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Simplified Matrix Focusing Imaging Algorithm for Ultrasonic Nondestructive Testing
Chinese Journal of Mechanical Engineering volumeÂ 35, ArticleÂ number:Â 19 (2022)
Abstract
Full matrix focusing method of ultrasonic phased array has been proved with advantages of good signaltonoise ratio and imaging resolution in the field of Ultrasonic NDT. However, it is still suffering from the timeconsuming data acquisition and processing. In order to solve the problem, two simplified matrix focusing methods are provided in the paper. One provided method is a triangular matrix focusing algorithm based on the principle of reciprocity for the multichannel ultrasonic system. The other provided method is a trapezoidal matrix focusing algorithm based on the energy weight of the different channel to the focusing area. Time of data acquisition and computational is decreased with the provided simplified matrix focusing methods. In order to prove the validity of two provided algorithms, both sidedrilled holes and oblique cracks are used for imaging experiments. The experimental results show that the imaging quality of the triangular matrix focusing algorithm is basically consistent to that of the full matrix focusing method. And imaging quality of the trapezoidal matrix focusing algorithm is slightly reduced with the amount of multichannel data decreasing. Both data acquisition and computational efficiency using the triangular matrix focusing algorithm and the trapezoidal matrix focusing algorithm have been improved significantly compared with original full matrix focusing method.
1 Introduction
In recent years, ultrasonic phased array technology has been developed rapidly [1,2,3,4]. And ultrasonic phased array technology has been gradually applied in the field of industrial nondestructive testing [5,6,7,8]. The phased array imaging algorithms have been widely studied [9,10,11,12]. Holmes et al. [13] proposed the concept of fullmatrix focus capture (FMC), and set up the total focus method (TFM) algorithm using FMC. Compared with the conventional ultrasonic phased array testing, the geometric features of defects are more clearly with TFM technology. And the image quality of TFM technology is significantly better than the traditional phased array imaging algorithm with steering and focusing time delays. Wu et al. [14] extended the total focusing imaging method to the multimode case with both direct and reflection waves. Pan et al. [15] used the total focusing imaging method to simulate the cracks quantitatively. Wu et al. [16] studied primary reflection detection mode based on the optimized direct wave detection, and provided the multimode composite full focus imaging method. Aiming at the problem of internal defect detection of aluminum container weld, Wang et al. [17] proposed an ultrasonic array full focus imaging method based on oblique incidence.
However, TFM imaging is a timeconsuming method due to the large amount of full matrix data [18,19,20,21], and it is difficult to obtain realtime image [22,23,24,25]. In order to improve the efficiency of TFM, Ran [26] used FPGA to analyzing the principle of full focus synthetic aperture imaging algorithm, and designed software to optimize the algorithm for results. Jin et al. [27] introduced the wavenumber algorithm to improving computational efficiency for ultrasonic fullmatrix imaging in multi layered medium. Postprocessing speed of full matrix data is increased significantly with the provided method. The binary particle swarm optimization (BPSO) algorithm is applied to optimize the array layout. The BPSO algorithm produced favorable results in the case of smallscale sparse array [28]. Hu et al. [29, 30] optimized the locations of active array elements in the sparse array with the genetic algorithm.
Two simplified matrix focusing methods are proposed in the paper. One is the triangular matrix focusing algorithm based on the reciprocity principle of the transmitter and receiver channel, named as triangular matrix focusing method. The other one is the trapezoidal matrix focusing algorithm based on the energy weight of the distance from different channels to the focusing point, named as trapezoidal matrix focusing method.
In Section 2, the data acquisition and imaging algorithm of full matrix focusing method is introduced, and the simplified imaging methods are provided. In Section 3, sidedrilled holes and oblique crack defects imaging experiments are carried out. The experimental results are analyzed to compare the imaging quality and computational efficiency with three image algorithms.
2 Focus Imaging Algorithm
2.1 Full Matrix Focused Data Acquisition
Ascan data is acquired and arranged in a matrix for imaging. Taking a linear array with n elements as an example, the 1st element is fired and all elements are received. The acquired data is named as A_{11}, A_{12}, ..., A_{1n}. And the acquired data is placed in the first row. Then next element is excited in turn until all the nÃ—n Ascan data are acquired as listed in Eq. (1).
2.2 Full Matrix Focusing Imaging Algorithm
Imaging algorithm according to the full matrix data is shown schematically in Figure 1. The imaging region below the transducers is discretized into many focus points in alignment. Taking focus point P as an example, the distances between the point P and two elements i, j can be calculated. The amplitude A_{ij}(t_{0}) in the signal A_{ij} can be determined according to the corresponding travel time t_{0}. The amplitudes from all Ascan signals is superimposed with the same rule. Thus the digital amplitude of the focus point P \(I_{{\text{P}}} \left( {x,z} \right)\) can be calculated according to Eq. (2).
where c is the wave speed in the material.
2.3 Triangular Matrix Focusing Imaging Algorithm
The fullmatrix focus imaging algorithm needs all Ascan data acquisition and superposition, so it is a timeconsuming method. Considering the reciprocity principle of the multichannel acoustic system, the transmit and receive channels are interchanged. A good consistency in Ascan amplitudes is maintained. The A_{ij} signal (the i element is transmitted, and the j element is received) should have a good consistency with the A_{ji} signal (the j element is transmitted, and the i element is received). Then, the full matrix data shown in Eq. (1) is a symmetric matrix. It is sufficient that the imaging algorithm only use the upper triangular matrix signal, as shown in Eq. (4). It is called as the triangular matrix focusing imaging algorithm according to Eq. (5).
Comparing Eqs. (1) and (2) to Eqs. (4) and (5), the amount of data acquisition and calculation is reduced from nÃ—n to nÃ—(n+1)/2. Then the amount of both data transfer, storage and calculation will be almost reduced by half.
2.4 Trapezoidal Matrix Focusing Imaging Algorithm
In order to improve the computational efficiency, the signal energy weights of different channels to the focus point are also considered. Synthetic aperture focusing technique (SAFT) is very useful method. The data on the main diagonal of the matrix is used in SAFT. The transmitting element is same as the receiving element. For a given focus point as shown in Figure 2, the Ascan amplitude is weaken with the distance from the transmitter to the receiver element increasing. Taking path1, path2 and path3 in Figure 2 as an example, with the travelled distance increasing from the focus point, the Ascan amplitude of the receiving element i, j and n is weaken gradually.
It is easy to find that the data with larger energy weight are distributed near the diagonal of the triangular matrix. Therefore, the data near to the diagonal of the triangular matrix is reserved, and data far from the diagonal of the triangular matrix is ignored. Then, the trapezoidal matrix focusing imaging algorithm is determined as shown in Eqs. (6) and (7).
where \(i < k < N\).
After obtaining the superposition amplitude of each focusing point, the original image needs to be normalized for the digital display. The 256 colors bar is often used in the ultrasonic NDT field. The superposition amplitudes are normalized from âˆ’Â 127 to 128 using Eqs. (8) and (9).
If \(I_{{\text{P}}} \left( {x,z} \right) > 0,\)
If \(I_{{\text{P}}} \left( {x,z} \right) \le 0,\)
3 Experimental Verification
Both symmetric sidedrilled holes and asymmetric oblique cracks are used in the experiment. The experiments are analyzed imaging quality and calculation efficiency with different matrix focusing imaging algorithm. The data acquisition and imaging experiments are based on an AOS 64Ã—64 ultrasonic phased array system and independentresearchanddevelopment imaging software. The sampling frequency of each channel is 100MSPS. A linear array with 5 MHz center frequency and 0.6 mm element center distance is used in the experiment. Due to the large amount of data collected using the full matrix focus method, only 32 channels and elements are used in the data acquisition.
Symmetric sidedriller holes with 1.5 mm diameter are fabricated in an aluminum alloy specimen, as shown in Figure 3(a). Three asymmetric cracks with different tilt angles of 30Â°, 45Â° and 60Â° are fabricated at the bottom of the low carbon steel, as shown in Figure 3(b).
3.1 Comparison of Signal Consistency and Energy Weight
Considering the reciprocity principle of the Ascan signals, the transmitter and receiver channels are interchanged. Some pairs of Ascan signals are selected randomly from the full matrix data, as shown in Figure 4. Figure 4(a) shows that the signal of the symmetric sidedrilled holes, and Figure 4(b)â€“(d) shows the signal of asymmetric oblique cracks. It can be seen from the results that the A_{521} signal always keeps good consistency with A_{215} signal. Both phase and amplitude of A_{521} and A_{215} are almost same, and signal amplitude in the near field part is slightly different. The rule can also be found in any pair of A_{ij} and A_{ji} channels in the full matrix data. Therefore, the full matrix data is a symmetric matrix.
As shown in Figure 5, Ascan signals are sidedrill hole response with 6th element fired and 6th, 15th, 23rd element received. It can be seen that, as the distance from the transmitter to the receiver increasing, the amplitude of sidedrill hole response is gradually decreased and time delay is increased. The data energy weight is lower as far away from the main diagonal of the matrix.
3.2 Comparison of Imaging Quality
Comparison of different matrix focus imaging methods is shown in Figures 6, 7, 8 and 9. Figure 6 shows the different matrix focus imaging of symmetric defects, and Figures 7, 8 and 9 show the different matrix focus imaging of asymmetric cracks with oblique angle of 30Â°, 45Â° and 60Â°. It is difficult to identify the difference of the four algorithms in those imaging results observationally. The holes and crack tip signals with different oblique angles can be found clearly. And the crack oblique angle can be evaluated according to the missing position of bottom reflection to the crack tip position as shown in Figures 7, 8 and 9. These patterns can be found in a good consistency with original full matrix focusing method, provided triangular matrix focusing method and provided the trapezoidal matrix focusing method.
In order to analyze consistency of original full matrix focusing method, provided triangular matrix focusing method and provided the trapezoidal matrix focusing method, Ascan signals of sidedrilled holes and oblique cracks are shown in Figure 10. It can be seen that amplitude and phase of the synthesized Ascan signals are in good consistency with the three matrix focusing methods.
The signaltonoise ratio (SNR) of echo signal are calculated separately. The peak value of the signal is regarded as V_{s}, and the average amplitude of noise as V_{n}. Then the SNR is calculated according to the signal peak value and the average amplitude of noise using Eq. (10).
Echo signal SNR of oblique cracks and sidedriller holes are shown in Figure 11 with original full matrix focusing method, provided triangular matrix focusing method and provided the trapezoidal matrix focusing method. Obviously, the SNR of the triangular matrix focusing algorithm is almost the same as that of the full matrix focusing algorithm. And the slightly difference is found in that of the trapezoidal matrix focusing algorithm. As decreasing the number of Ascan data (k from 16 to 12), SNR of the trapezoidal matrix is decreased.
The computational efficiencies of original full matrix focusing method, provided triangular matrix focusing method and provided the trapezoidal matrix focusing method are figured out as shown in Figure 12. Bscan image with 115Ã—150 pixels is used to compare the imaging time with the three methods. It can be seen that the full matrix focusing is a timeconsuming method. The calculation time of the triangular matrix focusing method is cut in half to that of full matrix focusing method. The trapezoidal matrix focusing method spends shorter time as decreasing the number of Ascan data (k from 16 to 12). Therefore, the computation efficiency can be improved greatly with the help of both provided triangular and trapezoidal matrix algorithms.
4 Conclusions

(1)
Based on the reciprocity principle, the triangular matrix focusing algorithm is provided to simplify the full matrix focusing method. Furthermore, the trapezoidal matrix focusing algorithm is proposed based on the signal energy weight.

(2)
The experimental results of sidedrilled holes and oblique cracks show that, three imaging methods keep a great agreement in both Bscan images and synthesized Ascan signals. The signaltonoise ratio of the triangular matrix is almost same as full matrix focusing algorithm. And the signaltonoise ratio of the trapezoidal matrix are slightly lower as deceasing the amount of Ascan data.

(3)
The triangular and trapezoidal matrix algorithms can achieve the good imaging quality and increase the computation efficiency greatly.
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Acknowledgements
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Funding
Supported by the National Natural Science Foundation of China (Grant No. 51905070).
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XZ was in charge of the whole trial; ZM wrote the manuscript; JZ assisted with sampling and laboratory analyses and optimized the manuscript. All authors read and approved the final manuscript.
Authorsâ€™ Information
Xinyu Zhao, born in 1979, is currently an associate professor at School of Materials Science and Engineering, Dalian Jiaotong University, China. He received his Ph.D. degree in materials processing and engineering from Harbin Institute of Technology, China, in 2009. His research interests include welding, nondestructive testing and evaluation technology and equipment.
Zemin Ma, born in 1995, is currently a master candidate at School of Materials Science and Engineering, Dalian Jiaotong University, China.
Jiaying Zhang, born in 1988, is currently an associate professor at School of Materials Science and Engineering, Dalian Jiaotong University, China. She received her Ph.D. degree in materials processing and engineering from Harbin Institute of Technology, China, in 2018. Her research interests include nondestructive testing and evaluation technology.
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Zhao, X., Ma, Z. & Zhang, J. Simplified Matrix Focusing Imaging Algorithm for Ultrasonic Nondestructive Testing. Chin. J. Mech. Eng. 35, 19 (2022). https://doi.org/10.1186/s10033022006828
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DOI: https://doi.org/10.1186/s10033022006828
Keywords
 Ultrasonic NDT
 Phased array
 Full matrix focusing
 Imaging algorithms