Greedy breadth-first-search based chain classification for nondestructive sizing of subsurface defects

Applied Soft Computing 40 (2016) 260–273

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Greedy breadth-first-search based chain classification for nondestructive sizing of subsurface defects S. Shuaib Ahmed ∗ , B. Purna Chandra Rao, T. Jayakumar Nondestructive Evaluation Division, Indira Gandhi Center for Atomic Research, Kalpakkam, TN 603 102, India

a r t i c l e

i n f o

Article history: Received 28 March 2014 Received in revised form 13 November 2015 Accepted 18 November 2015 Available online 8 December 2015 Keywords: Machine learning Radial basis functions Neural networks Support vector machine Greedy breadth-first-search Eddy current testing

a b s t r a c t Machine learning including neural networks are useful in eddy current nondestructive evaluation for automated sizing of defects in a component or structure. Sizing of subsurface defects in an electrically conducting material using eddy current response is challenging, as the skin-effect and radial extents of magnetic fields are expected to strongly influence. Moreover, the information about all defect characteristics such as length, width, depth, and height is available within an eddy current image. Inspired by the recent developments in machine learning for multidimensional classification and their promise, this paper proposes chain classification for sizing of defects. Chain classification enables incorporation of dependency between the class variables which can enhance the performance of the machine learning algorithms. The best sequence among the class variables has been optimized using a greedy breadthfirst-search (GBFS) algorithm and systematic studies have been carried out using the GBFS. Two well established machine learning classification algorithms, namely, radial basis function neural network and support vector machine have been used in chain classification. Coupling the chain classification with the GBFS, an approach for automated sizing of defects has been proposed. From modeled as well as experimentally obtained eddy current images, it has been established that the proposed approach can successfully size subsurface as well as surface defects. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Machine learning, neural networks, and fuzzy systems play a vital role in eddy current (EC) nondestructive evaluation (NDE), for automated and accurate sizing of defects. Sizing is critical for accessing severity of defects and in turn, ensuring structural integrity and safety of components in many areas of complex engineering, including nuclear, aerospace, and defense industries [1–5]. Due to the classical skin-effect, the eddy currents decay exponentially with depth in a material and the area of interrogation by an EC probe in the material is larger than the actual probe size due to the radial extent of eddy currents [6]. As a result of these effects, there is a possibility that a smaller size defect present in a material, close to an EC probe (surface defects) can produce an identical response to a larger defect, buried deeper (subsurface defects) in the material. This is expected to influence the reliability of interpretation and sizing of defects during time-critical decision making by human experts. Automated sizing of defects using EC response is an active area of research in NDE [7].

∗ Corresponding author. E-mail address: [email protected] (S.S. Ahmed). http://dx.doi.org/10.1016/j.asoc.2015.11.032 1568-4946/© 2015 Elsevier B.V. All rights reserved.

Machine learning algorithms such as artificial neural networks (ANN) and its variants such as probabilistic neural networks (PNN) and radial basis function neural networks (RBFNN) were widely used for defect sizing using EC signals/images. Song and Shin [8] used a PNN for classifying four different types of defects. A numerical model was used to predict EC signals of 200 defects in an Inconel-600 tube. 22 features are extracted from the EC signals, and given as input to the PNN. This method was reportedly classified the heights of the defects with 91% classification accuracy. Rao et al. [9] proposed ANN for estimating the heights of surface defects in welds. From multi-frequency numerical simulations of EC signals, 8 features were extracted and used as input for the ANN. The ANN was able to detect and size defects with height 0.40 mm with a maximum deviation of 0.08 mm. Thirunavukkarasu et al. [10] proposed a RBFNN for estimating the height of surface defects. The generalization, interpolation, and extrapolation capabilities of the RBFNN were investigated using artificial defects in stainless steel plates. Reportedly, cross validation using the RBFNN was successfully able to evaluate the defects with height 1.2 mm with the maximum deviation of 35 ␮m. Recently, Bernieri et al. [11] studied sizing of subsurface defects in aluminum plates using ANN and support vector machine (SVM). In this work, 200 numerically simulated subsurface defects (with minimum defect dimension of 3.0 mm in

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length and 1.0 mm in height and 1.0 mm below surface) were used for sizing of the length, depth and height of defects. They were able to estimate the length with a mean absolute error of 0.048 and 0.120 for depth and height, respectively. Sizing multiple defect characteristics such as length, width, depth, and height of subsurface defects is scarce in the literature [8–11]. This involves prediction of multiple (multiclass) class variables and it is a new concept in machine learning, called multidimensional classification [12,13]. This can be approached by developing one independent classifier for each class variable. However, this approach will not capture dependencies among the class variables. Recent studies have revealed that incorporating dependency improves the performance of classification [12–17]. Chain classification is a method to incorporate dependency. It is a method of sequencing class variables where previous predictions in the sequence are used for present classification [14,15,17]. Sizing of defects using chain classification requires thorough investigation to determine an optimal sequence as well as for efficient classification, which is addressed in this paper. The primary objective of this paper is to develop classification algorithms with four class variables for automated and accurate sizing of subsurface defects. This paper proposes the chain classification coupled with greedy breadth-first-search (GBFS) algorithm to address dependency. Systematic studies have been carried out using GBFS algorithm with RBFNN and SVM to assess an optimal sequence for the chain classification that explicitly represent the dependency structure between the class variables. The efficacy of chain classification has been demonstrated with numerically modeled as well as experimental EC images of defects in stainless steel plates. 2. Eddy current testing Eddy current (EC) testing is an important NDE technique widely used in aerospace, nuclear, petrochemical, and other industries. Almost all heat exchangers (boilers, steam generators, condensers) and load-baring structures of aircrafts are inspected using this technique, for detection and sizing of wall thinning, fatigue cracks, pitting corrosion, stress corrosion cracks, hydrogen embrittlement, denting and deposits. The main reasons behind this widespread use are higher sensitivity to surface as well as subsurface defects, high testing speeds (up to 10 m/s), repeatability, ease of operation, versatility and data storage possibility. The most popular applications of this technique include detection of defects in plates, tubes, rods, bars, multi-layer structures, discs, welds, blades, and other regular as well as irregular geometries; material sorting; heat treatment adequacy assessment; proximity sensing; and coating thickness measurements [6]. Eddy current testing technique works on the principle of Faraday’s law of electromagnetic induction. In this technique, a coil (also called probe or sensor) placed over an electrically conducting material, e.g. stainless steel, aluminum, etc., as shown in Fig. 1 is excited with sinusoidal alternating current to induce eddy currents in the material. According to Lenz’s law, the induced eddy currents will produce a secondary magnetic field in the direction opposite to the primary magnetic field of the coil. This interaction results in impedance change. The change also arises due to defects such as cracks in the material [6]. The flow of eddy currents in the material is not uniform in the depth and lateral directions. The eddy currents are quite dense at the surface as compared to deep inside due to the skin-effect. Theoretical standard depth of penetration of eddy currents, ␦, the depth at which the surface EC density has fallen to 37%, describes the skin-effect: ı=



1 f

(1)

261

where f is excitation frequency,  is magnetic permeability and  is electrical conductivity. The locus of changes in impedance or induced voltage during the movement of EC probe over the material is called an EC signal. As illustrated in Fig. 1, the probe response or signal from a line scan of a probe is displayed on a complex plane with resistance as abscissa and inductive reactance as ordinate. In general, maximum magnitude of an EC response is the magnitude corresponding to maximum interaction region of a probe with defects which is highlighted in Fig. 1. The phase angle , at the maximum magnitude of the response is also displayed in Fig. 1. A series of parallel line scans (raster scan) of the probe over the material surface produces an EC image that gives a better perspective of defect with the information on spatial extent. Interestingly, it is also possible to simultaneously excite an EC probe with currents at several frequencies and obtain a response that has the information from entirely different interrogating depths. Multi-frequency technique is very important for sizing of subsurface defects. As can be seen from Fig. 1, four important characteristics of subsurface defects are length, width, depth and height. The information about length, width, depth and height of defects is embedded together in an EC image and cannot be readily separated. Typical EC images of a defect at 4 different frequencies are shown in Fig. 2. It can be observed from Fig. 2 that the length and width of the EC response of the defect are oversized than the actual length and width due to convolution of probe footprint with the defect and also due to the divergence of electromagnetic field in the thickness region. The image blur depends on the excitation frequency as well as the size of the defect. As a result, defect sizing will be inaccurate. In general, length and width of defects can be approximately estimated from EC images with image processing and the knowledge of probe footprint while determination of depth and height is rather complex. For sizing of defects, several researchers preferred classification algorithms to continuous mapping [2,3,8,11]. 3. Chain classification Defect sizing is achieved through supervised classification by representing features extracted from EC images as input. The target is to predict the size of length, width, depth and height of a defect. The targets are represented as multiple (output) class variables and each class variable, for example, length, is by itself a multiclass classification [8]. Three possibilities exist for classification of multiple class variables as follows: • Decomposing each class variable into a distinct multiclass classification (independent classification). • Representing a new class variable, formed from all possible combinations of every class variable (compound classification). • Sequencing the class variables where previous predictions in the sequence are used for present classification (chain classification). Independent classifications cannot incorporate dependency between the class variables and the expressive power of classification algorithms is restricted. Compound classifications incorporates dependency, however, it will lead to data scarcity for each new class and a high potential to overfit them. Chain classification is a sequence of independent classifications combined with dependency [14,15,17]. This is an attractive method for sizing of defects from EC images with multiple class variables. In chain classification, d independent classification functions (g1 (.), g2 (.), . . ., gd (.)) are trained, representing d class variables. The process starts with the function g1 ({xi }N ), where x ∈  ⊂ Rm is a i=1 N

vector of input features, predicting class values {ci1 }i=1 for N input instances. Then, during execution of the function g2 (.), along with N

x, the class values c1 is also provided as input: g2 ({xi }N , {ˆci1 }i=1 ), i=1

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Fig. 1. EC testing of an electrically conducting plate for detection of subsurface defects.

N

that predicts class values {ci2 }i=1 . In this way, the input feature x

along with all predicted class values c1 , c2 , . . ., ck−1 is provided to the function gk (.). The chain process completes with the function N

N

N

gd ({xi }N , {ˆci1 }i=1 , . . ., {ˆcid−1 }i=1 ) predicting the class values {cid }i=1 . i=1 The classification function g(.) can be any supervised classification algorithm such as ANN, SVM, decision trees and k-nearest neighbours. A typical chain classification for subsurface defect sizing is illustrated in Fig. 3. In this study, two diverse representations and learning algorithms, namely, RBFNN and SVM have been considered in chain classification. RBFNN has demonstrated its capability of classification in various domains including fault detection [18] and gene classification [19]. RBFNN consists of three layers, an input layer, a hidden layer, and an output layer. The hidden layer is formed from centroids that are determined by performing cluster analysis on the input layer. Determining the number of clusters is a user defined parameter called clustering factor, ˛, which is a fraction of the size of input

data. For better generalization, and reliable classification, lower ˛ is preferred. The hidden layer is activated by the Gaussian activation function governed by the spread parameter, . The output layer corresponds to the possible output class values and computed by minimizing the error between the hidden and the output layer. RBFNN can be used for multiclass classification. RBFNN is described in detail elsewhere [20]. SVM has also received considerable attention for binary and multiclass classification in diverse areas of science and engineering such as fault diagnosis [21], speech recognition [22] and power prediction [23]. The basic concept of SVM is to find a decision boundary in the form of a hyperplane that bisects binary classification training data to its respective class. A constraint is posed to the decision boundary that it should contain a large margin between the classes. Formally, it leads to a convex optimization problem that is solved in its dual form. For nonlinear separable data, such as EC image features, use of kernel functions such as RBF, polynomial and sigmoid are highly preferred. The kernel functions are governed by

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Fig. 2. Typical EC images of defect (length: 20.0 mm, width: 2.0 mm, depth: 2.0 mm, height: 1.0 mm) at 1, 5, 10 and 25 kHz.

Fig. 3. Typical chain classification for subsurface defect sizing.

the parameters such as ı for RBF, degree of polynomial, d for polynomial, and ˇ and  parameters for sigmoid. Detailed description of SVM, along with its properties can be found elsewhere [24,25]. For defect sizing, RBFNN and SVM have been considered as the classification algorithms in the present chain classification study. The central question remains with chain classification is the determination of an optimal sequence of class variables to incorporate in the chain. This is an optimization problem which aims at identifying a sequence that maximizes the performance of the chain. 4. Greedy breadth-first-search (GBFS) algorithm for determination of an optimal sequence The determination of an optimal sequence for chain classification is an optimization problem that can be represented as a search tree and it is different from traditional search strategies [26] in the following aspects: • Initially, the target (optimal sequence) is unknown. • Performance evaluation (requires computation of classification algorithm) at each node (node expansion) of the search tree is computationally intensive than a mere visit to the node. • Performance is a vector (a component for each class variable) and depending on the path and level of the search tree, the evaluation function modifies different components of the performance vector. • Performance evaluation at a node of the search tree provides only partial information about the target.

263

To address the above optimization problem, in this paper a modified breadth-first-search (BFS) called, greedy breadth-first-search (GBFS) is proposed. GBFS can optimize the sequence rapidly than traditional uninformed search strategies. The objective of the GBFS is to avoid tracing poor performing sequences that possibly degrade the performance and further, propagate the degradation in the chain. An example search tree for the GBFS is illustrated in Fig. 4. Three class variables {C1 , C2 and C3 } are taken in the example and represented as {Vc1 , Vc2 , Vc3 }. Vc1 is considered as the root node of the search tree and the first level and the second level chain is also displayed in Fig. 4. In the example, initially, Vc1 is visited and expanded, followed by the visit and expansion of {Vc1 → Vc2 } and {Vc1 → Vc3 }. {Vc1 → Vc2 } represents the class values of C1 that are included during prediction of C2 . The searching process continues, following the BFS. In the example, the performance of {Vc1 → Vc2 → Vc3 } is computed by classification, only if, the performance of C2 in {Vc1 → Vc2 } is greater or equal to the performance of C2 during independent classification. Similarly, the performance of {Vc1 → Vc3 → Vc2 } is computed by classification only if the performance of C3 in {Vc1 → Vc3 } is greater or equal to the performance of C3 in {Vc1 → Vc2 → Vc3 }, if it is expanded, otherwise it is compared to independent classification of C3 . The fact that GBFS algorithm ensures optimal sequence has been shown in Appendix. The pseudo code for GBFS is given in Fig. 5. Input to the GBFS algorithm is a class variable represented as a root node of the search tree. The children of this node are first level chain formed from all possible remaining class variables and so on. The algorithm initiates with a queue, for implementation of the BFS, and a bestPerformance tracker, containing the performance of each class variable. Any comparison is carried out only to the concerned individual class variable. The tracker bestPerformance is updated at each node expansion. A node is expanded using computePerformance(.) only if the performance of its parent is equal to the value of the bestPerformance tracker for the concerned class variable. At the end all the paths which improved performance over independent classification are returned. The performance evaluation is discussed in the next section. 5. Performance evaluation Each expansion of node requires an evaluation of performance of a classifier. Fig. 6 gives a pseudo code for the function computePerformance(.). It takes a node Vci for evaluation of a class variable ci and adds all the class variable of its parents in the search tree as input along with the input features. Then it performs a 10-fold cross validation using a classifier such as ANN, SVM or RBFNN. With cross validation the performance at that node can be computed using accuracy in the scale of 0–1 as the metric as defined in Fig. 6. Apart from evaluation for individual class variable, this paper also evaluates results for all the class variables together using weighted mean accuracy (WMA). Given a test set with N instances D

N

and D dimensional of input feature S = {(xn , {cnd }d=1 )}n=1 , where each class variable c d ∈ Y d = {y1d , y2d , . . ., ykd } and a predicted set d

D

of class {pdi }d=1 , then the WMA can be defined as the following: WMA =

1  kd  ı1 (cnd , pdn ) D N k i i=1 n=1 d=1

where

ˇ1 (cnd , pdn )

N

D

 =

1,

if cnd = pdn

0,

otherwise

(2)

.

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Fig. 4. Classifier chain tree and sequence of expansion of nodes using BFS and GBFS.

The WMA can be better explained using a toy example of actual and predicted class for two class variable c1 and c2 . Let c1 ∈ {A, B, C} and c2 ∈ {X, Y} i.e. number of possible classes for class variable c1 is k1 = 3 and c2 is k2 = 2. Let the number of instances where a classifier algorithm predicted accurate be 7 and 6 for c1 and c2 , respectively. Let the total number of instances be 10. Then the WMA of this toy example is =

1 10

 3 5

 2

×7 +

5



×6

= 0.66

6. Approach for defect sizing Coupling the chain classification with the GBFS, a new approach has been proposed for sizing of defects from Eddy current images and it is shown in Fig. 7. In this approach, RBFNN and SVM have been used as classification algorithms in chain classification. The clustering required for hidden layer determination of RBFNN has been carried out by traditional k-means algorithm. Polynomial function has been used as the kernel of SVM. As EC images are essentially high dimensional data, classification using them is computationally expensive. Therefore, certain characteristic features have been obtained for dimensionality reduction and redundancy elimination. These features have been used as input for chain classification in the proposed approach. Machining of defects and generation of EC images of defects in large numbers by experiment for training RBFNN and SVM is time consuming and too expensive. It is also difficult to introduce buried subsurface defects with known dimensions, e.g. 2.0 mm height defect located at 2.0 mm below surface in a 5.0 mm thick plate. Therefore, numerically modeled EC images have been used for training purpose and experimental EC

images of defects with known size, fabricated using electric discharge machining (EDM) have been used for testing and validation. 6.1. Dataset description 6.1.1. Training dataset Eddy current images for training have been generated by modeling the governing equations of the EC phenomenon using CIVA software version 9. CIVA is a benchmark software for modeling of eddy current, ultrasonic, and radiography nondestructive evaluation techniques. Numerical modeling of CIVA is based on semi analytical methods using the dyadic Greens functions. The efficiency of numerical model by CIVA eddy current module has been validated by a series of experiments [27,28]. The modeled EC probe has two coils, one for excitation and other for reception of EC response. The excitation coil has been shielded with a cup type ferrite core. The cross sectional view of the modeled probe is shown in Fig. 8. Defects of four different lengths (20.0, 25.0, 30.0, and 35.0 mm) have been modeled in an austenitic stainless steel AISI type 304 plate of 5.0 mm thickness. Width of defects has been varied as 1.0, 2.0, and 3.0 mm. Using CIVA, EC images of defects have been obtained at two different excitation frequencies (5 kHz and 10 kHz). At these frequencies the depth of penetration of eddy currents of approximately 5.0 mm and 4.0 mm is expected. A dataset of 300 defects has been modeled with combinations of different heights (0.5, 1.0, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 2.8, 3.0, 3.3, 3.7, 3.8, and 4.0 mm) and depths (1.0, 1.2, 1.3, 1.4, 1.7, 1.8, 2.0, 2.2, 2.6, 2.7, 3.0, 3.2, and 3.7 mm). It is noteworthy that among the 300 modeled defects, 236 are completely embedded in the material (subsurface) and do not open on either side of the plate and 64 defects open to the surface opposite to the probe.

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Fig. 5. The pseudo code of GBFS algorithm for optimal sequence determination.

6.1.2. Testing dataset For testing and validation, experimental images have been obtained using an eddy current instrument. This system consists of a sine wave generator for exciting the EC probe with frequencies in the range of 500 Hz to 80 kHz. The sine wave generator output is fed to the power amplifier that drives the EC probe. A lock-in amplifier

is used for measuring the in-phase and quadrature components of the receiver coil induced voltage sinusoids. The measured in-phase and quadrature components from 2 different frequencies are digitised using a NI PCI-6220 data acquisition (DAQ) card and software developed using LabVIEW. An x–y scanner is used to move the EC probe over the plate surface to make linear scan (signal) or a few

Fig. 6. The pseudo code for performance evaluation.

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Fig. 8. Cross sectional view of the modeled probe.

Fig. 7. Proposed approach for sizing of defects.

line scans in a raster (image). The movement of the probe is controlled using a stepper motor and software developed in LabView. Photograph of the EC instrument and the probe used is shown in Fig. 9. For experimental investigation, 9 defects have been fab-

ricated by EDM process on a 5.0 mm thick stainless steel plate specimen. Care has been taken such machined defect dimensions are nearly same as that used in the model. During raster-scan imaging, centre of the EC image i.e. ROI (region of interest) is aligned with the centre of the defect. The scan pitch has been fixed as 1.0 mm along both x and y directions. The distance covered by the EC probe along the length direction is 60.0 mm while it is 70.0 mm across the defect. The region surrounding a detected feature in all four directions above a threshold of 10 mV is segmented and considered as ROI. The dimensions of the fabricated defects are given in the Table 1. The depth of defects can be obtained by subtracting the height from the thickness of the plate (i.e. 5.0 mm), as the defects are open to the other side surface. The typical CIVA modeled EC images of defect-1 at 5 kHz and 10 kHz are shown in Fig. 10 while

Fig. 9. Photograph of (a) the EC instrument used for test data generation and (b) the EC probe.

Fig. 10. CIVA modeled EC images of defect (depth: 3.0 mm, height: 2.0 mm, length: 25.0 mm, width: 2.0 mm) when EC probe is excited with (a) 5 kHz and (b) 10 kHz.

S.S. Ahmed et al. / Applied Soft Computing 40 (2016) 260–273 Table 1 Dimensions of defects fabricated for testing and validation.

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Table 3 The features extracted from EC images for a defect at 5 kHz and 10 kHz.

Defect no.

Length, mm

Width, mm

Depth, mm

Height, mm

Feature number

Feature

Defect-1 Defect-2 Defect-3 Defect-4 Defect-5 Defect-6 Defect-7 Defect-8 Defect-9

25.0 25.0 25.0 30.0 30.0 30.0 35.0 30.0 35.0

2.0 2.0 3.0 2.0 2.0 3.0 2.0 2.0 2.0

3.0 2.0 2.0 3.0 2.0 2.0 3.0 1.0 2.0

2.0 3.0 3.0 2.0 3.0 3.0 2.0 4.0 3.0

F1

Distance from maximum peak magnitude to 25% of peak magnitude along length direction Distance from maximum peak magnitude to 50% of peak magnitude along length direction Distance between extreme peak magnitudes along length direction Total area of defect response covering within 25% of peak magnitude in the image Total area of defect response covering within 50% of peak magnitude in the image Distance between extreme peak magnitudes along width direction Maximum magnitude in the EC image at 5 kHz and 10 kHz Phase angle at the maximum magnitude in the EC image Ratio of maximum magnitudes of EC image at 5 kHz and 10 kHz

F2 F3 F4 F5 F6

experimentally obtained images of this defect are shown in Fig. 11. As can be seen, a good agreement exists between the experimental and modeled images.

F7 F8 F9

6.2. Feature extraction The defects have been categorized into 4 classes for depth and length and 3 classes for height and width, as shown in Table 2. Sets of features have been extracted from the images for classification of length, width, depth and height. Table 3 enlists the features extracted for an excitation frequency. From the EC images, the geometrical distance covered from maximum peak magnitude to degrade 25%, and 50% of peak magnitude has been taken as features for classification of length and width of the defects. The geometrical distance covered between two extreme peaks in an EC image has also been taken as a feature for classification of length and width as shown in Fig. 12. The magnitude and phase angles from the EC images have been extracted as features for the classification of depth and height. Additionally, for defect height classification, the ratio of maximum magnitude at two frequencies has also been extracted. Fig. 13 displays the maximum magnitude for different classes of depths at the two frequencies. It can be observed from

Fig. 13, there is a significant overlap between the different classes and this highlights the complexity of defect sizing from the EC measurements. 7. Results and discussion 7.1. Independent classification In order to understand the sensitivity of the classification algorithms, with respect to their parameters (˛ and  for RBFNN and d for polynomial kernel of SVM), an analysis has been performed. Using the sensitivity analysis, it is possible to optimize the algorithm. The performance has been evaluated using accuracy on a scale of 0–1 during stratified 10-fold cross validation of training data, and the average of 10 independent runs is reported. The higher is the accuracy the better is the performance. Fig. 14 illustrates the

Fig. 11. Experimental EC images of defect (depth: 3.0 mm, height: 2.0 mm, length: 25.0 mm, width: 2.0 mm) when EC probe is excited with (a) 5 kHz and (b) 10 kHz.

Table 2 Description of classes for subsurface defect sizing. Length, mm

Class

Width, mm

Class

Depth, mm

Class

Height, mm

Class

20.0 25.0 30.0 35.0

L1 L2 L3 L4

1.0 2.0 3.0

W1 W2 W3

<1.5 1.5–2.5 2.5–3.5 >3.5

D1 D2 D3 D4

<2.0 2.0–3.5 >3.5

H1 H2 H3

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Fig. 12. Description of features extracted from an EC image.

performance of the RBFNN under different settings of its parameters, namely, the clustering factor, ˛ and the spread . In particular, ˛ has been varied from 0.05 to 0.65 with an interval of 0.05. The variation in accuracy to variation in ˛ has been studied for different fixed  such as 0.2, 0.4, 0.6, 0.8 and 1.0. From Fig. 14, it can be observed that for a constant , the initial performance improves significantly with increase in ˛. As ˛ increases further, the accuracy remains nearly constant. For lower , the accuracy tends to remain nearly constant throughout for width and depth, but decreases for length and height. For height, even for  = 1.0, the performance decreases with increase in ˛ after a short constant period. On the contrary, when ˛ is fixed, the curve gets closer as  increases, especially for ˛ in the range between 0.2 and 0.3. It can be interpreted as the accuracy tends to produce stable performance as  increases. Based on these observations, ␣ has been set to 0.25 and  has been set to 1.0 and these values have been used throughout the rest of the analysis. Sensitivity analysis of SVM has been carried out by varying the polynomial kernel parameter, d and is shown in Fig. 15. From Fig. 15, it can be observed that there is an initial increase in accuracy when d increases from 1 to 3. With further increase in d, the accuracy tends to be nearly constant until d is 6 and then it decreases. As d is maximum in 3, this has been chosen and has been used throughout the rest of the analysis. With these optimal parameters, Performance of RBFNN and SVM has been compared with WMA Table 4 gives the comparison of RBFNN and SVM for defect sizing. It can be observed from Table 4, that the SVM has produced the highest WMA.

7.2. GBFS algorithm for an optimal sequence identification Fig. 16 illustrates the typical result of GBFS using SVM for depth as a root node of the search tree. L, W, D and H denote the independent classification results on the training data for length, width, depth and height, respectively. D → W denotes the independent classification of depth and its results have been included during the classification of width. Similarly, D → W → H denotes the independent evaluation of depth, its results have been included during the classification of width and further results of depth and width have been included for the classification of height. From Fig. 16, it can be noted that the search starting with depth as root node, expanded 11 nodes using SVM, in contrast to BFS, which would have required computation of 16 nodes. The number of nodes expanded for RBFNN and SVM with each class variable as root node is given in Table 5. In total, the GBFS has terminated with expansion of 28 nodes for RBFNN and 31 nodes for SVM. It would have

Table 4 Comparison of performance of independent RBFNN and SVM for defect sizing on training data. Learning algorithm

RBFNN SVM

Accuracy

WMA

Length

Width

Depth

Height

0.9793 0.9216

0.9020 0.9436

0.9207 0.9876

0.8580 0.9072

0.9200 0.9420

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Fig. 13. Maximum magnitudes of defects at 5 and 10 kHz for different classes of depths (a) D1, (b) D2, (c) D3, and (d) D4.

Table 5 Comparison of number of nodes expanded by GBFS using RBFNN and SVM, and BFS. Class variable of root node

Length Width Depth Height Total

Number of nodes expanded for GBFS RBFNN

SVM

9 6 9 4 28

4 9 11 6 30

Number of node expansion required for BFS

16 16 16 16 64

required 64 expansions using either algorithm with BFS without greedy. Table 6 summarizes the sequences that improved the performance of the chain classification over independent classification

by either algorithm. Interestingly, many sequences as listed in Table 6, has improved the accuracy over independent classification for RBFNN as well as SVM. This can be interpreted as the class variables are dependent on one another and cannot be isolated. However, D → W → H → L is the sequence which has produced the best results, improving the WMA to 0.9335 and 0.9646 for RBFNN and SVM, respectively. The optimal sequence D → W → H → L can also be explained from the eddy current perception. Defect depth influences the probe impedance change (magnitude and phase) more predominantly than that of length, width, or height. By determining the class of the depth at first, the algorithm has the tendency to classify more accurately the width, height and length of the defect. Table 7 compares the performance of chain classification and independent classification. It can be observed from Table 7 that the classification for length and width has produced a marginal

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Fig. 14. 10-fold cross validation performance of RBFNN for classification of (a) length, (b) width, (c) depth, and (d) height for different ˛ and .

improvement in accuracy to 0.9807 from 0.9793, and to 0.9073 from 0.9020 respectively, using RBFNN. Accuracy has increased to 0.9560 from 0.9216 and to 0.9505 from 0.9436 for length and width, respectively, using SVM. It can also be noted that the accuracy of the length using RBFNN is greater than that of SVM in chain classification. Substantial improvement in classification has been achieved for height and it is evident by both the algorithms. It has improved to 0.9140 from 0.8580 for RBFNN and to 0.9595 from 0.9072 for SVM. Thus, the results clearly establish that the chain classification results superior performance for defect sizing than independent classification.

7.3. Chain classification on test data The performance of the proposed approach has been tested using EC images obtained experimentally. The procedure for obtaining the images is described in detail in Section 6.1.2. Table 8 reports the performance of independent classification and chain classification on test data. Chain classification has successfully classified all the defects from the EC images. But independent classification has misclassified Defect-1, Defect-4 and Defect7. It is noteworthy that Defect-1, Defect-4 and Defect-7 are deeper (located 3.0 mm below the surface) among all the defects

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Table 6 Comparison of performance of RBFNN and SVM in chain sequence with different defect characteristics as root node of GBFS. Defect characteristics

Length

Width

Depth

Height

Sequence

Independent L→W L→W→H L→H W→L W→H W→L→H W→H→L D→L D→W D→H D→L→H D→W→L D→W→H D→W→H→L D→H→L H→L

WMA RBFNN

SVM

0.9200 0.9202 0.9281 0.9247 – 0.9295 – – 0.9204 0.9211 0.9244 0.9252 – 0.9331 0.9335 – –

0.9420 – – – 0.9425 0.9498 0.9500 0.9587 0.9464 0.9435 0.9437 0.9483 0.9465 0.9547 0.9646 0.9523 0.9511

Fig. 15. 10-fold cross validation performance of SVM for classification of all defect characteristics for different polynomial kernel parameter, d.

considered. This success is attributed to the influence of dependency and its effective incorporation by the chain classification. For testing the robustness of the chain classification, eddy current images of 3 surface defects (depth: 0.0 mm) and 3 near-surface defects (depth: 0.2 mm) have been tested. The heights of the defects have been varied to 1.0, 2.0 and 3.0 mm. The length and

width of the defects have been kept constant at 30.0 mm and 2.0 mm, respectively. The EC images of a surface and a near surface are shown in Fig. 17. Chain classification with RBFNN and SVM has successfully classified depth and height as class D1 and H1, respectively. Thus, the proposed approach can be used to size both surface and subsurface defects.

Fig. 16. Typical GBFS using SVM with depth as the root node of the search tree with 11 expanded nodes.

Table 7 Comparison of performance of independent as well as chain classification. Learning algorithm

Independent RBFNN Chain RBFNN (D → W → H → L) Independent SVM Chain SVM (D → W → H → L)

Accuracy Depth

Width

Height

Length

0.9207 0.9207 0.9876 0.9876

0.9020 0.9073 0.9436 0.9505

0.8580 0.9140 0.9072 0.9595

0.9793 0.9807 0.9216 0.9560

272

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Table 8 Comparison of performance of independent and chain classification on test dataset. Defect no.

Defect-1 Defect-2 Defect-3 Defect-4 Defect-5 Defect-6 Defect-7 Defect-8 Defect-9 a

True class

L2,W2,D3,H1 L2,W2,D2,H2 L2,W3,D2,H2 L3,W2,D3,H1 L3,W2,D2,H2 L3,W3,D2,H2 L4,W2,D3,H1 L3,W2,D1,H3 L4,W2,D2,H2

Chain classification

Independent classification

RBFNN

SVM

RBFNN

SVM

L2,W2,D3,H1 L2,W2,D2,H2 L2,W3,D2,H2 L3,W2,D3,H1 L3,W2,D2,H2 L3,W3,D2,H2 L4,W2,D3,H1 L3,W2,D1,H3 L4,W2,D2,H2

L2,W2,D3,H1 L2,W2,D2,H2 L2,W3,D2,H2 L3, W 2, D 3, H 1 L3,W2,D2,H2 L3,W3,D2,H2 L4,W2,D3,H1 L3,W2,D1,H3 L4,W2,D2,H2

L2,W2,D3,H2a L2,W2,D2,H2 L2,W3,D2,H2 L3,W2,D3,H2a L3,W2,D2,H2 L3,W3,D2,H2 L4,W1a ,D3,H1 L3,W2,D1,H3 L4,W2,D2,H2

L2,W2,D3,H2a L2,W2,D2,H2 L2,W3,D2,H2 L3,W3a ,D3,H1 L3,W2,D2,H2 L3,W3,D2,H2 L4,W3a ,D3,H1 L3,W2,D1,H3 L4,W2,D2,H2

Misclassified data.

Fig. 17. EC images of (a) surface and (b) near-surface defect (length: 30.0 mm, width: 2.0 mm and height: 1.0 mm).

8. Conclusion Automated non destructive sizing of subsurface defects from eddy current images using machine learning has been addressed in this paper. This paper uses chain classification to incorporate dependency among the class variables (length, width, depth and height); this is otherwise assumed to be independent. For optimization of the best sequence to be used in the chain, an algorithm called greedy breadth-first-search (GBFS) has been proposed in this paper. GBFS with radial basis function neural network and support vector machine is capable of rapid sequence optimization for chain classification by avoiding the search of poor sequences which degrade the classification performance. The major conclusions drawn from the study are as follows: • Chain classification with SVM appears very good for sizing of subsurface defects, especially, the depth and the height. This algorithm has ensured a WMA of 94.2% as compared to 92.0% achieved by RBFNN. • D → W → H → L is found to be the optimal sequence, by both the RBFNN and SVM. The same sequence determination by two diverse algorithms strongly indicates the dependency structure during sizing. • Chain classification has been able to significantly incorporate the dependency existing among the class variables and has successfully resulted in sizing of defects located even 3.0 mm below the surface, from the response obtained through numerical modelling as well as through experimental measurements. • Chain classification has able to successfully classify, depth and height of the surface as well as near-surface defects, confirming its robustness.

Studies with experimental EC measurements of various fabricated defects validate the effectiveness and efficiency of the chain classification, indicating that it is a promising method for automated and accurate sizing of defects. Further investigations deal with studies related to oriented as well as irregular shaped defects.

Acknowledgements The authors wish to acknowledge the help received from Mr. Anil Kumar Soni, DGFS-PhD fellow, NDE Division, Indira Gandhi Center for Atomic Research (IGCAR) and Dr. S. Thirunavukkarasu, Scientific Officer, NDE Division, IGCAR. One of the authors Dr. Shuaib thanks IGCAR for providing the fellowship to undertake this research work in IGCAR.

Appendix A. Appendix The proposed Greedy breadth-first-search (GBFS) algorithm ensures optimal sequence which can be shown with following statements: Let Pv2 (V1 → V2 ) be the performance evaluation for the class variable v2 where the predictions of v1 are given as the input for the prediction of v2 . Let Pm (V1 ,2 ) be the overall performance evaluation, where the class variables v1 and v2 are independently predicted. Then the following relationships hold: if

PV 2 (V1 → V2 ) ≥ PV 2 (V2 )

⇒

Pm (V1 → V2 ) ≥ Pm (V1 , V2 )

(A1)

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and if

PV 2 (V1 → V2 ) < PV 2 (V2 )

⇒ Pm (V1 , V2 ) ≥ Pm (V1 → V2 )

(A2)

Let {V1 → V2 → · · · → Vi → Vj → · · · → VD } be an optimal sequence for a system. Let us assume the performance PVj (Vi → Vj ) < PVj (Vj ). Then following (A2) the following can be stated Pm (V1 → V2 → · · · → Vi , Vj → · · · → VD ) ≥ Pm (V1 → V2 → · · · → Vi → Vj → · · · → VD ) which is a contradiction to the statement {V1 → V2 → · · · → Vi → Vj → · · · → VD } is the optimal sequence. Hence, for overall sequence to be optimal all the class variables should increase in the performance. Thus, GBFS ensures optimality by avoid tracing poor performing sequences that possibly degrade the performance and further, propagate the degradation in the chain. References [1] S.-F. Chuang, J.P. Basart, J.C. Moulder, The application of wavelets and fuzzy logic to eddy current flaw detection in steam generator tubes, in: D.O. Thompson, D.E. Chimenti (Eds.), Review of Progress in Quantitative Nondestructive Evaluation, Springer US, 1998, pp. 775–781. [2] F. Lingvall, T. Stepinski, Automatic detecting and classifying defects during eddy current inspection of riveted lap-joints, NDT E Int. 33 (1) (2000) 47–55. [3] R.N. de Mesquita, D.K. Ting, E.L. Cabral, B.R. Upadhyaya, Classification of steam generator tube defects for real-time applications using eddy current test data and self-organizing maps, Real-Time Syst. 27 (1) (2004) 49–70. [4] S. Ahmed, S. Thirunavukkarasu, B.P.C. Rao, T. Jayakumar, Competitive learning on cosine similarity method for classification of defects in two-layered metallic structures, Stud. Appl. Electromagn. Mech. 36 (2012) 183–191. [5] L. Udpa, S.S. Udpa, Eddy current defect characterisation using neural networks, Mater. Eval. 48 (9) (1990) 342–347, 353. [6] B.P.C. Rao, Practical Eddy Current Testing, Alpha Science Int’l Ltd., 2006. [7] B.P.C. Rao, Eddy current testing: basics, J. Non Destr. Test. Eval. 10 (3) (2011) 7–16. [8] S.-J. Song, Y.-K. Shin, Eddy current flaw characterization in tubes by neural networks and finite element modeling, NDT E Int. 33 (4) (2000) 233–243. [9] B.P.C. Rao, B. Raj, T. Jayakumar, P. Kalyanasundaram, An artificial neural network for eddy current testing of austenitic stainless steel welds, NDT E Int. 35 (6) (2002) 393–398. [10] S. Thirunavukkarasu, B.P.C. Rao, T. Jayakumar, P. Kalyanasundaram, B. Raj, Quantitative eddy current testing using radial basis function neural networks, Mater. Eval. 62 (12) (2004) 1213–1217.

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