\}$$

(27)

where $f$ and $T$ denote the excitation signal frequency and period, respectively.

3.2. Simulation Results and Analysis

Select the probe excitation frequency of 20 MHz and the simulation calculation results of the received signal, as shown in Figure 6.

The analysis of the waveform diagram reveals that in the protective coating interface reflection echo, due to the thin thickness of the coating and the large difference in the acoustic properties of the two layers of the coating, the ultrasonic waves will be reflected and transmitted between the coating and will be superimposed on each other to interfere with the waveform distortion, causing the final received signal to differ from the ideal echo signal. As shown in Figure 6, the waveforms of fireproof and anticorrosive coatings have been mixed, resulting in distortion of the entire waveform. Inability to effectively distinguish the time domain location of the echo signal peaks for individual coatings, the signal must be reconstructed after decomposition using the algorithm studied in this paper. Then, the time-domain information of each coating is obtained according to the amplitude of the reconstructed signal, and the thickness of each coating is calculated accordingly.

Initially, a feature dictionary and K–SVD algorithm were generated directly from the original signal without wavelet transform preprocessing for mixing and separation; the results are depicted in Figure 7. Comparing Figure 6 and Figure 7, it can be seen that while the mixed signals are successfully separated, the echo information of the two coatings obviously contains only the echo information of the anticorrosive layer, and the echo information features of the fireproof layer are not effectively identified by the algorithm. It indicates that the unpreprocessed K–SVD algorithm is not suitable for the separation of multiple signals with high degree of overlap, so the application of the original algorithm in coating detection is limited.

The separation effect of the mixed signals is shown in Figure 8; the MP algorithm based on the extended Gabor dictionary parameterization was iterated 20 times to mix and separate the simulated experimental signals. Comparing Figure 6 and Figure 8, it can be concluded that the classical MP algorithm suppresses the signal partials at the echo edges but does not significantly separate the ultrasound detection signals from the thinner coating overlap in the time domain space. In addition, the MP algorithm, after parameter discretization expansion, not only occupies a large amount of memory, but also has a very lengthy operation time, which is especially restrictive for coating thickness detection.

Finally, the improved algorithm proposed in this paper is used to separate the coating ultrasonic detection mixed signals, with the time axis displaying the time domain characteristics of the separated signals. For the initial values in the proposed algorithm to be pre-set, set T = 15, τ = 0.3, and h = 60. The ultrasonic detection samples 1215 points, so the algorithm condition is met because the pre-set signal matrix partition number h is less than the calculated value of Equation (17). The time domain diagram of signal separation reconstruction is shown in Figure 9. Comparing Figure 6, Figure 7, and Figure 9, the reflected echoes at each interface of the coating can be clearly analyzed from Figure 9 without the influence of signal mixing. The time of the received echo is obtained at the peak of the echo, and the specific thickness of the corresponding coating can be calculated according to the propagation theory of acoustic waves. This algorithm took 9.902691 s to execute on an AMD Ryzen 7 5800 H computer with Radeon Graphics 3.20 GHz. The MP algorithm based on the Gabor dictionary executes in 193.7080 s on a computer with the same configuration, demonstrating that the improved optimization algorithm presented in this paper is significantly superior to the traditional sparse decomposition matching tracking algorithm in terms of computing efficiency and mixing and separation accuracy. The time domain features of the reconstructed signals of this paper’s algorithm are analyzed, and the detection thickness of each coating is computed, with the results tabulated in

Table 2. Coating thickness according to the algorithm of this paper.

Coating	Actual Thickness (mm)	Detection of Thickness (mm)	Relative Error (%)
Fireproof layer	0.1	0.103	3.0
Anti-corrosion layer	0.1	0.109	9.0

. According to Table 2, relative errors in the detection of protective coating thicknesses are within acceptable limits, and simulation experiments verify that the proposed adaptive feature dictionary and K–SVD algorithm based on the proposed algorithm are effective at separating mixed ultrasonic coating detection signals.

4. Experiment and Discussion

4.1. Specimen and Experimental Platform

The essence of the improved algorithm proposed in this paper is the ability to effectively separate the mixed signals. Whether it is the overlap between coating and coating echo signals or between coating and substrate echo signals, it is the same kind of problem, and the practical effect of the algorithm can be verified. Therefore, to simulate the signal mixing phenomenon in ultrasonic inspection of protective coating of assembled steel structures, a steel plate of the same material (300 mm × 300 mm × 5 mm) was designed and fabricated, and a layer of 0.2 mm thickness of anti-corrosion layer was coated on the plate. The ultrasonic water dip detection signal is mixed with the heed, which can be used to verify the practical effect of the improved algorithm to separate the mixed signals. The schematic and physical drawings of the specimen are shown in Figure 10.

The equipment used in this experiment is UTFSCAN−SURFACES & HOLES V6.0 developed and produced by Nanchang Hangkong University. During the specimen coating thickness inspection, an ultrasonic probe with a center frequency of 10 MHz was chosen, and the test block was inspected at multiple locations using the water immersion ultrasonic inspection method; the physical diagram of the experimental platform is shown in Figure 11. The left side of the figure shows the mechanical transmission device and the ultrasonic signal transmitting and receiving device, and the right side shows the control and processing system. After the control system issues the sweep command, the transmission device drives the signal transmitting receiver to detect and sample the specimen and then uploads the sampling signal to the processing system. The detection signal is received using an acquisition card with a sampling frequency of 200 MHz, and 1642 points are used to reconstruct the echo signal from its single location.

A small piece of the specimen was first cut for metallographic testing, and the coating thickness was visually inspected through a metallographic microscope, the results of which are shown in Figure 12.

In the metallographic test results depicted in Figure 12, the darkest color represents the anticorrosive layer coating, and the color on the right represents the substrate. In the resulting metallographic image, five random locations were chosen to measure the thickness of the anticorrosive layer. Then, the average thickness obtained from metallographic testing was considered as the true thickness of the anticorrosive coating, and the measurement results are summarized in

Table 3. Metallographic test results of coating thickness.

Test Points	Coating Thickness Measurement Results $\left(\mathsf{\mu }\mathbf{m}\right)$	Average Thickness $\left(\mathsf{\mu }\mathbf{m}\right)$
1	209.12	209.49
2	212.29
3	210.24
4	209.12
5	206.69

.

Because the detection echo signal contains two basic information, including the acoustic wave propagation time and echo amplitude, and the speed of sound of ultrasound affects the time domain information of the signal and, thus, the accuracy of coating thickness detection; therefore, the sound velocity in the coating must be measured precisely. Multiple locations on the specimen are chosen at random for ultrasonic water immersion sweeping; the obtained echo signals are mixed and separated by the algorithm presented in this paper; and the location of the coating-substrate interface echoes are separated based on the reconstructed signals. The difference between the peak positions of the two waves is approximately 40 sampling points. The sampling frequency is 200 MHz, so the sampling time interval is 0.005 μs. Thus, the one-way average sound time difference between the detected coating and the substrate is 0.1 μs, and when combined with the average coating thickness of 209.49 μm provided by Table 3 for metallographic inspection, the average sound speed in the corrosion layer can be calculated as 2095 m/s.

4.2. Signal Decomposition Reconfiguration

A random 10 mm × 10 mm area of the coated specimen was swept using ultrasonic water immersion inspection equipment. Due to the experimental constraints, the coating specimens were ultrasonically inspected from the back to avoid the influence of interfacial echoes from the water immersion-focused inspection specimens on the test results. The test results of a randomly taken point are shown in Figure 13.

As can be seen in Figure 13, the pulse width of the interfacial echo obtained by focusing the water immersion probe onto the specimen surface is too wide and will completely drown out the coating detection signal, so the coating detection signal is extracted between the primary and secondary reflection echoes at the steel interface. The specimen’s thickness is measured with vernier calipers, and when combined with the speed of sound in the steel, the primary and secondary reflected echoes at the interface of the steel can be calculated as depicted in Figure 13 at positions 1 and 2. Multiple interface echo signals of the anticorrosive layer and the primary echo of the steel are obviously mixed, and there are abnormal wave peaks between positions 1 and 2. Based on the sound velocity in the anticorrosive layer and the coating thickness of the preset specimen, it is known that these abnormal signal peaks should be the multiple interface echo signals of the coating. Now, the mixed signals are separated using the algorithm proposed in this paper, in which the ultrasonic wave wavelength $\lambda =209.5 \mathsf{\mu }\mathrm{m}$ propagated in the anticorrosive layer is calculated according to the equation $c=\lambda f$ based on the average ultrasonic wave speed of 2095 m/s in the anticorrosive layer and the excitation frequency of 10 MHz. Then, the detection resolution is about half wavelength, i.e., 104.75 μm. Additionally, 104.75 μm needs to be sampled approximately ten times at the sampling rate of 200 MHz. It takes about ten times of sampling, so the signal segmentation index in T = 10. Figure 14 compares the separation results of the MP algorithm, the OMP algorithm, and the mixing separation effect of the K–SVD algorithm without wavelet transform.

Comparing the results of the decomposed reconstructed experimental signals in Figure 14 reveals that the MP algorithm and the OMP algorithm based on the Gabor dictionary are ineffective at separating the ultrasonic water immersion detection signals of the mixed anticorrosive layer. Only the reflected echoes of the mixed steel interface and the reflected echoes of the coating interface are brought out morphologically, which suppresses the incoherent structural noise and does not separate the mixed echo signals on the time axis. In contrast, the K–SVD algorithm without wavelet transform has clearly separated wave peaks in the reconstructed time-domain map; however, after the calculation of the wave peak time−domain information, it is not the exact time−domain location of the reflected echoes at the steel and coating interfaces, indicating that the algorithm is affected by the non-correlated signals in the dictionary training of the optimal sparse coefficients. The algorithm proposed in this paper reconstructs the signal time domain map. The mixed interface echo signal separation effect is good, the wave peak time domain information is extracted, and the corresponding position signal comparison in the original signal is corresponding, proving that this algorithm can be used for the separation of the ultrasonic mixed detection signal of anticorrosive layer.

Ten groups of signals were randomly selected from the detection signals obtained in the swept area for processing, and the time domain information was extracted from the separated reconstructed signals to calculate the thickness of the preservative layer, the results of which are shown in

Table 4. Experimental detection results of the algorithm in this paper.

Serial Number	Detection of Thickness (mm)	Relative Error (%)
1	0.1938	7.49
2	0.2147	2.50
3	0.1990	4.99
4	0.2042	2.54
5	0.2147	2.50
6	0.2252	7.51
7	0.2147	2.50
8	0.2252	7.51
9	0.2147	2.50
10	0.1992	4.91

.

Comparing the calculated value to the metallographic detection value, the relative error of the coating thickness calculated by the algorithm of this paper after signal separation is within 8%, while the variation of the inspection relative error is mainly concentrated around three numbers of 2.5%, 4.9%, and 7.5%; it can be seen in combination with the certain variation of the coating thickness in the metallographic experiment that the coating is not absolutely balanced and uniform, and the variation of the relative error is reasonable. And the relative error value calculated in Table 4 verifies the reasonableness of the algorithm applied to the actual inspection in this paper.

5. Conclusions

In this paper, we propose a method for separating ultrasonic mixed signals using an adaptive feature dictionary and an improved K–SVD algorithm, which is used to process and analyze ultrasonic water immersion detection signals of assembled steel protective coatings and can effectively separate the overlapping coating signals. The research results of this paper are summarized as follows:

• The wavelet transform is used to identify the polar values of the detected signal as the input of dictionary learning, which can effectively avoid the influence of multiple features of the original signal on the signal mixing and separation, and the simulation and experiment show that the signal mixing and separation after wavelet transform are more accurate.
• The algorithm proposed in this paper, which trains the learning dictionary by signal extremes as features, greatly improves the algorithm operation efficiency while ensuring the accuracy of signal mixing and separation, and the simulation and experiment show that the detection relative error of the algorithm in this paper is within 9%, and the operation speed is 19 times higher than that of the MP algorithm.
• The algorithm developed in this paper is more intuitive than the MP algorithm for the separation of the overlapped signals in the time domain, and the location of the coating echoes can be obtained more intuitively from the separated signals. At the same time, simulation and experiment show that for 0.1–0.2 mm thickness of the coating, ultrasonic detection of the mixed signals can be effectively separated and identified. Therefore, the method has great potential for use in coating thickness detection.