High-Density Polyethylene Pipe Butt-Fusion Joint Detection via Total Focusing Method and Spatiotemporal Singular Value Decomposition

: High-density polyethylene (HDPE) pipes are widely used for urban natural gas transportation. Pipes are usually welded using the technique of thermal butt fusion, which is prone to manufacturing defects that are detrimental to safe operation. This paper proposes a spatiotemporal singular value decomposition preprocessing improved total focusing method (STSVD-ITFM) imaging algorithm combined with ultrasonic phased array technology for non-destructive testing. That is, the ultrasonic real-value signal data are first processed using STSVD filtering, enhancing the spatiotemporal singular values corresponding to the defective signal components. The TFM algorithm is then improved by establishing a composite modification factor based on the directivity function and the corrected energy attenuation factor by adding angle variable. Finally, the filtered signal data are utilized for imaging. Experiments are conducted by examining specimen blocks of HDPE materials with through-hole defects. The results show the following: the STSVD-ITFM algorithm proposed in this paper can better suppress static clutter in the near-field region, and the average signal-to-noise ratios are all higher than the TFM algorithm. Moreover, the STSVD-ITFM algorithm has the smallest average error among all defect depth quantification results.


Introduction
High-density polyethylene (HDPE) pipes, due to good chemical properties and mechanical strength, such as corrosion resistance, toughness, long service life, etc., have been widely used in urban natural gas supply system [1][2][3].In the process of urban gas transportation, several HDPE pipes need to be connected, and joint welding using the thermal butt-fusion technique is a common means of pipes connection [4,5].Butt-fusion joints are the weakest link of the HDPE pipes, which determine the safety performance of the whole pipeline system.During the butt-fusion welding process of HDPE pipe joints, improper adjustment of welding parameters (e.g., temperature, pressure, and duration of heating, etc.) and contamination of the material surface are prone to result in a number of manufacturing defects [6][7][8].These manufacturing defects include through-holes, inclusions, and lack of fusion.In addition, poor workmanship during on-site installation, repair, and maintenance of piping may also lead to defects [9,10].These defects of butt-fusion joints will increase the risk of structural failure or critical damage to the HDPE pipes, which is detrimental to the sustainability of the whole pipeline systems.It may even lead to gas leakage, resulting in safety accidents and economic losses.In order to safeguard the weld quality of HDPE pipe butt-fusion joints from structural failure, non-destructive evaluation (NDE) using a number of intelligent inspection tools is essential [11,12].Ultrasonic phased array inspection technology is a widely used method in NDE, known for its effectiveness in identifying defects and quantitatively characterizing HDPE materials [7,[13][14][15][16][17].The total focusing method (TFM), based on the full matrix capture (FMC) technique, has now become a golden rule for post-processing of ultrasonic phased array inspection.The TFM algorithm utilizes a delay and sum (DAS) beam synthesis method to focus the ultrasonic signal into a predefined grid region for imaging [18].However, the viscoelastic properties of HDPE materials cause scattering and absorption of ultrasonic waves during propagation, resulting in reduced signal amplitude, lower signal-to-noise ratio (SNR), and decreased quality in TFM images [19].
Singular value decomposition (SVD) has now been shown to be useful in the field of NDE for signal extraction as well as image optimization [20][21][22].After SVD processing, the defect signal is often distinguished from noise.This is due to the fact that the defect signal has a large energy share and corresponds to larger singular values, whereas noise corresponds to smaller singular values [23,24].Zhang et al. [25], in the detection of circular hole defects in noisy metallic materials, utilized high-order SVD to directly process FMC data.Then, the processed signal data are imaged with the ultrasonic reverse time migration imaging method, and the method can effectively suppress the structural noise.Rao et al. [26], in the detection of defects in side-drilled holes of HDPE piping materials, proposed an improved TFM algorithm utilizing FIR filters as well as block-wise SVD processing, which can effectively suppress near-field clutter and background noise.In the field of medical ultrasonic imaging, an advanced spatiotemporal singular value decomposition (STSVD) filtering method is used to optimize images [27].The 3D medical ultrasonic complex-valued signal data are first converted into a 2D Casorati matrix, including the spatial dimension (compounding the depth z-direction of the ultrasonic transducer and the x-direction of the number of samples) and the temporal dimension (temporal sampling), and are then subjected to SVD processing.Based on the difference in spatiotemporal coherence of the signal components, it is important to set both higher-order and lower-order cutoffs.By selecting the most favorable spatiotemporal singular values for reconstruction, it becomes possible to effectively suppress noise and artifacts [28,29].Rao et al. [30] applied the STSVD method to TFM imaging of HDPE piping materials.They initially demodulated the FMC real-valued signal data, converted them to complex-valued data similar to medical ultrasonic signals.Subsequently, they utilized STSVD filtering and implemented imaging with the TFM algorithm.This method is more conducive to noise suppression.However, there are still relatively few studies related to the optimization of TFM images of HDPE piping materials using STSVD.In addition, the directivity function of the array element in the sound field characterizes the mapping between the energy intensity of the ultrasonic waves and the direction of propagation [31].The TFM image is optimized by establishing a compensation factor based on the directivity function, taking into account the energy attenuation of ultrasonic waves.It has a better effect of suppressing the structural noise and improving the SNR in ultrasonic phased array inspection of high-attenuation metal coarse-crystalline materials [32,33].However, there are also fewer studies on the application of this method to ultrasonic phased array inspection of HDPE piping materials.
The contribution is that, to ensure the weld quality of HDPE pipe butt-fusion joints, this paper proposes a spatiotemporal singular value decomposition preprocessing improved total focusing method (STSVD-ITFM) imaging algorithm combined with ultrasonic phased array technology.The algorithm consists of three important steps, the first step is setting the enhancement factor to enhance the spatiotemporal singular values corresponding to the defective signal components when filtering the FMC real-valued signal data directly using STSVD.The second step is improving the TFM pixel point amplitude based on the composite modification factor, which is composed of the directivity factor and the energy attenuation factor corrected by adding the angle variable.Finally, the STSVD-filtered FMC data matrix is then processed for the improved TFM imaging.To evaluate the effectiveness of the algorithm proposed in this study, experiments are carried out on a test block made of HDPE material.The imaging results are then compared with the TFM algorithm, the STSVD processing TFM algorithm (STSVD-TFM), and the improved TFM (ITFM) algorithm introduced in this paper.
The remainder of this paper is structured as follows: The second section describes the theory of the algorithms, including the overall architecture of FMC data, the principle of STSVD filtering, the principle of the ITFM algorithm, the STSVD-TFM algorithm, and the implementation of the ITFM algorithm.The third section includes the experiments, including the HDPE test block and experimental equipment.It also mentions the extraction of ultrasonic signals from specific transmit-receive array element combinations in the FMC data, filtering them with STSVD.The fourth section includes the imaging results, comparing and analyzing images generated using the four algorithms with relevant imaging data.The final section includes the conclusions.

The Ultrasound Phased Array Data Based on FMC
The FMC technique is a more advanced way of acquiring signal data based on ultrasonic phased array probes.In a one-dimensional linear ultrasonic array with N array elements, the FMC technique sequentially excites each array element and receives the ultrasonic signal from all of them.This process follows a one-transmit-one-receive mode until the Nth array element both excites and receives the ultrasonic signal [34].The process is shown in Figure 1.
evaluate the effectiveness of the algorithm proposed in this study, experiments are out on a test block made of HDPE material.The imaging results are then compare the TFM algorithm, the STSVD processing TFM algorithm (STSVD-TFM), and t proved TFM (ITFM) algorithm introduced in this paper.
The remainder of this paper is structured as follows: The second section descri theory of the algorithms, including the overall architecture of FMC data, the princ STSVD filtering, the principle of the ITFM algorithm, the STSVD-TFM algorithm, a implementation of the ITFM algorithm.The third section includes the experime cluding the HDPE test block and experimental equipment.It also mentions the ext of ultrasonic signals from specific transmit-receive array element combinations FMC data, filtering them with STSVD.The fourth section includes the imaging comparing and analyzing images generated using the four algorithms with releva aging data.The final section includes the conclusions.

The Ultrasound Phased Array Data Based on FMC
The FMC technique is a more advanced way of acquiring signal data based on sonic phased array probes.In a one-dimensional linear ultrasonic array with N arr ments, the FMC technique sequentially excites each array element and receives th sonic signal from all of them.This process follows a one-transmit-one-receive mod the Nth array element both excites and receives the ultrasonic signal [34].The pro shown in Figure 1.The acquired ultrasonic signal is represented by a 3D real-valued matrix of sion (N, N, NS), denoted as u(xt, xr, tNs).Where xt and xr are the positions of the tran and receiver of the array element, respectively.t is the propagation time, and N number of sampling points.Then, the corresponding time series signal data at a moment can be expressed as follows: The 3D distribution of the FMC data is shown in Figure 2. The acquired ultrasonic signal is represented by a 3D real-valued matrix of dimension (N, N, N S ), denoted as u(x t , x r , t Ns ).Where x t and x r are the positions of the transmitter and receiver of the array element, respectively.t is the propagation time, and N S is the number of sampling points.Then, the corresponding time series signal data at a given moment can be expressed as follows: The 3D distribution of the FMC data is shown in Figure 2.

STSVD Filtering Processing
The biggest difference between the STSVD and the conventional SVD process that the STSVD focuses on the 2D Casorati matrix consisting of spatial and tempora ples.The data acquired using the FMC technique can be considered as a coupling ultrasonic signal in the spatial and temporal dimensions, so the FMC data matrix is formed into a two-dimensional Casorati matrix of dimension S [27,29].Processi matrix S using STSVD: where U = [u1, u2, …, ui, …, uN 2 ] is the left singular vector of the matrix S, also know spatial singular vector, it has dimension (N 2 , N 2 ) and any ui denotes a column vec , , , , , and arranged in descending order, r is the rank of the matrix S the rest of the elements are 0. V = [v1, v2, …, vi, …, vNs] is the right singular vector matrix S, also known as the temporal singular vectors of dimension (NS, NS), and denotes a column vector of dimension NS × 1.This is based on the key assumptions that the components of the ultrasonic sign not uniformly sensitive to spatiotemporal coherence and that the spatiotemporal c ence of the signal can be characterized by temporal singular vectors V. Therefore, s temporal filtering of signal data using STSVD makes the enhancement of defects a as the suppression of noise possible.The static clutter in the near-field region exhibit amplitude and low spatiotemporal coherence, which corresponds to lower-order sin values.In contrast, noise exhibits low amplitude and higher spatiotemporal cohe corresponding to higher order singular values [28,30].The spatiotemporal filtering o nal is realized by setting low-order and high-order cutoffs to filter out the singular v and singular vectors of a defective signal.The cutoff is chosen based on the singular order in the singular value spectrum.
Filtering the singular values corresponding to defective signals based on the order and high-order cutoff: where I ST is a diagonal matrix whose elements consist of 0 and 1, in which the elem between the low-order cutoff rst and the high-order cutoff rgt are 1 and the rest are

STSVD Filtering Processing
The biggest difference between the STSVD and the conventional SVD processing is that the STSVD focuses on the 2D Casorati matrix consisting of spatial and temporal samples.The data acquired using the FMC technique can be considered as a coupling of the ultrasonic signal in the spatial and temporal dimensions, so the FMC data matrix is transformed into a two-dimensional Casorati matrix of dimension S [27,29].Processing of matrix S using STSVD: where is the left singular vector of the matrix S, also known as a spatial singular vector, it has dimension (N 2 , N 2 ) and any u i denotes a column vector of dimension N 2 × 1. ∆ is an affine matrix of dimension (N 2 , N S ) with diagonal elements and arranged in descending order, r is the rank of the matrix S, and the rest of the elements are 0. V = [v 1 , v 2 , . .., v i , . .., v Ns ] is the right singular vector of the matrix S, also known as the temporal singular vectors of dimension (N S , N S ), and any v i denotes a column vector of dimension N S × 1.This is based on the key assumptions that the components of the ultrasonic signal are not uniformly sensitive to spatiotemporal coherence and that the spatiotemporal coherence of the signal can be characterized by temporal singular vectors V. Therefore, spatiotemporal filtering of signal data using STSVD makes the enhancement of defects as well as the suppression of noise possible.The static clutter in the near-field region exhibits high amplitude and low spatiotemporal coherence, which corresponds to lower-order singular values.In contrast, noise exhibits low amplitude and higher spatiotemporal coherence, corresponding to higher order singular values [28,30].The spatiotemporal filtering of signal is realized by setting low-order and high-order cutoffs to filter out the singular values and singular vectors of a defective signal.The cutoff is chosen based on the singular value order in the singular value spectrum.
Filtering the singular values corresponding to defective signals based on the low-order and high-order cutoff: where I ST is a diagonal matrix whose elements consist of 0 and 1, in which the elements between the low-order cutoff r st and the high-order cutoff r gt are 1 and the rest are 0. To enhance the strength of the signal of the defect, set the enhancement factor µ. Figure 3 shows the singular value spectrum of FMC data of an HDPE material after STSVD processing.The positions of r st and r gt are labeled, and the matrix ∆ ST that corresponding to the selected defective signal is composed of the singular values between them.The red curve represents the singular value matrix ∆ ST µ enhanced by µ.It is worth noting that, unlike the processing done in the literature [30], this paper utilizes STSVD to process the FMC real-valued data directly, which leads to a larger difference between r st and r gt , but does not have too much effect on the filtered results (see Sections 3.2 and 4.1 below for details).
curve represents the singular value matrix ( ) Δ enhanced by μ .It is that, unlike the processing done in the literature [30], this paper utilizes STS the FMC real-valued data directly, which leads to a larger difference betw but does not have too much effect on the filtered results (see Sections 3.3 and details).Then, the inverse STSVD calculation is performed to obtain the filtered trix S ST : ( ) Finally, the spatiotemporally filtered 2D Casorati matrix S ST is conve FMC data matrix u ST (xt, xr, tNs) with dimensions (N, N, NS).

Improved Total Focusing Method
The TFM algorithm is derived from the beam synthesis method DAS time-delayed superposition of the acquired FMC ultrasonic signal data, whi thesized and focused to produce an imaging effect in a specified grid region in Figure 4 [18].The pixel amplitude ITFM(x, z) of any imaging point P(x, z) be expressed as follows: , , , where H denotes the Hilbert transform.Then, the pixel point amplitude ob forming the TFM operation on the FMC data that has been processed usi filtering is as follows: , , , When array elements excite ultrasonic waves in the sound field, there of mapping between its energy intensity and the propagation direction of th which can be characterized by the directivity function.Then, build the dir Dt,r(x, z) based on the following function [31,33]: Then, the inverse STSVD calculation is performed to obtain the filtered Casorati matrix S ST : Finally, the spatiotemporally filtered 2D Casorati matrix S ST is converted into a 3D FMC data matrix u ST (x t , x r , t Ns ) with dimensions (N, N, N S ).

Improved Total Focusing Method
The TFM algorithm is derived from the beam synthesis method DAS.It involves a timedelayed superposition of the acquired FMC ultrasonic signal data, which is then synthesized and focused to produce an imaging effect in a specified grid region, as illustrated in Figure 4 [18].The pixel amplitude I TFM (x, z) of any imaging point P(x, z) in the grid can be expressed as follows: where H denotes the Hilbert transform.Then, the pixel point amplitude obtained by performing the TFM operation on the FMC data that has been processed using the STSVD filtering is as follows: When array elements excite ultrasonic waves in the sound field, there is some kind of mapping between its energy intensity and the propagation direction of the sound beam, which can be characterized by the directivity function.Then, build the directivity factor D t,r (x, z) based on the following function [31,33]: where d 0 is the width of a single array element.θ t,p and θ r,p are the angles between the array elements t, r, and P(x, z) in the OZ direction of the coordinate axis.And λ denotes the wavelength.
( ) , Then, the pixel point amplitude obtained by performing the ITFM ope FMC data that have been processed using the STSVD filtering is as follows:

Experiments
In this paper, an HDPE plate test block is selected for ultrasonic phase detection experiments, which is made of PE100, and its specification is 400 m × 80 mm.The physical and structural diagrams of the test block are show The HDPE test blocks contained three sizes of through-hole defects located to 50 mm in depth.There are 9 defects each of 1 mm and 2 mm diameter, a The attenuation of ultrasonic energy is related to the distance from the array element to the imaging point, so the energy attenuation factor can be approximated as follows [33]: where d t,P and d r,P are the distance between the array elements t, r, and P(x, z), respectively.From the directivity function, it can be seen that the attenuation of ultrasonic energy is also related to the direction of propagation of the acoustic beam.The cosine of the angle between the array element and the imaging point serves as an angular variable for correcting E t,r (x, z): where E t,r ′ (x, z) is the corrected energy attenuation factor.c is the speed of ultrasonic waves propagating in the HDPE materials.The composite modification factor M t,r (x, z) is established based on D t,r (x, z) and E t,r ′ (x, z): The improved pixel point amplitude matrix I ITFM (x, z) is obtained by weighting I TFM (x, z) by M t,r (x, z): Then, the pixel point amplitude obtained by performing the ITFM operation on the FMC data that have been processed using the STSVD filtering is as follows:

Experiments
In this paper, an HDPE plate test block is selected for ultrasonic phased array defect detection experiments, which is made of PE100, and its specification is 400 mm × 40 mm × 80 mm.The physical and structural diagrams of the test block are shown in Figure 5.The HDPE test blocks contained three sizes of through-hole defects located from 10 mm to 50 mm in depth.There are 9 defects each of 1 mm and 2 mm diameter, and the depth interval between adjacent defects is 5 mm.Defects with a diameter of 3 mm contain 5 defects, with a depth interval of 10 mm between adjacent defects.In this paper, the one-dimensional linear ultrasound array is used for the detection experiments, and the placement of the phased array probe is shown in Figure 5a.Its associated acquired and processed ultrasonic parameters are shown in Table 1.In this paper, the one-dimensional linear ultrasound array is used for the detection experiments, and the placement of the phased array probe is shown in Figure 5a.Its associated acquired and processed ultrasonic parameters are shown in Table 1.

A-Scan Signal Analysis
The FMC data contain the ultrasonic echo A-Scan signal from all combinations of transmit-receive arrays that carry complete information about the defects.Figure 6 demonstrates the ultrasonic echo signal of the (16, 48) array element combination (the 16th array element excites the signal and the 48th array element receives) for 1 mm diameter throughhole defects in the HDPE test block.Figure 6 shows the time-domain information of this echo signal, revealing significant static clutter in the ultrasonic near-field area with a wide range of amplitude.The presence of through-hole defects can be observed based on the echo of the ultrasonic signal.Particularly, the adverse impact of interference on through-hole defects no. 1 and no. 2 in the near-field region due to its proximity to the probe can be observed.This interference leads to a degradation of the echo signal from the defective area.However, in the no. 3 to no. 5 through-hole defects, which are situated far from the near-field region, the echo signal is significantly heightened, thus aiding in the identification of the defects.As the depth of defects increases, the viscoelastic attenuation of the HDPE test block also increases.This results in a decrease in the amplitude of the echo signal for no.6 and the later through-hole defects.Specifically, the amplitude decrease is more significant for the no.7, no. 8, and no. 9 through-hole defects.In the case of the no. 9 through-hole defect, the echo signal is completely overshadowed by surrounding noise.
Processes 2024, 12, 1267 of this echo signal, revealing significant static clutter in the ultrasonic near-field a a wide range of amplitude.The presence of through-hole defects can be observe on the echo of the ultrasonic signal.Particularly, the adverse impact of interfer through-hole defects no. 1 and no. 2 in the near-field region due to its proximit probe can be observed.This interference leads to a degradation of the echo sign the defective area.However, in the no. 3 to no. 5 through-hole defects, which are far from the near-field region, the echo signal is significantly heightened, thus a the identification of the defects.As the depth of defects increases, the viscoelastic ation of the HDPE test block also increases.This results in a decrease in the ampl the echo signal for no.6 and the later through-hole defects.Specifically, the am decrease is more significant for the no.7, no. 8, and no. 9 through-hole defects.In of the no. 9 through-hole defect, the echo signal is completely overshadowed by su ing noise.

STSVD Signal Filtering
The key to filtering the FMC signal data using STSVD lies in the selection of order cutoff rst and high-order cutoff rgt.rst corresponds to the signal component w spatiotemporal coherence, which controls the effective suppression of static clutte responds to the signal component with high spatiotemporal coherence, which ens integrity of the defective signal.To verify the performance of rst for the suppre static clutter and rgt for the ability to characterize the integrity of the defects, this designed to filter the ultrasonic echo signal using a (16, 48) array element combina 1 mm diameter through-hole defects with different values.The signal filtering is

STSVD Signal Filtering
The key to filtering the FMC signal data using STSVD lies in the selection of the low-order cutoff r st and high-order cutoff r gt .R st corresponds to the signal component with low spatiotemporal coherence, which controls the effective suppression of static clutter.R gt corresponds to the signal component with high spatiotemporal coherence, which ensures the integrity of the defective signal.To verify the performance of r st for the suppression of static clutter and r gt for the ability to characterize the integrity of the defects, this paper is designed to filter the ultrasonic echo signal using a (16, 48) array element combination for 1 mm diameter through-hole defects with different values.The signal filtering is shown in Figures 7 and 8.  Figure 7 illustrates the signal filtering for r st = 5, 10, 15, and 20, respectively.Only the singular values before the value of r st are eliminated, and the remainder are used to reconstruct the matrix ∆ ST .By comparing with the original ultrasonic echo signal (Figure 6), the static clutter amplitude is reduced in Figure 7, and it can be seen that the low-order cutoff is beneficial to static clutter suppression.Notably, in Figure 7c,d, the amplitudes of the no. 1 and no. 2 through-hole defects are significantly improved, which improves the recognition of the defects.Figure 8 illustrates the filtering with the values of r gt set to 250, 300, 350, and 400, respectively.The reconstruction of the matrix ∆ ST is performed by retaining the singular values between the first value and the r gt value.Loworder singular values are retained, resulting in static clutter that is not well suppressed.Changing the selection of higher-order singular values affects the integrity of the defects and their surrounding noise.The high value of r gt allows a large number of singular values containing noisy information to be used for matrix ∆ ST reconstruction.
Setting the r st value to 21 and the r gt value to 387 for reconstructing the singular value matrix ∆ ST , the obtained STSVD filtered processed (16,48) array element combination ultrasonic echo signal is shown in Figure 9, where the blue curve represents the echo signal after filtering using the singular value matrix ∆ ST intercepted by r st and r gt .The red curve represents the echo signal after filtering using the singular value matrix ∆ ST µ enhanced by µ.As can be seen from Figure 9, both filtering methods have better suppression of static clutter in the near-field region compared to the original echo signal (Figure 6).Each defect can be completely characterized, and both can increase the amplitude of the no. 1 and no. 2 through-hole defects.In addition, the singular value matrix ∆ ST µ enhanced by µ can improve the overall signal amplitude level without losing signal components when reconstructing the echo signal.Δ .By comparing with the original ultrasonic echo signal (Figure 6), the static clutter amplitude is reduced in Figure 7, and it can be seen that the low-order cutoff is beneficial to static clutter suppression.Notably, in Figure 7c,d, the amplitudes of the no. 1 and no. 2 through-hole defects are significantly improved, which improves the recognition of the defects.Figure 8

Comparison of Imaging Results
The imaging results of different algorithms for the HDPE test block with 1 mm, 2  (16,48) combination of array elements ultrasonic echo signal using STSVD filtering.The blue curve represents filtering using the singular value matrix ∆ ST , and the red curve represents filtering using the singular value matrix ∆ ST µ .Furthermore, the locations of the defects are labeled in the figure.

Comparison of Imaging Results
The imaging results of different algorithms for the HDPE test block with 1 mm, 2 mm, and 3 mm diameter through-hole defects are shown in Figure 10, Figure 11, and Figure 12, respectively.It should be clarified that the ∆ ST µ processing is applied in all the filtering of FMC inspection data using STSVD, which utilizes µ to enhance the filtered defective signal singular value matrix.In the imaging of through-hole defects with 1 mm diameter, r st is set to 21 and r gt to 387.In the imaging of through-hole with 2 mm and 3 mm diameter, r st is set to 10 and r gt to 360.For the setting of low-order and high-order cutoff, it is only applied for the HDPE test block in this paper.the imaging results for a 1 mm diameter through-hole defects shown in Figure 10, the conventional TFM algorithm (Figure 10a) can present the locations of all defects.Due to the viscoelastic attenuation property of HDPE materials, the defect amplitude gradually decreases with depth, which is more obvious with the no.6 to no. 9 through-hole defects.There is a large number of static clutter in the near-field region of higher amplitude at the top of the image, which is consistent with the ultrasonic echo signal demonstrated in Figure 6.In addition, the no. 1 through-hole defect is relatively heavily contaminated by noise due to its proximity to static clutter.Figure 10b shows the imaging result of the TFM algorithm after the STSVD filtering process, and it can be seen that the static clutter amplitude at the top of the image is reduced.However, the no. 1 through-hole defect is still contaminated with noise and the amplitude of the no.7 to no. 9 through-hole defects is substantially lower than that of the TFM Image. Figure 10c shows the imaging result of the ITFM algorithm with lower Static clutter amplitude than the TFM and STSVD-TFM images.The no. 5 to no. 9 through-hole defects amplitudes are elevated, especially for the no.7 to no. 9 defects where the amplitude is more significantly elevated compared to the TFM and STSVD-TFM images.The ITFM algorithm considers the impact of directivity and ultrasonic energy attenuation, and in this way improves the TFM algorithm.Nevertheless, numerous artifacts emerge below a depth of approximately 35 mm, due to the boost of the amplitude of the defects at that depth, which also causes the amplitude of the background noise to rise to a high level.Figure 10d shows the imaging results of applying STSVD filtering processing to the ITFM algorithm.The STSVD-ITFM algorithm reduces the amplitude of the static clutter on the basis of the ITFM image, which is better suppressed.It also reduces the background noise at a depth of about 35 mm in the ITFM image and suppresses artifacts.Relatively, the no.6 to no. 9 through-hole defects' amplitudes at this depth are also lower than the ITFM image.However, these defects' amplitudes are still higher than TFM and STSVD-TFM images.

Analysis of Relevant Imaging Data
The pixel point amplitudes of three sizes of defects at a 50 mm depth location und different algorithms are extracted and depicted as curves for comparison, as shown Figure 13.As expected, the ITFM algorithm processed images with the highest defect pixel point amplitudes.Due to the presence of a large number of artifacts in the image this depth, the pixel point amplitude in the non-defective regions is higher than the r of the image.The STSVD-ITFM algorithm reduces the pixel point amplitude in the no defective regions of the ITFM image.The defective region amplitude is higher than STSVD-TFM image and is similar to the TFM image.The overall level of pixel point a plitude in the 1 mm defective region is higher than that of the TFM image.The STSV TFM algorithm reduces the pixel point amplitude of non-defective regions in the TF image to a significantly lower level.This reduction also results in an overall lower pi point amplitude of the defective region at that depth compared to the other three imag Figures 11 and 12 show the imaging of 2 mm and 3 mm diameter through-hole defects, where the performance of the imaging algorithms is nearly identical to the 1 mm diameter imaging results.Conventional TFM images (Figures 11a and 12a) still have higher amplitude static clutter above them, and the defect amplitude gradually decreases with increasing depth.The STSVD-TFM algorithm reduces the amplitude level of the static clutter, but the amplitude of the no.7 to no. 9 through-hole defects in Figure 11b and the no. 4 to no. 5 defects in Figure 12b are lower than the corresponding TFM images.The ITFM algorithm effectively reduces static clutter while increasing the amplitude level of defects at deeper locations.However, it also increases the amplitude of background noise surrounding the defects.A large number of artifacts began to appear at a depth of about 43 mm (Figures 11c and 12c).The STSVD-ITFM algorithm can enhance the advantage of suppressing static clutter based on the ITFM algorithm, while improving the disadvantage that the ITFM algorithm can introduce a lot of noise.Although the amplitude of defects at deeper locations is reduced, it is still higher than the STSVD-TFM image.
In summary, the conventional TFM algorithm is vulnerable to static clutter in the near-field region and viscoelastic attenuation.While the STSVD filtering can decrease the amplitude of static clutter, it is important to note that the amplitude of defects in deeper locations affected by viscoelastic attenuation should be lower.The ITFM algorithm shows better suppression of static clutter as well as strong carry fading ability but is less robust to background noise.The STSVD-ITFM algorithm provides the best suppression of static clutter and improves the drawback of the ITFM algorithm's poor robustness to background noise.However, the fading resistance is weaker than the ITFM algorithm, yet still higher than the STSVD-TFM algorithm.

Analysis of Relevant Imaging Data
The pixel point amplitudes of three sizes of defects at a 50 mm depth location under different algorithms are extracted and depicted as curves for comparison, as shown in Figure 13.As expected, the ITFM algorithm processed images with the highest defective pixel point amplitudes.Due to the presence of a large number of artifacts in the image at this depth, the pixel point amplitude in the non-defective regions is higher than the rest of the image.The STSVD-ITFM algorithm reduces the pixel point amplitude in the non-defective regions of the ITFM image.The defective region amplitude is higher than the STSVD-TFM image and is similar to the TFM image.The overall level of pixel point amplitude in the 1 mm defective region is higher than that of the TFM image.The STSVD-TFM algorithm reduces the pixel point amplitude of non-defective regions in the TFM image to a significantly lower level.This reduction also results in an overall lower pixel point amplitude of the defective region at that depth compared to the other three images.To quantitatively evaluate the imaging performance of each algorithm and to measure the image quality, comparisons are made by calculating the signal-to-noise ratio (SNR).The SNR is calculated as follows [34]: where Imax is the maximum amplitude of the defective region and Iaverage is the average amplitude of the non-defective region.The higher the SNR value, the better the quality of the image.The sum of the SNR for each defect in Figures 10-12 is calculated based on their pixel point amplitude.This sum is then divided by the total number of defects to find the average SNR.It is worth noting that the average amplitude of the non-defective region in the images is calculated by ignoring all the defect amplitudes and areas.The calculated average SNR results are shown in Figure 14 and Table 2.  To quantitatively evaluate the imaging performance of each algorithm and to measure the image quality, comparisons are made by calculating the signal-to-noise ratio (SNR).The SNR is calculated as follows [34]: SNR = 20 log 10 I max I average (13) where I max is the maximum amplitude of the defective region and I average is the average amplitude of the non-defective region.The higher the SNR value, the better the quality of the image.The sum of the SNR for each defect in Figures 10-12 is calculated based on their pixel point amplitude.This sum is then divided by the total number of defects to find the average SNR.It is worth noting that the average amplitude of the non-defective region in the images is calculated by ignoring all the defect amplitudes and areas.The calculated average SNR results are shown in Figure 14 and Table 2.
Both the ITFM and STSVD-ITFM algorithms show some improvement in average the SNR compared to the remaining two algorithms.The STSVD-ITFM algorithm has the highest average SNR in the imaging detection of 1 mm diameter through-hole defects.The STSVD-ITFM algorithm improves by 2.46 dB compared to the TFM algorithm, by 3.3 dB compared to the STSVD-TFM algorithm, and by 0.30 dB compared to the ITFM algorithm.The ITFM algorithm has the highest average SNR for the detection of 2 mm and 3 mm diameter cross-hole defects.The ITFM algorithm improved by 2.07 dB and 6.81 dB compared to the TFM algorithm, by 3.05 dB and 7.13 dB compared to the STSVD-TFM algorithm, and by 0.41 dB and 1.33 dB compared to the STSVD-ITFM algorithm, respectively.The average SNR of the STSVD-ITFM algorithm is improved by 1.66 dB and 5.48 dB compared to the TFM algorithm and by 1.64 dB and 5.80 dB compared to the STSVD-TFM algorithm, respectively.
where Imax is the maximum amplitude of the defective region and Iaverage is the amplitude of the non-defective region.The higher the SNR value, the better the qu the image.
The sum of the SNR for each defect in Figures 10-12 is calculated based on th point amplitude.This sum is then divided by the total number of defects to find t age SNR.It is worth noting that the average amplitude of the non-defective regio images is calculated by ignoring all the defect amplitudes and areas.The calculat age SNR results are shown in Figure 14 and Table 2.The depth of the defect can be evaluated based on where the largest pixel point magnitude is located within its region.The results of depth quantification of different imaging algorithms for each size defect are shown in Table 3, Table 4, and Table 5, respectively.In the analysis of defect depth quantification, the imaging algorithms exhibit similar average error values, with the STSVD-ITFM algorithm demonstrating the smallest average errors.Specifically, the average error for 1 mm defects is approximately 0.7 mm, while for 2 mm and 3 mm defects, it is around 1.0 mm and 1.3 mm, respectively.It is observed that as the defect size increases, the quantitative error also increases, with the maximum error value remaining below 1.4 mm.

Conclusions
To ensure the weld quality of HDPE pipe butt-fusion joints, a STSVD-TFM imaging algorithm combined with ultrasonic phased array technology is proposed in this paper.FMC real-valued signal data are filtered using STSVD and used for improved ITFM algorithm imaging.Inspection of 1 mm, 2 mm, and 3 mm diameter through-hole defects in a HDPE material test block, distributed in depths of 10 mm to 50 mm, by extracting ultrasonic A-Scan signals of 1 mm diameter defects and comparing the imaging results of each algorithm, it is proved that STSVD is feasible to filter the FMC real-valued signal data directly.The ITFM algorithm can increase the amplitude of defects at deeper locations, but it introduces a lot of clutter and is less robust to background noise.The STSVD-ITFM algorithm has a better ability to suppress static clutter, which improves the disadvantage of the ITFM algorithm's poor robustness to background noise.The defect amplitude at deeper positions is higher than the STSVD-IFM algorithm.The STSVD-ITFM algorithm improves the average SNR by 3.30 dB, 1.66 dB, and 5.48 dB as compared to the TFM algorithm.In addition, the average errors of the STSVD-ITFM algorithm are smaller than the other three algorithms in all defect depth quantification results.The average errors are 0.67 mm, 0.92 mm, and 1.24 mm, respectively.supervision, Q.W.; project administration, Q.W; funding acquisition, Q.W.All authors have read and agreed to the published version of the manuscript.

Figure 1 .
Figure 1.Process of acquiring ultrasonic signal data using FMC technology.

Figure 1 .
Figure 1.Process of acquiring ultrasonic signal data using FMC technology.

Figure 3 .
Figure 3. Singular values spectrum of FMC data of an HDPE material processed us

Figure 3 .
Figure 3. Singular values spectrum of FMC data of an HDPE material processed using STSVD.

Figure 4 .
Figure 4. Schematic of the TFM imaging area.

Figure 4 .
Figure 4. Schematic of the TFM imaging area.

Processes 2024 ,Figure 5 .
Figure 5. HDPE test block physical and structural diagrams: (a) physical diagram and (b) structural diagram.

Figure 5 .
Figure 5. HDPE test block physical and structural diagrams: (a) physical diagram and (b) structural diagram.

Figure 7 .
Figure 7. Filtering of (16, 48) array element combination ultrasonic echo with different r st values 5, 10, 15 and 20 respectively: (a) r st = 5; (b) r st = 10; (c) r st = 15; and (d) r st = 20.The filtering is achieved by reconstructing the matrix ∆ ST using only the singular values after the r st value.

Figure 7
Figure 7 illustrates the signal filtering for rst = 5, 10, 15, and 20, respectively.Only the singular values before the value of rst are eliminated, and the remainder are used to reconstruct the matrixST illustrates the filtering with the values of rgt set to 250, 300, 350, and 400, respectively.The reconstruction of the matrix ST Δ is performed by retaining the singular values between the first value and the rgt value.Low-order singular values are retained, resulting in static clutter that is not well suppressed.Changing the selection of higher-order singular values affects the integrity of the defects and their surrounding noise.The high value of rgt allows a large number of singular values containing noisy information to be used for matrix ST Δ reconstruction.Setting the rst value to 21 and the rgt value to 387 for reconstructing the singular value matrix ST Δ , the obtained STSVD filtered processed (16, 48) array element combination ultrasonic echo signal is shown in Figure 9, where the blue curve represents the echo signal after filtering using the singular value matrix ST Δ intercepted by rst and rgt.The red curve represents the echo signal after filtering using the singular value matrix ( ) ST μ Δ enhanced by μ .As can be seen from Figure 9, both filtering methods have better suppres- sion of static clutter in the near-field region compared to the original echo signal (Figure

Figure 8 .
Figure 8. Filtering of (16, 48) array element combination ultrasonic echo with different r gt values 250, 300, 350 and 400 respectively: (a) r gt = 250, (b) r gt = 300, (c) r gt = 350, and (d) r gt = 400.The filtering is achieved by reconstructing the matrix ∆ ST retaining the singular values between the first value and the r gt value.

Figure 9 .
Figure 9. Processing of (16, 48) combination of array elements ultrasonic echo signal using STSVD filtering.The blue curve represents filtering using the singular value matrix

Figure 9 .
Figure 9. Processing of(16, 48) combination of array elements ultrasonic echo signal using STSVD filtering.The blue curve represents filtering using the singular value matrix ∆ ST , and the red curve represents filtering using the singular value matrix ∆ ST µ .Furthermore, the locations of the defects are labeled in the figure.

Figure 13 .
Figure 13.Pixel point amplitude curves for each defect at 50 mm depth.

Figure 14 .
Figure 14.Comparison results of average SNR for three sizes of defects for each algorithm.

Figure 13 .
Figure 13.Pixel point amplitude curves for each defect at 50 mm depth.

Figure 14 .
Figure 14.Comparison results of average SNR for three sizes of defects for each algorithm

Figure 14 .
Figure 14.Comparison results of average SNR for three sizes of defects for each algorithm.

Table 1 .
Associated acquired and processed ultrasonic parameters for the experiment.

Table 1 .
Associated acquired and processed ultrasonic parameters for the experiment.

Table 2 .
Average SNR for three sizes of defects for each algorithm.

Table
Depth quantitative results of 3 mm diameter through-hole defects (no. 1 to no. 5).