Defect detection of metallic samples by electromagnetic tomography using closed loop fuzzy pid controlled iterative landweber method

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Nondestructive Testing and Evaluation ISSN: (Print) (Online) Journal homepage: www.tandfonline.com/journals/gnte20 Defect detection of metallic samples by electromagnetic tomography using closed-loop fuzzy PID-controlled iterative Landweber method Pu Huang, Xiaofei Huang, Zhiying Li & Yuedong Xie To cite this article: Pu Huang, Xiaofei Huang, Zhiying Li & Yuedong Xie (12 Jan 2024): Defect detection of metallic samples by electromagnetic tomography using closed-loop fuzzy PID-controlled iterative Landweber method, Nondestructive Testing and Evaluation, DOI: 10.1080/10589759.2024.2304256 To link to this article: https://doi.org/10.1080/10589759.2024.2304256 Published online: 12 Jan 2024 Submit your article to this journal Article views: 20 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=gnte20 NONDESTRUCTIVE TESTING AND EVALUATION https://doi.org/10.1080/10589759.2024.2304256 Defect detection of metallic samples by electromagnetic tomography using closed-loop fuzzy PID-controlled iterative Landweber method Pu Huang, Xiaofei Huang, Zhiying Li and Yuedong Xie Key Laboratory of Precision Opto-mechatronics Technology of Education Ministry, School of Instrumentation and Opto-Electronic Engineering, Beihang University, Beijing, China ABSTRACT ARTICLE HISTORY Received 27 September 2023 Electromagnetic tomography (EMT) uses the mutual inductance of Accepted January 2024 the coil to visualise the conductivity distribution of interesting regions Since the conductivity of defects and metal samples are KEYWORDS different, the metal samples with defects can be treated as binary- Electromagnetic valued material distributions This paper investigates the closed- tomography; defect loop fuzzy proportional, integral and derivative (PID)-controlled detection; fuzzy PID iterative Landweber method The whole method includes fuzzy controller; Landweber PID controller, the Landweber reconstruction method, and the method; image Dirichlet-to-Neumann map Specifically, the differential signal reconstruction between the mutual inductance of the coil and the feedback signal is used as the input of the fuzzy PID controller The fuzzy controller can automatically adjust three parameters (Kp, Ki and Kd) of PID controller Subsequently, the output of the PID controller can serve as the input of the Landweber algorithm to reconstruct the distribution of conductivity Furthermore, the Dirichlet-to- Neumann map is used to calculate the mutual inductance, acting as the feedback signal based on the reconstruction conductivity distribution Finally, both the numerical simulation and experi­ ments are applied to verify the proposed method The results indicate that the proposed method can reconstruct the image with a clear edge, and the average correlation coefficient can reach 0.792 Introduction During the service process, metal materials are prone to defects due to corrosion, compression and wear, resulting in potential safety hazards Non-destructive testing can provide effective defect information without damaging the structure of metal mate­ rials [1] Compared to other technologies, electromagnetic non-destructive testing has the advantages of non-contact and high sensitivity Traditional electromagnetic non- destructive testing equipment generally adopts a single sensor, and the sensor or metal plate needs to be moved during testing, which takes a long time and requires the corresponding mechanical devices [2–5] It not only easily causes detection errors but also cannot meet the requirements of real-time performance Compared with the single CONTACT Yuedong Xie yuedongxie@buaa.edu.cn © 2024 Informa UK Limited, trading as Taylor & Francis Group P HUANG ET AL sensor structure, the multi-sensor detection can improve the detection accuracy and efficiency by increasing the number of sensors [6,7] Electromagnetic tomography (EMT) is a kind of non-destructive testing technology, which has the advantages of non-contact, visualisation and fast imaging [8,9] It is widely used in the field of defect detection, multiphase flow measurement, biomedical and other fields The EMT system mainly consists of sensor arrays, the data acquisition system and image reconstruction algorithm In fact, the ultimate goal of EMT technology is to obtain the spatial distribution of materials with different conductivities There are two key problems, i.e the forward problem and the inverse problem, that need to be solved The inverse problem of EMT is to solve the distribution of conductivity according to the measured voltage and prior sensitivity matrix, that is image reconstruction [10–13] Due to the fact that the electromagnetic sensitive field of EMT exhibits nonlinear property, it is difficult to directly solve it Meanwhile, the EMT inverse problem belongs to the Fredholm integral equation of the first kind Its solution is ill-conditioned, which limits the development of EMT In practice, image reconstruction algorithms of EMT technol­ ogy can be divided into iterative and non-iterative algorithms Non-iterative algo­ rithms include linear back projection (LBP), Tikhonov regularisation and truncated singular value decomposition [14–16] The LBP algorithm is simple and has fast imaging speed, but the accuracy of image reconstruction is relatively low It is suitable for online rapid qualitative imaging, but cannot provide accurate quantitative infor­ mation The regularisation method is an effective method to overcome the ill-posed problem of EMT However, the parameters of the regularisation method are selected based on experience The iterative algorithms mainly include Landweber iteration, algebraic reconstruction technique (ART), simultaneous iterative reconstruction tech­ nique (SIRT) and so on [17–19] The ART and SIRT are commonly used algebraic iteration methods, which require a higher number of iterations to achieve better image reconstruction results The Landweber iterative algorithm is based on the principle of steepest descent and is the most commonly used iterative algorithm for solving the inverse problem of EMT However, it requires multiple iterations and has a slow rate of convergence In recent years, Zhang et al investigated the compatible multi-template supervised descent method to monitor the structural information of CFRP (Carbon Fiber Reinforced Polymer) [20] Liu et al designed a novel L-type sensor and three-layer array eddy current sensor combined with LBP method to inspect the defect of the wheel [21,22] Moreover, Liu et al investigated image reconstruction algorithms combining deep learning and optimised fully connected net to learn image reconstruction of EMT [23] Wang et al investigated the sparse regularisation method to improve the EMT image reconstruction quality [24] Ma and Soleimani researched the dual-plane magnetic induction tomography method to locate the damage of composite parts [25,26] Besides that, Soleimani improved the reconstruction quality of EMT image using the Kalman filtering method [27] Teniou et al proposed the constrained Landweber algorithm to improve image reconstruc­ tion, which uses both boundary data and the foreground–background fractions [28] Wang et al developed a novel EMT system based on FPGA (Field Prog ram mable Gate Array), which uses TMR (Tunnel Magneto Resistance) sensors instead of tradi­ tional coils [29,30] Meanwhile, the improved Landweber iterative algorithm is NONDESTRUCTIVE TESTING AND EVALUATION investigated to improve the quality of image reconstruction [31] Tamburrinoa et al proposed non-iterative monotonic imaging algorithm for defect detection, and the method can be simplified using the geometric symmetry characteristics of the detected object [32] In this paper, a closed-loop fuzzy proportional, integral and derivative (PID)- controlled iterative Landweber reconstruction method is proposed The whole recon­ struction method includes a fuzzy PID controller, the Landweber method and the Dirichlet-to-Neumann map The differential signal between the measurement and the feedback signal is fed into the fuzzy controller, and the fuzzy controller can adjust the parameters of PID The output of PID acts as the input of the Landweber algorithm to reconstruct the distribution of conductivity Based on the distribution of conductivity, the boundary mutual inductance can be calculated by the Dirichlet-to-Neumann map, which is the feedback signal The closed-loop structure can improve the quality of reconstructed images The proposed method can achieve three-dimensional imaging for metallic defects, which is conducive to quantitative evaluation of defect size and thus avoids the occurrence of accidents in practice Fundamental methods In EMT systems, image reconstruction algorithms reconstruct the field distribution based on boundary measurement values and sensitivity matrices The factors that affect the quality of image reproduction mainly include two parts: software and hardware systems The hardware system mainly includes the rationality, accuracy and anti- interference ability of each part of the system design The software mainly includes image reconstruction algorithms, whose performance directly determines the final ima­ ging quality and is the core of EMT If EMT is approximated as a linear system, its forward problem can be expressed as Equation (1) U ¼ SG (1) The greyscale value G of the reconstructed image can be obtained by Equation (2) if the inverse matrix of S is assumed to exist G ¼ SÀ 1U (2) However, the inverse matrix of S cannot be directly obtained in the inverse problem of EMT due to the ill-condition The image reconstruction algorithm can also be seen as a process of approximating the inverse matrix of S, so the imaging accuracy is limited to some extent [33] The Landweber iterative algorithm has good imaging accuracy, which is widely applied to image reconstruction for EMT The Landweber iterative algorithm transforms the original problem into an optimisation problem, which can be expressed as Equation (3) f Gị ẳ kSG Uk2 (3) Equation (3) can be converted to finding the minimum value of equation (4) P HUANG ET AL Figure The sketch map of the proposed PID-controlled iterative Landweber method f Gị ẳ SG UịTSG Uị ẳ GTSTSG 2GTSTU ỵ UTUị (4) Therefore, the Landweber algorithm takes the negative direction of the gradient as the optimisation search direction, and the iterative formula can be expressed as, � G0 ¼ STU (5) Gkỵ1 ẳ Gk kSTSGk UÞ Fuzzy PID-controlled iterative Landweber method Figure illustrates the sketch map of the proposed fuzzy PID-controlled iterative Landweber method for image reconstruction in EMT The mutual inductances of coils are measured, and the matrix form of the discretized boundary map (Dirichlet-to- Neumann map) is established The measured signal is compared with the feedback signal and is further fed into the fuzzy controller, which can be used to adjust the parameters of PID to yield an input for the Landweber method The Landweber method reconstructs the image based on the measured mutual inductances Subsequently, the reconstructed image is normalised to obtain a feedback Dirichlet-to Neumann map for comparison with the measured signal The iteration termination is determined by the difference between the reconstructed conductivity distribution of the current iteration and the previous one When the difference of the reconstructed conductivity distribution is less than the threshold, the reconstructed conductivity distribution can be the output The closed-loop structure can ensure the convergence of the iterations, and the proposed robust method can be achieved for conductivity distribution 3.1 Fuzzy PID controller The proposed reconstruct algorithm adopts fuzzy PID controller to reduce diver­ gence between the reconstructed image and the measured Dirichlet-to-Neumann image The fuzzy PID control utilises fuzzy logic to optimise the parameters of the PID controller in real time based on certain fuzzy rules The fuzzy PID controller NONDESTRUCTIVE TESTING AND EVALUATION can overcome the disadvantage of traditional PID parameters that cannot be adjusted in real time Specifically, the deviation is input into the controller and is fuzzificated into the fuzzy set using the membership function Fuzzy reasoning is applied by following fuzzy rules to yield a fuzzy set of output Finally, it is defuzzified to update the PID coefficients The three parameters of the PID con­ troller are updated during each iteration The input and output of the fuzzy PID controller can be expressed as, qNNiị ẳ KpiịẵpNNiị pNNi 1ị ỵ KIiịpNNiị (6) ỵ KDiịẵpNNiị 2pNNi 1ị ỵ pN�Nði À 2Þ� where i is the number of iterations KpðiÞ, KIðiÞ and KDðiÞ denote the proportional, integral and derivative parameters of PID controller in ith iteration pN�NðiÞ is the difference between the measured mutual inductance matrix Mmeasure and the feedback mutual inductance matrix Mfb, which is the input of the fuzzy PID controller qNNiị ẳ qNNiị qNNi 1Þ is the output of the fuzzy PID controller The input of fuzzy PID controller are e1ðiÞ and e2ðiÞ, which are related with pN�NðiÞ and can be expressed as, e1ðiÞ ¼ kpN�NðiÞk2 (7) e2iị ẳ e1iị 2e1i 1ị þ e1ði À 2Þ (8) In fact, the Gaussian membership function has the characteristics of continuity and smoothness, which can improve the accuracy of the membership function and make the output of the fuzzy controller more accurate Meanwhile, the Gaussian membership function can effectively solve ambiguity and uncertainty, such as noise interference Hence, the Gaussian membership function is adopted, and the Gaussian membership values for these two inputs are, � �2 À e1À a1ðhÞ 1i; hị ẳ e 1hị (9) � �2 À e2À a2ðhÞ (10) σ ðhÞ 2i; hị ẳ e where 1i; hị and 2i; hị are the membership function of the two fuzzy PID controller inputs e1ðiÞ and e2ðiÞ in the hth fuzzy value a1ðhÞ and a2ðhÞ are the central value of the hth Gaussian curve of e1ðiÞ and e2ðiÞ Besides that, σ1ðhÞ and σ2ðhÞ are the standard deviations of the hth Gaussian curve The fuzzy rules are adopted to adjust the parameters of PID controller, i.e Kp, KI and KD Firstly, the range of input of fuzzy controller (e1ðiÞ and e2ðiÞ) and PID parameters can be estimated by FEM (Finite Element Method) numerical solution The fuzzy sets should be able to cover the entire range of inputs and outputs Furthermore, the number of rules can be determined to cover all possible input conditions A large number of fuzzy rules can lead to overfitting, while a small number of rules may reduce the control accuracy Finally, according to the performance of the system, the parameters of fuzzy control P HUANG ET AL rules, such as membership function parameters, need to be adjusted using FEM numer­ ical solution Assuming that the hth fuzzy value contains a total of m combinations, the membership degree of the hth fuzzy value is Xm μoutði; hị ẳ 1i; hjịị2i; hjịị j¼1 (11) � �2 � �2 Pm À e1À a1ðhÞ À e2À a2ðhÞ e σ ðhÞ e σ ðhÞ jẳ1 where outi; hị is the membership value of output parameters, and μ1ði; hðjÞÞ and μ2ði; hðjÞÞ denote the membership value of e1ðiÞ and e2ðiÞ in the hðjÞ fuzzy value The centroid method is used for defuzzification, which can be expressed as, Xn , Xn outiị ẳ outi; hịhị μoutði; hÞ (12) h¼1 h¼1 where n is the number of fuzzy values and λðhÞ is the hth fuzzy value of output The μoutðiÞ is the output of the fuzzy controller in the ith iteration Hence, KpðiÞ, KIðiÞ and KDðiÞ of PID controller in the ith iteration can be calculated as, Pn � Pn > >>> KPiị ẳ Pouti; hịPhị outi; hị >>>< hẳ1 Pn � h¼1 KIiị ẳ Iouti; hịIhị Pn outi; hị (13) >>> hẳ1 �h¼1 >>> Pn Pn >: KDiị ẳ Douti; hịDhị outi; hị hẳ1 hẳ1 where Pouti; hị, Iouti; hị and μDoutði; hÞ are the hth membership values of PID con­ troller parameters KpðiÞ, KIðiÞ and KDðiÞ in the ith iteration λPðhÞ, λIðhÞ and λDðhÞ are the corresponding fuzzy values 3.2 Iterative Landweber method based on a fuzzy PID controller in the closed-loop control The Dirichlet-to-Neumann map can be expressed as a mutual inductance matrix mea­ sured on the coil Mmeasure when the conductivity distribution of sensing region is σðzÞ � M @ΩðzÞ� measure : φðzÞj@Ω ! σðzÞ �� (14) @n @ where z ẳ x ỵ yi is the coordinate of point ðx; yÞ @Ω is the boundary of sensing region, and n is the unit normal vector of region boundary Apart from that, M0 denotes the mutual inductance matrix of coil when the conductivity of sensing region Ω is conductivity of metal samples The difference signal ΔM ¼ Mmeasure À M0, i.e variation of mutual inductance matrices, is the input of the NONDESTRUCTIVE TESTING AND EVALUATION reconstruction algorithm In addition, the Dirichlet-to-Neumann map can be calculated by the finite element model Since the reconstructed image of metallic samples with defects can be regarded as binary image, if only step function is used to achieve binarization, it will lead to closed-loop instability Considering that the output of sigmoid function ranges from to and the sigmoid function has the characteristic of smooth output, it can be applied to binariza­ tion of the reconstructed conductivity distribution The sigmoid function can be writ­ ten as, GBiị ẳ 1ỵe (15) À aðGN ðiÞÀ 0:5Þ where GBðiÞ and GNðiÞ are the reconstructed conductivity distribution before and after normalisation a is the coefficient of the sigmoid function Figure depicts the flow chart of the proposed closed-loop PID reconstruction algorithm, and specific details are as follows: (1) Mutual inductance matrices of metal samples M0 and measured field Mmeasure are obtained by EMT sensor Furthermore, the difference signal ΔM ¼ Mmeasure À M0 is calculated, which is the input of the proposed method Meanwhile, the para­ meters of fuzzy PID controller are initialised (2) The ith feedback mutual inductance matrices Mfb can be calculated according to ith reconstructed conductivity distribution using the Dirichlet-to-Neumann map The corresponding mutual inductance variation ΔM0 is obtained by subtracting M0 from Mfb Above all, the ith mutual inductance difference matrix pN�NðiÞ between the ΔM and ΔM0 is acquired, which is input to the fuzzy PID controller and outputs ΔqN�NðiÞ based on Equation (6) (3) The Landweber algorithm is applied to reconstruct conductivity distribution, and the conductivity distribution GBðiÞ can be normalised by the sigmoid function The iterations can be terminated until the k GBðiÞ À GBði À 1Þ k2 is smaller than threshold T, and the final reconstructed conductivity distribution GBðiÞ is output Simulation and discussion In this section, the simulations are carried out to determine the parameters and evaluate the proposed method As shown in Figure 3, the EMT sensor adopts nine planar coils, which are arranged in a structure of × The sensor is located above the copper samples with defects The air region surrounds the sample and the sensor to ensure the boundary conditions The parameters of sensor array are listed in Table The materials of samples are adopted as copper in simulations As shown in Figure 4, four typical different defect distributions are involved to verify the performance of the proposed reconstruction method Specifically, four typical conductivity distributions are single central defect, single non-central defect, two edge defects and four edge defects The thickness of the samples is mm, and the depth of defect is mm In addition, the red region in Figure is copper with 58 MS/m The blue region is defect, and the conductivity of defect is P HUANG ET AL Figure The flow chart of the proposed closed-loop PID reconstruction algorithm 0.1 S/m The conductivity of copper is 58 MS/m, and the conductivity of defect (air) is close to MS/m Hence, the conductivity of defects is about 10 orders of magnitude smaller than that of copper In other words, as long as the conductiv­ ity of the defect is set small enough, the impact on the reconstruction results is negligible The position and number of defects are different, which includes single central defect, single non-central defects, two defects, and four defects In order to evaluate the quality of the reconstructed image, the correlation coefficient (Cor) is applied and defined as, NONDESTRUCTIVE TESTING AND EVALUATION Figure The planer EMT sensor in FEM Table The dimension of the coil in the EMT sensor Inner and outer radii of the coil mm/5 mm Height of the coil mm Spacing between coils 10 mm Lift-off mm Figure Four typical conductivity distributions iẳ1 Pn GRiị GRịGTiị GTị Cor ẳ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifi (16) Pn � Pn iẳ1 GRiị GRị iẳ1 GTiị GTị where GR represents the grey value of the reconstructed image and GT is the actual grey value of the image n denotes the number of pixels in the image In addition, G�R and G�T are the average grey values of GR and GT, respectively The correlation coefficient ranges from to The closer the correlation coeffi­ cient of the reconstructed image is to 1, the better the quality of the reconstruc­ tion image If the correlation coefficient is much less than 1, the quality of reconstruction will be worse 10 P HUANG ET AL 4.1 Determination of parameters in fuzzy PID controller and smooth function The reconstruction image deviation e1ðiÞ and cumulative deviation e2ðiÞ are used as inputs of the fuzzy PID controller The PID parameters KpðiÞ, KIðiÞ and KDðiÞ are the outputs of the control system The fuzzy inference method is used to adaptively adjust PID parameters online to meet the error requirements of the reconstructed images Firstly, it can observe the dynamic characteristics of the system through closed-loop operation or simulation and repeatedly debug the parameters to determine the PID control parameters based on the impact of each parameter on the system until a satisfactory response occurs These parameters are set as the initial parameters of the adaptive fuzzy PID controller Since this range matches the input of the fuzzy controller, the PID parameters can be quickly adjusted The parameters of the algorithm can be determined through extensive simulation testing e1 and e2 belongs to ½À 0:12; 0:12�, Kp is with ½À 1; 1�, KI belongs to ½À 1; 1�, Kd belongs to ½À 0:1; 0:1� Due to the simple operation and low memory consumption, Gaussian membership functions are chosen as membership functions for input and output variables Taking the distribution of four defects as an example, we determine the number of fuzzy subsets through FEM numerical simulation Figure illustrates the reconstructed results of four defects with different numbers of fuzzy subsets When the number of fuzzy subsets is five, the reconstructed result converges after approximately 40 iterations and the accuracy is relatively low As the number of fuzzy subsets increases, the accuracy of imaging gradually improves However, the larger the number of fuzzy subsets, the larger the computation In addition, it can also easily lead to overfitting When the number of fuzzy subsets is seven, the distribution of the four defects can be seen more accurately and the computation amount is relatively small Here, a total of seven fuzzy subsets were selected as Gaussian membership functions, namely NB, NM, NS, ZO, PS, PM and PB Moreover, the letters N, B, M, S, ZO and P denote negative, big, small, zero and positive, respec­ tively For instance, since the range of e1 is ½À 0:12; 0:12�, the fuzzy values of NB, NM, NS, ZO, PS, PM and PB are −0.12, −0.08, −0.04, 0, 0.04, 0.08 and and 0.12, respectively Besides that, the Mamdani fuzzy inference model is used in the fuzzy inference of the system, and the centroid method is applied for the defuzzification of the fuzzy PID controller In fact, the parameter a of sigmoid function is also analysed, which is related to the effect of smooth segmentation Figure shows the sigmoid function with different values a As can be seen from Figure 6, the larger the parameter a, the steeper the sigmoid function Due to the difference in segmentation functions, the Figure The reconstructed results of four defects with different numbers of fuzzy subsets (a) fuzzy subsets, (b) fuzzy subsets, (b) fuzzy subsets and (b) 11 fuzzy subsets NONDESTRUCTIVE TESTING AND EVALUATION 11 Figure The sigmoid function with different values of a Figure The image reconstruction results of the proposed method with different values of a image reconstruction results of the proposed method with different a are depicted in Figure It can be observed that the increased slope improves reconstruction image quality in the centre region As shown in Figure 8, the correlation coeffi­ cient of the reconstruction image Cor firstly increases and then decreases with the increase in coefficient a When coefficient a equals 40, the correlation coefficient of the reconstructed image reaches its maximum, i.e 0.9 When a = 80, it is easy to cause oscillation in the reconstructed image, resulting in a position shift and a decrease in the correlation coefficient of the reconstructed image Hence, in the subsequent algorithm validation, the selection of coefficient a is selected as 40 for the segmentation function If the smooth segmentation is not introduced, severe oscillations appear, and it makes the closed loop unstable In this case, it is necessary to choose the appropriate algorithm parameters to ensure the stability of the closed-loop struc­ ture When the appropriate parameters are adopted, the reconstructed result of the single central defect without the smooth segmentation is shown in Figure The reconstructed area is correct, but the edges are not smooth Moreover, the correlation coefficient is only 0.7981, which is lower than the results with the smooth segmentation 12 P HUANG ET AL Figure The correlation coefficient of the reconstruction image with different values of a Figure The reconstructed result of the single central defect without the smooth segmentation (a) Three views and (b) vertical view 4.2 Image reconstruction using simulation data In order to evaluate the proposed reconstruction method, three other typical algorithms, i.e LBP, Landweber and L1 regularization method, are also used for comparison The maximum iteration number of Landweber and L1 regularization is set to 500, and the �T factor is calculated as S S Meanwhile, the termination criterion of the proposed method and other iterative algorithms (Landweber and L1 regularization) is kΔGBðiÞk < 0:05 The reconstructed images of different conductivity distributions are shown in Figure 10, and the corresponding correlation coefficients of reconstructed images are listed in Table NONDESTRUCTIVE TESTING AND EVALUATION 13 Figure 10 The reconstructed images of four distributions using different algorithms in simulation Table The correlation coefficient of reconstruction images using different algorithms in simulation Algorithms Single central defect Single non-central defect Two edge defects Four edge defects LBP 0.2764 0.6470 0.4710 0.3853 L1 regularization method 0.8399 0.8256 0.6623 0.3963 Landweber 0.8848 0.7991 0.8236 0.3979 Proposed method 0.9037 0.8382 0.8528 0.7443 The advantage of LBP is fast computation speed, but it cannot reconstruct clear contour boundaries Besides that, it is not possible to accurately obtain good reconstruction results for central defects due to the uneven distribution of sensitivity fields The quality of reconstructed images using Landweber and L1 regularisation algorithms has significantly improved, especially in the distribution of central defects, edge defects and two central defects The correlation coefficients are large than 0.8 As shown in Figure 11, the middle position of the imaging region has a higher sensitivity Hence, the defect region will gather toward the center as the number of iterations increases, forming artefacts in the middle region for center four defects reconstruction Compared with other reconstruction meth­ ods, the reconstructed results using the fuzzy PID-controlled iterative Landweber method have clear contour boundaries Correspondingly, there is also a small increase in the correlation coefficient of the reconstructed image obtained by the proposed method More importantly, fuzzy PID-controlled iterative Landweber method can achieve an accurate imaging of four defects and eliminate artefacts The reason for the phenomenon is that the closed-loop structure weakens the impact of high sensitivity in the central region by adjusting the mutual inductance data of the reconstructed conductivity distribution 14 P HUANG ET AL Figure 11 Sensitivity distribution of EMT sensors Experiment 5.1 Experiment setup In order to verify the proposed method, the experiment is also carried out using the EMT system Specifically, the EMT system shown in Figure 12 mainly consists of a host PC, multi-channel eddy current equipment, 9-coil sensor array, and metal samples with defects The EMT measurement equipment uses digital excitation signals and digital signal demodulation in FPGA to improve data speed and signal-to-noise ratio, which can meet the requirement of real-time measurements Meanwhile, the measurement speed of the equipment can reach 131 frames/s and the signal-to-noise ratio (SNR) can be reach 65 dB for stable measurement In our previous research, the parameters of nine-coil sensor array are optimised to improve the uniformity of the sensitive field by orthogonal tests and response surface methodology [34] Specifically, the inner and outer radii of coils are mm and mm The Figure 12 EMT experiment setup NONDESTRUCTIVE TESTING AND EVALUATION 15 lift-off and height of coil are mm and mm, and the distance between adjacent coils is 10 mm Moreover, the turns of coils are 200, and the excitation current is set as 6A The copper plates with different defect distributions are manufactured to verify the proposed method Consistent with the defect distribution in the simulation, the four typical defect distributions are single central defect, single non-central defect, two edge defects and four edge defects The thickness of the samples is mm, and the depth of defect is mm 5.2 Image reconstruction using experiment data Experiments are carried out using EMT system and 9-coil sensor array In the proposed reconstruction algorithm, the coefficient a of segment function is set to 40, which is estimated by multiple simulation tests The termination criterion of the iterative algo­ rithm is also kΔGBðiÞk < 0:05, and the maximum iteration number is 500 The factor of �T Landweber and L1 regulation method is calculated as S S As depicted in Figure 13, four metal samples with different defect distributions are reconstructed by different methods Similar to the results of reconstruction using simu­ lation data, the LBP method only can roughly describe the contour of defects and has a low imaging accuracy Moreover, the central defect cannot be distinguished well For simple distributions, such as single defect distributions (i) and (ii) and double defect (iii), the L1 regularization and Landweber can obtain the location and size of defects, only slightly unclear at the edges of the defect The fuzzy PID-controlled iterative Landweber method improves the contour of defect imaging Compared with the traditional Landweber, the reconstruction results of the proposed method contain fewer artefacts in distribution (iv) Table lists the correlation coefficients of the reconstructed images Figure 13 The reconstructed images of four distributions using different algorithms in the experiment 16 P HUANG ET AL Table The correlation coefficient of reconstruction images using different algorithms in the experiment Algorithms Single central defect Single non-central defect Two edge defects Four edge defects LBP 0.2715 0.6507 0.4310 0.3752 L1 regularization method 0.7939 0.7896 0.7223 0.3913 Landweber 0.8109 0.7901 0.7326 0.3929 Proposed method 0.8188 0.8145 0.8091 0.7354 using different methods The average correlation coefficient of the proposed method is 0.792 For simple defect distribution, the imaging quality of the proposed method is slightly higher than that of the imaging method For complex defect distributions (four defects), the results of other imaging methods contain central region artefacts The closed-loop structure can eliminate artefacts and obtain more accurate imaging results Compared to the traditional Landweber algorithm, the proposed method can reconstruct the distribution of four defects well, and the imaging results have improved by 87.2% using the proposed method Taking the distribution of four defects as an example, the image reconstruction quality and parameter changing trend of the proposed method is further analyzed As shown in Figure 14, the three parameters of the PID controller gradually decrease and eventually reach during the iteration process Correspondingly, the correlation coefficient of the reconstructed image gradually increases in a zigzag pattern When the number of iterations reaches about 20, the change in correlation coefficient is very weak and the reconstruction result reaches stability and converges The above phenomenon can be explained as the closed-loop negative feedback structure continuously eliminating residuals Fuzzy control adjusts the three parameters of PID based on the magnitude of the differential signal As the number of iterations increases, the three parameters of PID control eventually converge to and the reconstruction result ultimately reaches stability In addition, the initial values and fuzzy rules of PID control for different defect reconstructions are the same and the reconstruction results for both simple and complex defect distributions been improved, which indicate that the proposed method has robustness Figure 14 The image reconstruction quality and parameters’ changing trend: (a) parameters of PID controller and (b) correlation coefficient NONDESTRUCTIVE TESTING AND EVALUATION 17 In fact, the fuzzy PID-controlled iterative Landweber method needs to calculate the positive problem, i.e Dirichlet-to-Neumann map The process requires a certain amount of time, thereby reducing the speed of computation In future research, we will focus on fast computation of forward problems to make the proposed reconstruction algorithm more applicable Conclusion This paper proposed an image reconstruction algorithm for defect detection in EMT The method mainly consists of fuzzy PID controller, Landweber method, smooth segmenta­ tion and Dirichlet-to-Neumann map, which can effectively improve the quality of image reconstruction Specifically, the differential signal between the measurement signal of coil array and feedback signal is applied as the input of the fuzzy PID controller The fuzzy controller can automatically adjust the three parameters of the PID controller and accelerate convergence rate of closed-loop structure Subsequently, the Landweber algo­ rithm invert the output signal of the PID controller into reconstructed conductivity distribution Furthermore, the Dirichlet-to-Neumann map is used to calculate feedback signal Both numerical simulations and experiments are used to evaluate the proposed method Compared with the traditional image reconstruction algorithm, artefacts can be effectively reduced, and the correlation coefficient of four defect distribution can be improved to 87.2% using the proposed method Besides that, the proposed method is suitable for both simple and complex defect distribution, which demonstrates that the method has robustness Disclosure statement No potential conflict of interest was reported by the author(s) Funding This work was supported by the Fundamental Research Funds for the 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