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Resolution Enhancement in Ultrasonic TOFD Imaging by Combining Sparse Deconvolution and Synthetic Aperture Focusing Technique (SparseSAFT)
Chinese Journal of Mechanical Engineering volumeÂ 35, ArticleÂ number:Â 94 (2022)
Abstract
The shallow subsurface defects are difficult to be identified and quantified by ultrasonic timeofflight diffraction (TOFD) due to the low resolution induced by pulse width and beam spreading. In this paper, SparseSAFT is proposed to improve the time resolution and lateral resolution in TOFD imaging by combining sparse deconvolution and synthetic aperture focusing technique (SAFT). The mathematical model in the frequency domain is established based on the l_{1} and l_{2} norm constraints, and the optimization problem is solved for enhancing time resolution. On this basis, SAFT is employed to improve lateral resolution by delayandsum beamforming. The simulated and experimental results indicate that the lateral wave and tipdiffracted waves can be decoupled with SparseSAFT. The shallow subsurface defects with a height of 3.0 mm at the depth of 3.0 mm were detected quantitatively, and the relative measurement errors of flaw heights and depths were no more than 10.3%. Compared to conventional SAFT, the time resolution and lateral resolution are enhanced by 72.5 and 56% with SparseSAFT, respectively. Finally, the proposed method is also suitable for improving resolution to detect the defects beyond dead zone.
1 Introduction
Ultrasonic testing is widely used to inspect material properties and estimate defects quantitatively [1, 2]. Timeofflight diffraction (TOFD) is an ultrasonic testing technique with high inspection speed and accurate size measurement, and Bscan image is the common mode for presenting detection results [3,4,5,6,7]. However, ultrasonic signals are received in the wide range due to beam spreading, inducing that the tipdiffracted waves are displayed as hyperbolic arcs in the Bscan image. Meanwhile, the detectable range by TOFD is more than 7.0 mm in general [8]. The shallow subsurface defects located in dead zone are hard to be identified because of the superposition between lateral wave and tipdiffracted waves [8,9,10]. Therefore, it is necessary to improve lateral resolution and time resolution in TOFD detection image by combining signal processing techniques.
The deconvolution techniques are beneficial to improving time resolution. Compared to conventional Weiner filter, the Weiner filter followed by autoregressive spectral extrapolation is more suitable to separate overlapped signals by the specific combination of multiple frequency windows [11, 12]. This method was employed by ultrasonic TOFD technique to quantitatively evaluate the nearsurface notch with 3.0 mm depth [13]. In contrast, the sparsity reflecting the limited number of spikes in reflection sequence is the powerful prior information in ultrasonic testing. Some sparsitybased methods are developed to obtain the sparse signals with high time resolution by introducing l_{1}norm constraint rather than extending the frequency band [14,15,16,17,18,19]. For example, the sparsity constraint is imposed on the overlapped TOFD signals from defect, realizing the improvement of time resolution and the quantitative estimation of 0.5 Î¼s time difference [18].
The delayandsum (DAS) beamforming algorithms, e.g., synthetic aperture focusing technique (SAFT), are wellknown for improving the lateral resolution in ultrasonic imaging [20]. Meanwhile, SAFT is employed to enhance signaltonoise ratio (SNR) and quantify defects by imaging the ultrasonic signals with low directivity [21,22,23,24,25,26]. Based on the relative position between defect and TOFD probes [27], the lateral resolution of Bscan images is improved to identify defects by using SAFT. On this basis, the combination of SAFT and phase coherence imaging (PCI) provides better lateral resolution and SNR level [28]. It should be noted that the methods mentioned above focus on the enhancement of lateral resolution or time resolution. When the tobedetected defects are located in the dead zone, the ideal lateral resolution and time resolution are difficult to be obtained by the single method simultaneously. It is necessary to combine different methods for realizing resolution enhancement.
In this paper, SparseSAFT is proposed to enhance the resolution in TOFD imaging. First, the sparsity of reflection sequence is applied to processing the TOFD signals. Then, the l_{1} and l_{2} norm constraints are combined to construct the objective function in the frequency domain, weakening the influence of the pulse width and beam spreading of original signals. Finally, SAFT is used to obtain the highresolution image, and to size the defects in dead zone. Section 2 presents the theories of TOFD inspection and SparseSAFT. The simulated and experimental results are described in Section 3. Section 4 is the discussion, and the conclusions are presented in Section 5.
2 Theories
2.1 TOFD Inspection
The schematic diagram of ultrasonic TOFD inspection is presented in Figure 1(a), where the transmitter (T) and receiver (R) are a pair of probes connected with the wedges having the same angle [28,29,30,31]. The received signal y(n) is modeled as
where * represents convolution, w(n) is input pulse, r(n) is reflection sequence, and n(n) is the noise.
The Bscan is obtained by using the TOFD probes scanning with a step of Î”s from position 1 to position M, where the probe center spacing (PCS) is 2S. The point coordinates of transmitter and receiver at position i (1â‰¤iâ‰¤M) are respectively defined as (iÎ”s, 0) and (iÎ”s+2S, 0), and the corresponding received signal is denoted as y^{i}(n). Therefore, the Bscan image, i.e., signal matrix B_{raw}=[y^{1}(n), â€¦, y^{i}(n), â€¦, y^{M}(n)], is consisted of Mnumber Ascan signals, as schematically presented in Figure 1(b).
As shown in Figure 1(b), the received signal y^{i}(n) contains lateral wave and tipdiffracted waves. There is a nearsurface dead zone with a depth of D due to the pulse width of lateral wave, as given by
where t_{p} is the pulse width of lateral wave, and c_{l} is the longitudinal wave velocity in specimen.
The dead zone leads to the difficulty in calculating the defect depth d and height h with Eqs. (3) and (4).
where Î”t_{1} and Î”t_{2} are the time differences between the lateral wave and the diffracted waves from upper and lower tips, respectively.
In addition, the diffracted waves from defect tips are displayed as hyperbolic arcs in the Bscan image presented in Figure 1(b). The low lateral resolution influences the quantitative estimation of defect depth and size.
2.2 SparseSAFT
In this paper, SparseSAFT is proposed to enhance resolution by combining sparse deconvolution and SAFT. First, the TOFD inspection model is converted into the mathematical model in the frequency domain to construct an objective function.
The expression of Eq. (1) in the frequency domain is given by
where Y(f), R(f), W(f), and N(f) are the corresponding frequency spectrums of y(n), r(n), w(n), and n(n), respectively.
The frequency spectrum R(f) is estimated by Wiener filter.
where W^{*} is the conjugate of W, the noise desensitizing factor Q is often approximated as 0.01max(W(f)^{2}) [11, 19].
Meanwhile, the discrete reflection sequence r(n) can also be expressed as Eq. (7) [11, 19].
where r_{i} is the amplitude at the ith point; N is the length of r(n), and t_{i} is the corresponding time interval between the ith point and the 1st point.
The expression of Eq. (7) in the frequency domain is given by
Therefore, Eq. (9) is deduced by combining Eqs. (6) and (8).
Eq. (9) is rearranged as Eq. (10) according to the equivalence of the real parts and imaginary parts on both sides.
where A=[real(R(f_{1})),â€¦, real(R(f_{N})), img(R(f_{1})), â€¦, img(R(f_{N}))]; C=[cos(2Ï€f_{1}t_{1}),Â â€¦ cos(2Ï€f_{1}t_{N}),â€¦,cos(2Ï€f_{N}t_{1}),Â â€¦,Â cos(2Ï€f_{N}t_{N}),Â sin(2Ï€f_{1}t_{1}),Â â€¦sin(2Ï€f_{1}t_{N}), â€¦, sin(2Ï€f_{N}t_{1}),Â â€¦,Â sin(2Ï€f_{N}t_{N})]; R=[r(t_{1}),â€¦,r(t_{N})]. real{} is the real part, and img{} is the imaginary part. More details are described in Ref. [13]. It is obvious that the problem given by Eq. (10) has multiple solutions.
Subsequently, l_{1} norm constraint is imposed on R based on the sparsity of reflection sequence. The highresolution signal can be obtained by solving the following problem.
The constraint problem in Eq. (11) is transformed into the unconstraint problem in Eq. (12) by combining l_{2} and l_{1} norm constraints.
where Î¼ is the regularization parameter.
On this basis, the interiorpoint method [32] is employed to solve Eq. (12), and the sparse signal matrix B_{sparse} is obtained by processing every Ascan signal in matrix B_{raw}.
Finally, SAFT is implemented on the matrix B_{sparse} to improve lateral resolution. The delay time Î”T_{i} corresponding to position i and an arbitrary point (x, z) in the region of interest is given by
The DAS calculation is performed on every processed Ascan signal X^{i}(n) to obtain the reconstructed image B_{SparseSAFT}(x, z).
The flow chart of SparseSAFT is presented in Figure 2.
3 Simulation and Experiments
3.1 Simulation
Figure 3 presents the schematic diagram of the carbon steel model established in the CIVA software platform, and the longitudinal wave velocity c_{l} in carbon steel is equal to 5890 m/s. The shallow subsurface cracks #1â€“#3 with height h = 3.0 mm were set at the depth d =3.0, 3.5 and 4.0 mm, respectively. TOFD inspection was implemented by using a pair of probes with 5 MHz center frequency, 70Â° refraction angle, and 30.0 mm PCS. Therefore, the depth of dead zone was 6.1 mm for the t_{p}=0.4 Î¼s.
Figures 4(a) and 5(a) present the original Bscan image and Ascan signals corresponding to the three cracks, respectively. On the one hand, the time differences between lateral wave and tipdiffracted waves are difficult to be determined due to signal superposition. On the other hand, the beam spreading induces the tipdiffracted waves presented as hyperbolic arcs in the image, influencing the identification of defect characteristics.
Figures 4(b) and 5(b) show the reconstructed image and Ascan signals by using conventional SAFT, respectively. The lateral resolution is enhanced in comparison with Figure 4(a), but there is no obvious improvement in time resolution. The arrival times of lateral wave and tipdiffracted waves are unable to be obtained from the image and Ascan signals.
On this basis, the SparseSAFT was employed to process the signal matrix B_{raw}. The frequency spectrum R(f) was estimated by Wiener filter, and norm constraints were imposed based on the sparsity for deducing the optimization problem given by Eq. (12). Subsequently, the interiorpoint method was applied to solve the problem, obtaining the sparse signal matrix B_{sparse}. Finally, SAFT was implemented by DAS calculation to reconstruct the highresolution image B_{SparseSAFT}. Figures 4(c) and 5(c) present the reconstructed image and Ascan signals by SparseSAFT. Compared to Figures 4(a) and (b), the time resolution and lateral resolution in Figure 4(c) are both improved significantly, and the lateral wave and tipdiffracted waves are distinguished clearly.
Table 1 lists the crack depths and heights calculated with Eqs. (3) and (4) based on the time differences Î”t_{1} and Î”t_{2} read from Figure 5(c). The relative measurement errors are no more than 9.3%.
3.2 Experiments
Figure 6 presents the carbon steel specimen used for experiments. An artificial defect with a height of 3.0 mm was machined at the depth of 3.0 mm. The longitudinal wave velocity c_{l} was 5890 m/s. Ultrasonic TOFD inspection was implemented using a pair of probes with 5.0 MHz center frequency and 30.0 mm PCS. Therefore, the defect was located in the dead zone (D=6.1mm).
Figure 7 presents the original Bscan image and the reconstructed images by conventional SAFT and SparseSAFT. The Ascan signals corresponding to the defect in reconstructed images are shown in Figure 8. The lateral resolution is improved by conventional SAFT in comparison with the original image. However, there is no enhancement in time resolution, as shown in Figure 8(a). In contrast, the time resolution and lateral resolution are both significantly improved in the reconstructed image by SparseSAFT. Figure 8(b) demonstrates that the lateral wave and tipdiffracted waves are feasible to identify from the decoupled signals. The flaw depth (d=2.83 mm) and height (h=3.31 mm) are calculated with Eqs. (3) and (4) by the time differences Î”t_{1} and Î”t_{2} in Figure 8(b). The relative measurement errors are no more than 10.3%.
4 Discussion
The low time resolution and lateral resolution restrict the identification and quantitation of shallow subsurface defects from TOFD images. In this paper, the l_{1} and l_{2} norm constraints are introduced to solving the frequencydomain optimization problem for improving time resolution, and SAFT is employed to enhance lateral resolution by DAS calculation. It should be noted that the DAS images by conventional SAFT are reconstructed based on the original ultrasonic Ascan signals. The signals with wide pulsewidth contain redundant information of nondefects, resulting in the overlapping of lateral wave and tipdiffracted waves. There is no improvement in time resolution by conventional SAFT, and the arrival times of the lateral wave and tipdiffracted waves are hard to be determined from the reconstructed images. In contrast, SparseSAFT is proposed to decouple overlapped signals and reconstruct images. The pulse widths of lateral wave and tipdiffracted waves are greatly reduced. In addition, the improvement of time resolution for array signals is beneficial to the enhancement of lateral resolution in reconstructed images with the delayandsum beamforming method, e.g., SAFT and TFM [33]. The introduction of sparse deconvolution increases the focusing effects of SAFT. The reconstructed images by SparseSAFT contain less redundant information, realizing the identification and location of nearsurface defects. Therefore, the combination of sparse deconvolution and SAFT is helpful for enhancing the time and lateral resolution of ultrasonic TOFD images and has higher resolution in comparison with conventional SAFT.
Experimental results demonstrate that SparseSAFT has the ability to realize the resolution enhancement in TOFD imaging as shown in Figures 7 and 8. On the one hand, the distinguishable time interval Î”t is as small as 0.11 Î¼s, which is reduced by 72.5% compared to the pulse width t_{p}=0.4 Î¼s. On the other hand, the fullwidth at halfmaximum (FWHM) for the flaw tip in Bscan image is decreased significantly by SparseSAFT. Figure 9 presents the transverse profile graphs corresponding to the lower tip of the artificial defect in the different reconstructed images. The measured FWHM in Figure 9(b) is 44% of that in Figure 9(a) processed by conventional SAFT, i.e., the lateral resolution is improved by 56%.
Simulated and experimental results indicate that SparseSAFT is suitable for identifying shallow subsurface defects by enhancing image resolution. Meanwhile, the small defects beyond dead zone can also be detected by SparseSAFT. Figure 10(a) presents another artificial defect with a height of 3.0 mm at the depth of 11.0 mm in carbon steel specimen. Two 5.0 MHz probes with 52.0 mm PCS are adopted to perform ultrasonic TOFD inspection. The original Bscan image and reconstructed image by SparseSAFT are shown in Figure 10(b) and (c), respectively. The time resolution and lateral resolution are enhanced significantly by SparseSAFT, and the time differences Î”t_{1}=0.76 Î¼s and Î”t_{2}=1.23 Î¼s are obtained from Figure 10(c). The calculated flaw depth and height are respectively equal to 11.0 and 3.2 mm, whose relative errors are no more than 6.7%.
5 Conclusions

(1)
SparseSAFT is proposed by combining sparse deconvolution and SAFT to weaken the influence of pulse width and beam spreading on the time resolution and lateral resolution in TOFD imaging.

(2)
Simulated and experimental results demonstrate that the time resolution and lateral resolution are respectively enhanced by 72.5% and 56% with SparseSAFT compared to conventional SAFT.

(3)
The depths and heights of the defects located in and beyond dead zone are determined by SparseSAFT with no more than 10.3% measurement error.
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Acknowledgements
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Funding
Supported by National Key Research and Development Program of China (Grant No. 2019YFA0709003), National Natural Science Foundation of China (Grant No. 51905079), Liaoning Revitalization Talents Program (Grant No. XLYC1902082).
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XS wrote the draft manuscript and conducted experiment; LL and SJ were in charge of the whole trial. All authors read and approved the final manuscript.
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Xu Sun, is currently a PhD candidate at School of Materials Science and Engineering, Dalian University of Technology, China. His research interests include ultrasonic signal processing.
Li Lin, is currently a Professor at School of Materials Science and Engineering, Dalian University of Technology, China. Her main research interests include nondestructive testing and evaluation for materials.
Shijie Jin, is currently an associate professor at School of Materials Science and Engineering, Dalian University of Technology, China. His research interests include nondestructive testing and evaluation for materials.
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Sun, X., Lin, L. & Jin, S. Resolution Enhancement in Ultrasonic TOFD Imaging by Combining Sparse Deconvolution and Synthetic Aperture Focusing Technique (SparseSAFT). Chin. J. Mech. Eng. 35, 94 (2022). https://doi.org/10.1186/s10033022007683
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DOI: https://doi.org/10.1186/s10033022007683