New Trends in Networked Control of Complex Dynamic Systems: Theories and Applications
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Bo Ye, Hongchun Shu, Min Cao, Fang Zeng, Gefei Qiu, Jun Dong, Wenying Zhang, Jieshan Shan, "A Bayesian Network Method for Quantitative Evaluation of Defects in Multilayered Structures from Eddy Current NDT Signals", Mathematical Problems in Engineering, vol. 2014, Article ID 405707, 15 pages, 2014. https://doi.org/10.1155/2014/405707
A Bayesian Network Method for Quantitative Evaluation of Defects in Multilayered Structures from Eddy Current NDT Signals
Abstract
Accurate evaluation and characterization of defects in multilayered structures from eddy current nondestructive testing (NDT) signals are a difficult inverse problem. There is scope for improving the current methods used for solving the inverse problem by incorporating information of uncertainty in the inspection process. Here, we propose to evaluate defects quantitatively from eddy current NDT signals using Bayesian networks (BNs). BNs are a useful method in handling uncertainty in the inspection process, eventually leading to the more accurate results. The domain knowledge and the experimental data are used to generate the BN models. The models are applied to predict the signals corresponding to different defect characteristic parameters or to estimate defect characteristic parameters from eddy current signals in real time. Finally, the estimation results are analyzed. Compared to the least squares regression method, BNs are more robust with higher accuracy and have the advantage of being a bidirectional inferential mechanism. This approach allows results to be obtained in the form of full marginal conditional probability distributions, providing more information on the defect. The feasibility of BNs presented and discussed in this paper has been validated.
1. Introduction
Detection and quantitative evaluation of internal defects in multilayered structures are an essential task in a range of technological applications, such as maintaining the integrity of structures, enhancing the safety of aging aircraft, and assuring the quality of products [1–3]. Defects are generally formed in multilayered structures by residual stress or physical or metallurgical processes, and they can increase in both number and size with time, due to fatigue and corrosion, causing damage and sometimes sudden structural failure. Quantitative evaluation of defect characteristics, such as size, shape, and orientation, is highly desirable and is an emergent technique [4]. Experimental measurement should take advantage of advanced nondestructive testing (NDT) technologies, and it will be extremely valuable if early and accurate detection of defects is possible, especially where defects are related to internal damage.
Over the last several years, a number of NDT techniques have been developed for the detection and characterization of defects in multilayered structures. They are based on eddy current (EC) [5], ultrasonic [6], terahertz ray [7], thermal [8], acoustic emission [9], and Xray [10] measurements of the various structures tested. In the inspection of multilayered metallic structures, radiographic inspection has problems associated with radiation protection, ultrasound methods suffer from the high attenuation and the low reliability, and in thermographic inspection there is a problem with measurement precision. It would appear that these techniques have certain limitations in inspecting defects in multilayered structures where it is only possible to obtain access to one side of the sample. In contrast, eddy current nondestructive testing (ECNDT) is relatively rapid and has the advantages of having high sensitivity, being noncontact, of low cost, and easily implemented for automated, online testing and is one of the rigorous, physicsbased approaches for identifying micro hidden defects [11, 12].
Characterizing defects from measurements of the change in the eddy current is generally considered as the inverse problem [13–15]. Thus, the results of the quantitative evaluation of defects are retrieved by inversion of the measured data representing the change in impedance of a coil as it scans the specimen [16, 17]. Since the physical model of ECNDT is often complicated and nonlinear, the inversion process is often ill posed [18]. The methods based on solving Maxwell equations of electromagnetic field using numerical analysis, such as finite element method (FEM), boundary element method (BEM), and volume integral method (VIM), have been successfully developed to simulate the response of the measurement system to different thickness aluminum alloy planar structures and, to some extent, more realistic defects with complex profiles [19]. Nevertheless, the inverse problems are not yet fully resolved, for even the quantitative evaluation of simple shaped defects from ECNDT signals [20].
More recently, quantitative evaluation of defects has been implemented by using many sophisticated algorithms. The template matching method is firstly advocated [21]. Measurements are recorded on calibration samples and specific defect signals are stored as a standard template such as EC signals shape, peak voltage, phase data, smoothness, convexity, unimodality, or existence of derivatives. The evaluation results are obtained from the comparison of the currently collected signals with each calibration defect. In this type of method, the features used to characterize the probe response are not sufficient for inspection and can easily become ineffective owing to noise, interference, and liftoff variation, leading to inaccurate results. Then, some researchers present modelbased approaches to evaluate defects from EC signals [22]. These methods iteratively solve the forward model to simulate the inspection process and predict the probe response. The inverse problem of quantitative evaluation of defects is formulated as an optimization problem, which seeks a set of defect characteristic parameters by minimizing an objective function, representing the difference between the model predicted signals and the measured signals. Such approaches usually involve significant computational effort, since the physical model needs to be solved repeatedly. Afterwards, methods based on neural networks and the least squares (LS) regression have been used to establish the relationship between the defect characteristic parameters and the observed data [23, 24]. These methods usually require a large volume of prior knowledge, space limitations, and database of signals from defects for training. However, the training samples are often difficult to obtain for real industrial applications and there are obvious errors when a sample has never been contained in training set. Recently, there has been much interest in use of probability density function estimation and Bayesian estimation methods for quantitative evaluation of defects [25–27]. These methods employ sampling techniques such as Markov Chain Monte Carlo (MCMC) and Bootstrap methods, for probability density function estimation of defect characteristic parameters to obtain not only the quantity but also the uncertainty characterization of the measurand. The main problem with these methods is that sampling techniques require a significant amount of time. Therefore, a general framework for quantitative evaluation of defects from EC signals is very desirable, which can rapidly and accurately give the evaluation of defect characteristics.
In this paper, we propose the use of Bayesian networks (BNs) [28] to evaluate defects quantitatively from EC inspection signals. In ECNDT, the output signals may be corrupted by noise and other anomalous signals, arising from liftoff, edge effects, high frequency, probe angle variations, and so forth. This will result in unreliable detection and inaccurate characterization of defect. Although some of noise may be eliminated or decreased by a number of preprocessing algorithms, the inherent uncertainty and stochastic nature of inspection still need to be dealt with. BNs are a useful method for EC inversion modeling, because of their capability of handling uncertainty and incorporating prior information, providing more accurate evaluation results. They offer a natural and compact tool for dealing with two problems that occur throughout EC inversion modeling, uncertainty and complexity, and also a basis for efficient probabilistic inference [29]. In ECNDT, BNs generalize not only the forward model but also the inverse model by using bidirectional probability inference. This is an important advantage compared with traditional methods (e.g., LS regression method) that encode only the values of the dependent variables, given the input variables. In this paper, BNs are applied to quantitatively evaluate the realistic multidimensional characteristic parameters of defects by probability inference. We mainly discuss how to construct BNs from the domain knowledge and the real research data and how to perform probability inference in BNs. The proposed BNs allow us to obtain the full probability distributions of the needed evaluation characteristic parameters of defects, which give the accurate evaluation of defects and quantify its uncertainty characterization. Experimental results show that the proposed method maintains higher estimation accuracy than the previous methods.
The remainder of the paper is organized as follows. Section 2 gives the general formulation of quantitative evaluation of defects in multilayered structures using BNs. Section 3 reviews the principle of BNs which are applied to evaluate defect characteristics. Section 4 presents experiments and results. Finally, Section 5 contains conclusions.
2. Problem Description
The eddy current problem can be described mathematically by partial differential equations in terms of the magnetic vector potential: where represents the magnetic vector potential, , is imaginary unit, is the angular frequency of the excitation current, is the magnetic permeability of the media involved (H/m), is the electrical conductivity (S/m), is the dielectric constant (F/m), and is the excitation current density (Amp/m^{2}).
The forward problem under consideration is the calculation of the field perturbation due to a volumetric defect assuming time harmonic operations and linear and nonmagnetic conductors [30]. The main goal of solving the forward problem is to predict the probe response signals, given the defect parameters. On the contrary, the inverse problem is described as the task of quantitative evaluation of defect characteristics from the measured EC signals [31].
In this paper, EC inspection is treated as a static, stochastic process. The defect characteristic parameters and the observed signals are regarded as continuous random variables. Initially, the probability distributions of all variables are assumed as multivariate Gaussian, with unknown means and variances (classical assumptions). We construct a multivariate Gaussian model for the process of ECNDT, represented as a simple graphical model within the BNs which provide a modeling language and associated inference algorithm for stochastic domains. Then, the following work is to obtain the posterior probability distributions of all unknown variables, using Bayesian inference. Bayesian inference is based on Bayes Theorem as [32] where is the observed data set, is the unknown variables needing estimation, is the prior probability of , is the likelihood function, which incorporates the statistical relationships in addition to the mechanistic relationships among and , and is the posterior probability of . Using BNs, we calculate the posterior distributions of , when the known data arrive. However, normally one is interested in giving point predictions and/or probability intervals for predictions. We may use the means of distributions of as predictions and variances for probability intervals.
A complete diagram of the quantitative evaluation of defects is shown in Figure 1. The quantitative evaluation procedure is summarized as follows. The domain knowledge about ECNDT is used to formulate the structure of BNs representing the inspection models. Then, it is trained using real research labeled data, consisting of the EC signals and the corresponding defect characteristic parameters. After training, the results produce a generative model, suitable for use in ECNDT systems, which is able to predict the signals corresponding to different defect characteristic parameters or estimate defect characteristic parameters from the EC signals in real time.
3. Bayesian Networks
BNs were initially developed in the late 1970s. Over the last decade, BNs have become an established framework for representing and reasoning with uncertain knowledge [33]. More recently, researchers have developed methods using BNs to analyze measured data. These techniques that have been developed are new and still evolving, but they have been shown to be remarkably effective for many reallife inversion problems such as diagnosis of space shuttle propulsion systems, situation assessment for nuclear power plant, and information retrieval [34].
3.1. Bayesian Networks Models
BNs are mathematical models, combining graphics and probabilities to express mutual relationships between variables [35]. Let be a set of random variables, with each variable taking values in some finite domain Dom. BNs over is a pair that represents a set of distributions over the joint space of . is a set of directed acyclic graphs (DAGs), whose nodes correspond to the random variables in , and whose structure encodes conditional independence properties about the joint distributions. Each node directly depends on its parents Par. is a set of parameters which quantify the network by specifying the conditional probability distributions (CPDs) . Given Par is a set of variables that renders and independent, we can decompose a joint probability density using the chain rule of probability
Note that does not need to include all elements of which indicate conditional independence between those variables not included in and , given that the variables in are known.
In this paper, we use these ideas in context with continuous variables and dependencies, where the probability distributions of all continuous variables are multivariate Gaussian distributions. Each variable is a multivariate Gaussian distribution , and its probability density function is where is the dimensional mean vector, is the variance matrix, is the determinant of , and denotes the transpose of . Then, the CPDs can be represented in BNs by using linear Gaussian conditional densities. Given its parents are known, in this representation, the conditional density of is where , is the regression coefficient of in the regression of on the parents of , , and is the conditional variance of , given , where is the unconditional variance of , is the matrix of covariance between and the variables in , and is the covariance matrix of [36].
3.2. Learning Bayesian Networks
Learning generally refers to learning the graphical structure or the parameters (CPDs) for that structure or both [37, 38]. The learning results are a set of techniques for data analysis that combines prior knowledge with real data to produce improved knowledge. In a real application, the domain knowledge is based on a special set of rules, which can be used to create BNs structures on a casebycase basis. It is clear that the models created in this way are strictly based on the special physical process, since the structure of the graph is automatically generated, given the rules and the background facts. During the quantitative evaluation of defects from ECNDT signals, the BNs structures can be constructed using the knowledge about ECNDT and defect characteristics. The structure construction approaches will be described in Section 4 in detail. In this section, we consider only the problem of using data to determine the probabilities of a given structure.
The problem of learning BNs in this case can be stated as follows. Given a training set of independent instances, find the of BNs that best matches . The assumption of the linear Gaussian conditional distributions of all variables in the BNs would be considered. The task of unknown parameters estimation is to find the maximum likelihood estimation (MLE) of mean and variance vectors [39]. The MLE method is versatile and easy implementation and can be applied to most models and different types of data. If Gaussian prior distributions are assumed over the parameters, the MLE coincides with the most probable values thereof. The normalized loglikelihood of the training set is a sum of terms: The MLE method maximizes by finding the value of : The loglikelihood function decomposes according to the structure of the graph, and hence the contribution to the loglikelihood of each node can be maximized independently.
For the multivariate Gaussian CPDs, the loglikelihood function is given by The family of distributions has two parameters: . Therefore, we maximize the likelihood over both parameters simultaneously, or if possible individually. Taking the partial derivatives of , with respect to each one of the parameters and setting it equal to zero yields Solving (9) simultaneously yields
Formally, we say that the maximum likelihood estimator for is .
3.3. Inference in Bayesian Networks
In general, the computation of marginal CPDs of interest is known as probabilistic inference [40, 41]. The main goal of inference is to estimate the values and their probabilities of the unknown nodes, given the values of the observed nodes. When observations are given, this knowledge is integrated into the network and all the probabilities are updated accordingly. If we observe the “leaves” of a generative model and try to infer the values of the causes, this is called diagnosis or bottomup reasoning. If we observe the “roots” of a generative model and try to predict the effects, this is called prediction or topdown reasoning. BNs can be used for both of these tasks and others. In ECNDT, bidirectional probability inference permits BNs to respond to both predictive inference and diagnostic inference. Thus, BNs will have the utmost flexibility for predicting the signals corresponding to different defect characteristic parameters or for estimating defect characteristic parameters from the EC signals in real time.
The junction tree method is a new iterative algorithm that efficiently combines dynamic discretization with robust propagation algorithms to perform inference in BNs [42]. A junction tree representing BNs is constructed by moralization and triangulation of ; that is, it connects together all parents who share a common child, then drops the directionality of the arcs, and selectively adds arcs to the moral graph to form a triangulated graph. In a junction tree, the basic nodes are represented as cliques which are maximal complete subgraphs of the triangulated graph. The separator is a path between two cliques and and the subset of and [43].
After building a junction tree “shell,” we define potentials over cliques and separators. A potential is a nonnegative function of its arguments, which can be interpreted as the probabilities over cliques and separators. We denote these potentials as and . Every CPD of the original BNs is associated with a clique, such that the domain of the distributions is the subset of the clique domain. The notation represents the domain of a potential . A Gaussian clique potential can be represented in either moment form or canonical form where is the mean vector, is the variance matrix, , , and . Thus, we must represent the initial potentials in canonical form, because they may represent conditional likelihoods, rather than probability distributions, whereas, one can convert to the moment form at the end of the calculation. Then, the belief potentials encode the joint distribution of the BNs according to where is the clique potentials and is separator potentials.
Inference in junction tree based architectures is performed by passing messages between the adjacent cliques. At the beginning of message passing, each separator is initially empty. During inference, each separator is updated to hold each of the potentials passed over the separator. The clique potentials are, on the other hand, left unchanged. When evidence is absorbed from to , the potential passed over the separator connecting and is calculated as where is the set of neighboring separators of a clique .
After a full round of message passing, the joint probability distributions of any clique in the junction tree can be computed as the combination of the potentials of and all the received potentials associated with neighboring separators:
From a consistent junction tree, the posterior marginal CPDs of a variable and the evidence can be computed from any clique or separator potential containing by eliminating all variables in except :
4. Experiments and Results
4.1. Measurement System Configuration
The automatic system based on ECNDT for quantitative evaluation of defects in multilayered structures is obtained by integrating the test device with a computer. A schematic diagram of the system is shown in Figure 2. The computer based system can thus increase the reliability of the detection. It provides fast and robust database methods for retrieving old inspection data, which is important in monitoring defect initiation and growth. In the system, reference structures and a reference probe have been used. By comparing the signals from the reference structures with those from the monitored special structures, the system can easily make a decision on material condition and usage state.
The system is readily decomposed into a few main components: an AC excitation generator, two eddy current probes (an inspecting probe and a reference probe), a demodulation module, a low pass filter, a data acquisition interface (A/D converter), a coordinate measuring machine (CMM), and a computer. A sinusoidal current source provides a current through coils with amplitude 1 A at a frequency 200 Hz. In the system, the rightcylindrical aircored coil probe has been used. The coil parameters are inner radius mm, outer radius mm, length mm, and liftoff 0.5 mm. Consisting of exciting coils and pickup coils, the probes scan over the surface of the specimen by using a CMM. A computer program is used to set the scan area and velocity. During measurements, the sensor’s output signals are amplified by a low cost, high accuracy instrumentation amplifier AD620. Then, the amplified signals are sent to a phase sensitive detector using XR orthogonal decomposition techniques to directly compute the horizontal components (real part, inphase components) and the vertical components (imaginary part, quadrature components) of impedance signals. These two components are filtered by a secondorder lowpass filter with a cutoff frequency of 20 Hz. A data acquisition program written in Labview collects data from the output of the filter via a National Instruments DAQPad 6016 bit analogtodigital converter. The computer is controlling the whole system and it is performing such tasks as automating the process of inspection, data acquisition and displaying, and applying some signal processing techniques to automate the process of defect detection and quantification.
4.2. Two Groups of Experiments and Results
To verify the feasibility of the proposed method for quantitative evaluation of defects in multilayered structures from ECNDT signals, comparative experiments are carried out. Multilayered samples resembling a part of the wing splice of the aircraft are analyzed. In particular, corrosion is a critical problem for inservice aircraft structures which may lead to a degradation of structure integrity and fatigue resistance, directly affect the durability of structure, and even result in loss of function. So detection of deeply buried defects in multilayered airframe joints is widely recognized as the urgent and difficult NDT problem. We assume that the shape and position of defects are known in advance. This paper mainly discusses the defect evaluation problem concerning estimation of defect dimensions in multilayered structures in order to simulate inspection of the internal corrosion and local interlayer air gaps in real structures.
Firstly, we considered a simple problem where only one parameter of defects in multilayered structures needs to be determined. The first experimental specimen (specimen number 1) is shown schematically in Figure 3. It is composed of three layers of aluminum with a total thickness of 7.5 mm. Each plate has electrical conductivity MS/m, magnetic permeability H/m, length 480 mm, width 80.0 mm, and thickness 2.5 mm. There are 7 holes with different diameters in the center of the middle plate. The parameters of holes are electrical conductivity S/m, magnetic permeability H/m, diameter 1, 2, 3, 4, 5, 6, and 7 mm, and depth 2.5 mm, respectively. A major goal is to utilize BNs for the defect diameter estimation. The skin depth is about 8.28 mm and indicates promising robustness for the inspection of inner defects in the multilayered structures.
In this experiment, the random variables are the defect diameter and the probe response signals . We assume that the other factor’s influence is very small. From a general knowledge of ECNDT, the probe response signals vary due to the difference of the defect diameter. It is clear that the link between and will lead to a BN given in Figure 4. It is a basic BN which contains two nodes. For the root node , the CPDs only contain the priori probability of each state. The CPDs of are then defined by the conditional probabilities over each state. As mentioned above, the distributions of and are initially assumed as multivariate Gaussian with unknown means and variances.
Then, we train this model using real research data and the results would be a generative model suitable for use in defect diameter estimation. A total of 210 complex valued EC data vectors containing signals corresponding to seven types of defects are available for the experiment (each type having 30 records). The data set is denoised by the wavelet packet analysis method with Shannon entropy. Then, the resulting signals and corresponding defect diameters constitute the labeled data which are sent to train the BN.
After the aforementioned operations, a new trained BN and inference engine are obtained. One thing we can do to verify that the model is reasonable is to draw samples from it and visually compare with the simulated data [44]. Once built, the model could be used to estimate defect diameters when the inspection signals are available. The main step is to enter each of inspection signals as evidence and calculate the posterior marginal CPDs of the node . We use the means of the node as prediction and the marginal CPDs for probability intervals. The BN method is also tested on signals containing artificial injected random noise. Real inspection signals are modified to contain additional 10% and 20% random noise and are used to evaluate the performance of the BN method in response to measurement noise. The performance of estimation is evaluated with the bootstrap crossvalidation method [45]. The results obtained from signals containing three different levels of artificial random noise (noisefree, 10%, and 20% noise) using the BN method are shown in Figure 5. Furthermore, we compare the estimated results obtained from the BN method with those from the LS regression method which is very simple and usually used as a benchmark for all the other quantitative evaluation methods [24]. Figure 6 and Table 1 show the comparison results obtained from signals containing three different levels of artificial random noise (noisefree, 10%, and 20% noise) using the BN and LS regression methods, respectively.

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Secondly, a more complex example is studied, where the signals are collected from a multilayer sample with defects varying diameter and depth. Figure 7 illustrates the second experimental specimen (specimen number 2). The specimen consists of three layers of aluminum with a total thickness of 10 mm (2.5, 5, 2.5 mm), electrical conductivity MS/m, magnetic permeability H/m, length 420 mm, and width 80 mm. There are 5 holes with different diameters and depths in the center of the middle plate. The holes parameters S/m, H/m, diameter, and depth are (1, 2.5), (2, 3.5), (3, 4.5), (4, 5.5), and (5, 6.5) mm, respectively.
In this example, the random variables are the defect diameter , defect depth , and the probe response signals . The defect diameter and depth are two independent influencing factors of the probe signals. Using this special relationship, we can obtain the BN structure given in Figure 8. Node and node are linked to node by an arc, respectively. Like the first example, the distributions of , , and are also assumed as multivariate Gaussian with unknown means and variances.
In the experiment, a dataset with 150 records is acquired during scanning. The dataset contains EC signals from 5 types of defects (each type having 30 records). As described in the previous experiments, the same procedures are carried out. The diameter and depth estimation results obtained from signals containing three different levels of artificial random noise (noisefree, 10%, and 20% noise) using the BN method are shown in Figures 9 and 10, respectively. The comparison results of the estimated diameter and depth obtained from signals containing three different levels of artificial random noise (noisefree, 10%, and 20% noise) using the BN and LS regression methods are shown in Figures 11 and 12 and Table 2, respectively.

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Tables 1 and 2 show quantitatively the estimation error and the corresponding methods. It can be seen that the BN method has a higher precision than the LS regression method. The LS regression method has difficulty performing more accurate estimation and the satisfactory solution becomes unlikely in the case of signals containing highlevel noise.
In contrast, the BN method can achieve greater accuracy and robustness, even in the case of signals containing highlevel noise. It provides not only the accurate estimation of defect characteristic parameters, but also the probability distributions of these values. With increasing noise level, the estimated variances also increase, whereas the estimated means deviations from their actual values increase very limitedly. This demonstrates that although the variances growth means increasing the uncertainty of the results, the estimated error of BNs increases only slightly with the increasing noise level. It shows that BNs are effective in the quantitative evaluation of defects, due to their remarkable characteristics such as the combination of the domain knowledge and the reasoning under uncertainty in AI. This eventually results in better generalization performance in the BN method than the LS regression method. In brief, the BN method is able to successfully cope with highly complex and illposed ECNDE inverse problems, where each defect is hidden deeply in a multilayered structure.
5. Conclusions
A new approach for quantitative evaluation of defects in multilayered structures from ECNDT signals using BNs has been proposed and investigated. BNs discussed in this paper are simple and powerful, and their structures and parameters are easily learned from the domain knowledge of ECNDT and experimental data. The generative model can predict the probe response signals given the defect characteristic parameters or estimate the defect characteristic parameters from the inspection signals by using probability inference. In this paper, we mainly describe how the problem of defect characteristic parameter estimation from ECNDT signals can be modeled using BNs, in which the general model accommodates the uncertainties of the EC inspection. Two experiments have been carried out. Compared with the LS regression method, BNs have the advantage of being a bidirectional inferential mechanism and higher accuracy and robustness. They allow results to be obtained in the form of full CPDs, accounting for all the information available (evidences). It can be seen that the estimation results are probabilities rather than a single value. More information about the estimated parameters can be deduced from their probabilities, such as the means and variances, which describe the value itself and the degree of uncertainty respectively. The experimental results demonstrate the feasibility and effectiveness of the BN method proposed in this paper. This also encourages the attempts to tackle the other evaluation problems.
In this paper, the multivariate Gaussian assumption of all variables has been taken. Nevertheless, in real problems where the distributions are nonuniform, a nonGaussian assumption also could be more suitable. In fact, the validity of BNs is not restricted to the particular case of multivariate Gaussian distributions but to more general distributions (Gamma, Poisson, etc.). BNs can handle the problem where each conditional distribution of each variable can be any distributions. However, these possibilities are out of the scope of this paper and are the topic of current and future work of the authors. It will be helpful to analyze the EC signals more accurately in detail and lead to more precise predictions.
In addition, it is necessary to note that, in this paper, the problem under investigation is simple. However, in most ECNDT problems, the more complex configuration (interacting defects, partially conducting defects, defects with nonregular shape as corrosions, real defects, etc.) is to be handled. Future work, hopefully, will be done to extend the proposed BN method to the more complex ECNDT problems where the more unknown parameters of defects need to be estimated. We consider that the independence of causal influence can be exploited in Bayesian networks to reduce both the time and space cost of inference. The increase of the computational costs with the increase of the unknown parameters is the computation of the decomposed additional local joint probability density inference.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work is supported by the National Natural Science Foundation of China Grant nos. 51105183 and 51307172, the Research Fund for the Doctoral Program of Higher Education of China Grant no. 20115314120003, the Applied Basic Research Programs of Science and Technology Commission Foundation of Yunnan Province of China Grant no. 2010ZC050, the Foundation of Yunnan Educational Committee Grant no. 2013Z121, the National College Student Innovation Training Program Funded Projects Grant no. 201210674014, and the Science and Technology Project of Yunnan Power Grid Corporation Grant no. KYN2013110.
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