J Nondestruct Eval (2017) 36:40DOI 10.1007/s10921-017-0420-x

The Application of Rough Sets Theory to Design of Weld DefectClassifiers

Tomasz Chady1 · Ryszard Sikora1 · Leszek Misztal1 · Barbara Grochowalska1 ·Bogdan Grzywacz1 · Michał Szydłowski2 · Paweł Waszczuk1 · Mariusz Szwagiel3

Received: 6 February 2017 / Accepted: 8 May 2017 / Published online: 17 May 2017© The Author(s) 2017. This article is an open access publication

Abstract The innovative method for weld defect classifi-cation based on rough set theory is presented in this study.The classification rules have been generated by processingof data base composed of 640 radiographic images refer-ring to certain welding process in aircraft industry. Theobtained accuracy of defect identification (from 88% up to100%, depending on class of defect and choice of classi-fier) can be evaluated as at least competitive or even betterone comparing to results referring to other type of fre-quently “exploited” classifiers, those mentioned in attachedoverview section. The identification of weld defects is thefinal operation which is premised by complicated “chain”of consecutive operations transforming primary radiographsto the form enabling calculation of conditional attributes.That is why brief description of process of transformationof primary radiographs to the forms which are suitable forattributes calculation is included in the paper.

Keywords Weld defects classification · Classifiers based onrough sets · Relevance of attributes · Radiographic imagesprocessing

1 Introduction

The radiographic NDT methods of welds inspection are com-monly used in order to evaluate the quality of welds. In period

B Tomasz Chadytchady@zut.edu.pl

1 Faculty of Electrical Engineering, West PomeranianUniversity of Technology, Szczecin, Poland

2 Faculty of Mechanical Engineering and Mechatronics, WestPomeranian University of Technology, Szczecin, Poland

3 Pratt & Whitney S.A., Rzeszow, Poland

since 1990 till now we can observe considerable efforts aimedat automatic analysis of radiographs. The current paper con-centrates on the idea and possible applications of promisingmethod for classification of welds defects based on roughset theory [1–4]. The usefulness of proposed method forcomputerized classification of weld flaws was confirmed byintroductory experiments and comparative data presented in[5,6]. The method allows to determine the reducts, i.e. mini-mal set of attributes which preserve ability of distinguishingthe classes of weld defects. Thus, the redundant attributescan be removed by means of formalized procedures. Thediscussion of other problems encountered during realizationof consecutive operations preparing input image data to theform which allows coming to decision on weld quality andclass of imperfection is marginally included in scope of cur-rent paper. Let us note that hundreds of publications yieldedduring last 20 years have been devoted to these problems.The automatic classification of weld defects is based on inputdata generated by chain of previously done operations, i.e.weld images acquisition and preliminary processing, seg-mentation, feature extraction, detection of flaws. Thus, evenfaultless classifying algorithms do not guarantee perfect finalresults for “poor” input data caused by errors and omissionscontributed by former transformations of primary image. Thestandard EN ISO 6520-1 distinguishes between five types ofwelding defects, i.e. cracks, slag inclusions, porosity, lack offusion, and continuous undercuts. The levels of acceptanceof welds containing those imperfections are determined instandard EN ISO 5817 and the other ones.

Depending on sizes and other parameters referring to geo-metric and texture properties of single defect or cluster ofdefects the classifying algorithm should recognize them asacceptable or not. If necessary, the weld flaws should beassigned to classes defined by standards mentioned above.

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In order to design classifying algorithms one has to choosethe set of parameters (descriptors, conditional attributes) asbasis for decision making process and select the type ofclassifier processing these parameters. The computationalcomplexity can be important agent influencing the choiceof classifying algorithm as well. If we take into accountquite substantial number of known type of classifiers andprocedures determining classifier parameters as well as datadrawn from published reports, where up to about 60 differ-ent attributes describing weld flaws were processed, then thefollowing conclusion seems to be obvious: design of classifi-cation algorithm occurs as task characterized by large numberof “degrees of freedom”. That is why researchers proposedvariety of approaches. It could be argued that almost all exist-ing data mining methods and algorithms for classificationhave been used by scientific teams struggling with maximiz-ing of classification accuracy. The still existing importanceof problem discussed in current paper is proved by tens, ifnot hundreds, papers and reports dealing with welds flawsdetection, segmentation and classification published everyyear since the early nineties of the last century to the presentdays.

2 The Brief Overview of Methods Used for WeldDefects Classification

The numerous approaches have been used for design of welddefects classifiers. It seems, that most of them were designedby means of methods belonging to class of statistical and/orAI ones. Basing on the arbitrary and rough review of attemptsto classification given in this section one is able to recognizewhat are the typical algorithms which are useful for automaticclassification of weld defects. Furthermore, the overviewprovides with brief information on accuracies of various clas-sifiers as well as number and “essence”of input parametersused in classification procedures. The short information onless or more complicated methods of determining of relevantand redundant parameters can be found as well. Neverthe-less, the advanced presentation of contemporary “state of art”ought to be the subject of comprehensive, dedicated publica-tion which does not fall in framework of current paper.

In recent paper [7] the Bayesian Networks (BN) basedon probability theory has been used as classifying tool.Choosing the invariant geometric attributes like in [8], i.e.compactness, elongation index, rectangularity, symmetry,deviation index to the largest inscribed circle, Euclidianlengthening, etc. and learning the BN with set composedof several hundreds of welds samples the promising resultshave been obtained. For “fifty/fifty” splitting of data intolearning and testing sets the defects, like cracks (CR), lack ofpenetration (LP), porosities (PO), slug inclusions (SI), wereclassified correctly with accuracy 90–97%.

The expectation-maximization (EM) statistical method ofclassification of weld defects is presented in [9]. The dataclustering EM algorithm is an iterative way finding maximumlikelihood or maximum a’posteriori estimates of parametersof statistical models, where model depends on latent vari-ables. The results yielded by EM algorithm were comparedto results obtained by artificial neural network (ANN) classi-fier with input represented by 4 principal components createdon the basis of several “elementary” features and learnedby error backpropagation technique. The presented exper-imental results showed that performance of ANN classifier(accuracy about 97%) is a little bit better comparing to resultsof EM method (accuracy about 92%).

In [8] problem of relationship between elementary geo-metric parameters of weld defects and classes of defectshas been considered. This analysis inclined authors to intro-ducing of “hybrid” parameters SGD (Shape GeometricDescriptor) and GFD (Generic Fourier Descriptor). Fur-thermore, the combined descriptor f(CFD,FGD) has beenproposed in order to discriminate better the problematicdefects. The use of hybrid descriptors is especially advisedwhen CBIR (Content-Base Image Retrieval) technique ofclassification is used. This technique consists in compari-son of currently examined weld defect to correctly classified“patterns” of defects collected in data base. It is worth to bementioned that procedure of creation of “hybrid” parameterson the basis of primary parameters is one among commonlyused advanced tools aimed at deletion of inconsistencies indecision tables.

The very interesting study of statistical approach to welddefects classification on the basis of texture features is pre-sented in [10]. The following two groups of widely usedtexture features are taken into consideration:(1) features based on the co-occurrence matrix, which givesa parameterized information of how often one grey valuewill appear in a specified spatial relationship to another greyvalue on the image (angular second moment, contrast, cor-relation, sum of squares, inverse difference moment, sumaverage, sum variance, sum entropy, entropy, difference vari-ance, difference entropy, information measures of correlation1, information measures of correlation 2, maximal correla-tion coefficient),(2) features based on 2D Gabor functions.

The selection of features for classification from the ini-tial set of 64 based on Gabor filtering and 148 basedon co-occurrence matrix were done by means of the SFS(Sequential Forward Selection) method and analysis ofROC (Receiver Operation Characteristics) curves. The paperstates that best texture features drawn from co-occurrencematrix were the mean of the difference entropy and themean of the difference variance, for a distance of d=3,while the best Gabor’s features were those obtained forscale p=6, and orientations: \,—, /. The classification was

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executed by means statistical classifiers, i.e. polynomial,Mahalanobis and nearest neighbor. The sensitivities (definedas TP/(TP+FN), where TP—number correctly classifieddefects, FN—number of flaws recognized as non-defect)yielded by above classifiers were about 91% except thosefor nearest neighbor procedure, where sensitivities were lessthan 80%. This evaluation has been obtained after examina-tion of almost 1400 samples including 200 with defects.

The AI techniques can be concerned as most populartools used to classification of weld defects. The classicexpert system with weld defects classification rules drawnfrom sophisticated decision tree is described in [11]. Theimplemented knowledge base was gathered from specialists,textbooks and international standards. The expert system canidentify the 11 faults in gas pipelines welded by shieldedmetal arc welding. The classification of flaws is carried out onthe basis of features concerning shape of defect (rectangular-ity factor, quotient of perimeter and area), orientation (size inX direction, size in Y direction), location (inside weld, centreof weld, edge of weld, base metal). The dimensions of classi-fied weld defect are compared with requirements of standards(API, ASME, DIN, BS, AWS, ABS, JIS) which allow to for-mulate final decision on weld acceptation or rejection. Thediscussed paper does not contain the numbers characteriz-ing accuracy of classification. Nevertheless, it can be foundconcluding statement that accuracy does not differ from thisyielded by top experts.

The other rule-based expert system for detection of flaws(however without procedures for classifying of them) hasbeen described in [12]. This system seems to be one amongpioneering, where attributes of transversal gray level line pro-files and “technology” of curve fitting have been used to findthe weld defective areas.

The fuzzy expert systems presented in [13,14] can betreated as classic examples of classifier based on fuzzy-logic(FL) approach. The so-called WM (Wang, Mendel) machinelearning method [15] was used to generation of fuzzy rulesfrom more than 100 examples. The choice of number of par-titions for input and output data was supported by means ofgenetic algorithm (GA). The other fuzzy rules based classifi-cation algorithm for attributes selected by means of principalcomponent analysis (PCA) was presented in [6]. Dependingon class the obtained MSE (Mean Square Errors) were inrange 0.02–0.05. The best results were achieved for cracks,which was the most separated group in case of PCA anal-ysis. This algorithm was included in software of IntelligentSystem for Radiographs Analysis (ISAR). On the beginningauthors of ISAR system took into consideration almost 60weld defect attributes: 22 representing geometrical and textu-ral properties, 6 representing brightness and 32 representingcentral, normalized and invariant Hu’s shape moments [16–18]. The “basic” classifier included in ISAR system wasthat in form of Multi-Layer Perceptron Neural Network

(MLPNN) with 2 hidden layers containing up to16 neurons[19]. The database for learning and testing contained about1500 samples of weld defects representing standard classes ofweld imperfections. The comparative examinations of resultsgiven by MLPNN classifier and classifiers based on k-meansclustering algorithms proved that accuracy was much betterin case of applying of ANN (depending on class and numberof samples for ANN learning and testing the proper classifi-cation coefficients were in the acceptable range 0.6–0.7).

Artificial neural networks are commonly chosen tools forclassification of patterns based on learning from examples.In researches reported in [20] the set of 43 descriptors cor-responding to texture measurements (angular 2nd moment,contrast, correlation, sum of squares, inverse differencemoment, sum average, sum variance, sum entropy, entropy,difference variance var., data from co-occurrence matrix) andgeometrical features (relative position to the weld bead, theaspect ratio, the length/area ratio, the area/bounding rectan-gle ratio, the roundness, the rectangle ratio, the Heywooddiameter and the relative angle to the weld bead) were deter-mined for each segmented object and put as classifier input.The classifiers were trained to classify each of the objects intoone of the defect classes or indicate lack of defect. Three-foldcross validation was carried out for evaluation of experimen-tal results. The paper contains comparison of results yieldedby ANN-classifier to results obtained by use of SVM (Sup-port Vector Machine) and k-NN (k-Nearest Neighbourhood)classifiers. The experiments were carried out for 411 seg-ments (objects) corresponding to worm holes (85 cases),porosity (94 cases), linear slag inclusion (42 cases), gas pores(13 cases), lack of fusion (57 cases), crack (26 cases) andnon-defects, i.e. false positives (94 cases). To reject redun-dant data the Sequential Backward Selection (SBS) methodhas been applied. The obtained accuracies were about 87%for both ANN and SVM classifiers and identical input datarepresented by 43 descriptors. After diminishing the numberof descriptors to 7 the ANN classifier preserved almost thesame accuracy as in case of 43 descriptors but accuracy ofSVM was considerably lower. For the k-NN classifier theresults were significantly inferior.

The interesting results concerning nonlinear ANN classi-fiers are presented in [21,22]. The ANN was composed of twolayers (intermediate and output one). The number of neuronsin intermediate layer was optimized, i.e. the net performanceand its training errors were taken into account to find theoptimal number of neurons. Like previously, the problem ofrelevance of input data has been examined. The criterion forchoice of relevant attributes was based on observations ofchanges in the network responses when a feature (attribute)used was substituted by its average value. The larger thedifference between these net responses, the larger attributerelevance. The dimension of input vector initially representedwith four components, has been diminished gradually to 3,

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2 and 1 components in function of the features selected byrelevance criteria. The paper shows how to apply the NNnetwork to determining the first principal nonlinear discrim-ination component and consecutive ones. The presented ideacan be treated as generalization of PCA (Principal Compo-nent Analysis) which in fact is widely used techniques for thesize reduction of multivariable data set. The criterion of rele-vance together with nonlinear classifiers proved that only fourof the six initial features were relevant for the classification ofdefects. The obtained accuracy was about 99–100% for train-ing set containing 125 samples described by 4 conditionalattributes each. Thus, the considered paper supports thesisthat “quality” of the conditional attributes is more impor-tant than their “quantity”. By the way: problem of relationsbetween parameters quantity and quality is discussed even in“historical” papers [23,24] where authors convince that nomore than 10 meaningful features is sufficient for accurateweld defect classification.

The problems associated with design of NN classifiers ofweld flaws are deeply discussed in [25]. The samples for 140non-defects, 126 slag inclusions, 87 porosities, 8 transver-sal and 14 longitudinal cracks were examined. The initialset of attributes was composed of 12 members representedarea, centroid (x and y coordinates), major axis, minor axis,eccentricity, orientation, Euler number, equivalent diameter,solidity, extent and position. The samples with defect andnon-defect ones were classified by outstanding experts. TheANN classifiers were learned by backpropagation methodusing Widrow–Hoff algorithm for multiple-layer networksand non-linear differentiable neuron transfer functions. Toeliminate the redundant attributes the PCA transformation ofinput data has been executed. In order to overcome problemof ANN overfitting the procedures of Bayesian regular-ization and bootstrap approaches have been implemented.Furthermore, the classic MSE (Mean Square Error) criterionevaluating accuracy of ANN classifier was supplementedwith additional term representing the mean of the sum ofsquares of the network weights and biases. The paper con-tains comparative results of classifications for Bayesian andbootstrap regularization as well as classic and supplementedMSE criteria of ANN performance. Authors of [25] con-clude that the proposed technique is capable of achievinggood results. The ANN (composed of 11 input neurons and20 neurons in hidden layer) optimized by means of supple-mented MSE criterion and PCA transformation of input datahas occurred to be the best classifier. Depending on class ofdefect the accuracy of classification fell in the range from72% (slug inclusion) up to 96% (cracks). The non-defectsamples were classified correctly with accuracy 92% andobtained mean accuracy was about 80%.

There are numerous reports dealing with classifiers inform of feedforward multilayer neural networks [26,27]where supervised learning of ANN was executed and error

backpropagation learning was often used. The classifiersbased on networks of ART (Adaptive Resonance Theory)type are presented in [28]. The networks of this type areexamples of classifiers which can be learned without super-vision, i.e. algorithm of learning generates patterns of defectson the basis of “distances” between vectors of attributesdescribing weld flaws. It leads finally to determining of cen-ters of clusters and so-called largest radiuses of similarityrecommended for each class of defects. Analysis of so-calleddemerit factors gave possibilities of creation of several pat-tern curves (2–9) for each class of defects. The examinationof classifier under consideration was carried out with use ofattributes drawn from processed gray level curves of weldsamples representing 100 transversal profiles for each con-cerned classes (lack of discontinuity, porosity, longitudinalcrack, slug inclusion, lack of fusion, lack of penetration,undercutting).

In [29] were compared accuracies of NN classifiers tothose based on pruned and non-pruned decision trees opti-mized by measure of entropy. The following types of NNwere used: a backpropagation net, a RBF (Radial Basis Func-tion) Net, a Fuzzy ARTMAP Net and LVQ (Learning VectorQuantization) Net. The 36 features were initially taken intoaccount. The best accuracies calculated for testing set wereobtained for RBF and backpropagation net - both in the range94–95% of correct classifications. The error rates of othertype classifiers were in range 9–12%.

At the end of this section it should be noted, that authorsof papers often neglect presentation of complete informationabout structure of algorithms and their parameters. More-over, the described examinations were not done for identicaldatabases. That is why comparisons of final accuracies ofmethods presented in accessible publications should be donevery carefully.

3 The Classification of Defects Based on RoughSets Theory

The processing of weld defect image yields set of parameter-ized attributes (see former section) which can be useful fordefect classification purposes. The data referring to images ofdefects can be gathered in decision table which can be treatedas representation of information system (SI). The developedin the last twenty years of previous century theory of roughsets [1–4] can be considered both as extension of classic setstheory and tool enabling successful design of classifiers.

The discretization of conditional attributes usually dimin-ishes the number of samples (rows) in decision table becauseparameters of certain subsets of elements (“classes ofabstraction”) belong to the same ranges created for val-ues of attributes by discretization. The modified decisiontable obtained due to discretization can contain redundant

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attributes as well. The so-called “B-indiscernibility relation”can be used as precisely defined tool enabling the rejection ofredundant attributes. Let SI =(U , A)be information system,and B ⊆ A, where U is considered set of samples (“uni-verse”), A, B is set (subset) of attributes respectively. Therelation:

IND(B) = {(x, y) ∈ U ×U : ∀α ∈ B, α(x)= α(y)}(1)

is called the „B-indiscernibility relation”. Thus, the elementsx, y from U which fulfill (1) are indescirnible because allvalues of attributes α belonging to subset B and represent-ing those elements are identical. If subset B induces classesof abstraction which are identical to those generated by Athen attributes of A in classification procedures can be sub-stituted with reduced number of attributes, i.e. by attributesbelonging to B.

The rough sets theory introduces the lower and upperapproximations of set. On this basis one can evaluate thelevel of „belief” concerning the membership of given objectto set D (representing decision, concept, idea, type of welddefect, etc.). The lower approximation:

BIND(B)D = {x ∈ U : [x]IND(B) ⊆ D} (2)

represents elements of universe U belonging to classes ofabstraction induced by attributes of subset B where all mem-bers of these classes fulfill conditions of membership to setD. The upper approximation of D is defined as:

BIND(B)D = {x ∈ U : [x]IND(B) ∩ D �= φ} (3)

and represents all classes of abstraction induced by B whichcontain any elements or even single element belonging to D.

The boundary region:

BN (B)(D) = BIND(B)D − BIND(B)D (4)

is composed of elements belonging to these classes ofabstraction where values of attributes from set B do notallow to decide if given element belong to D or not. Forexample, if we assume that decision D denotes “unaccept-able crack in weld” then exact rules of classification of thisdefect have to be based on lower approximation of D. Forintroductory selection of elements which potentially can bedefective (“cracked”) one can use knowledge defining theupper approximation of D.The elements (welds) of boundaryregion can be cracked or not. For boundary region elements,the task of overcoming of classification inconsistencies hasto be solved (through the introduction of new conditionalattributes, changing of parameters for attributes discretiza-tion, introduction of “artificial” parameters by creation of

functions defined on previously used parameters, etc.). Ofcourse, depending on particular requirements, one can agreefor certain level of faulty decisions using other solutions, likewidely known “simple voting” approach, probabilistic ruleinduction [30]. The other exemplary algorithms processinginconsistent Decision Tables can be found in [30–33]. Thepresented LEM2 (Learning from Examples Module, version2) rule induction algorithm uses rough set theory to handleinconsistent data set. LEM2 algorithm induces a set of certainrules from the lower approximation and a set of possible rulesfrom the upper approximation. The procedure for inducingthe rules is the same in both cases - see [32]. The algorithmimitates the classical greedy scheme yielding a local cover-ing of each examples from the given approximation usinga minimal set of rules [33]. The preliminary discretizationof attributes is not necessary when algorithm called MOD-LEM is used [32]. The MODLEM processes these attributesduring rule induction, when elementary conditions of a ruleare created. The MODLEM algorithm can operate accord-ing to its Entropy Version or Laplace Version. MODLEMuses rough set theory to handle inconsistent examples anddeterminates a single local covering for each approximationof the concept [32]. The search space for MODLEM is big-ger comparing to search space for LEM2, because LEM2creates rules basing on already discretized attributes. That iswhy rules obtained by MODLEM are “simpler and stronger”.The algorithm called EXPLORE extracts all decision ruleswhich satisfy requirements, referring to strength, level of dis-crimination, length of rules and syntax conditions of rules. Itcan be adapted to processing of inconsistent examples byusing rough set approach or by tuning a proper value ofthe discrimination level. Induction of rules is executed asresult of exploring the rule space with simultaneous imposingthe restrictions referring to above mentioned requirements.Procedure of exploration of the rule space is repeated foreach concept to be decided. Each concept can be a class ofexamples or one of its rough approximations in case of incon-sistent examples. The “kernel” of algorithms was built usingbreadth-first exploration of rule space, beginning from one-condition rules. Exploration of given branch is stopped if therequirements are satisfied or stopping condition is fulfilled(impossibility of fulfilling the requirements is attained [33]).

The existence of mentioned and several other algorithmsinclines to using of rough sets approach to classification ofweld defects. Thus, definitions (2), (3), (4) can be taken intoconsideration during organizing of process of inducing ofclassification rules, because depending on results of roughselection (determination of upper approximation) the fartherclassification procedures can be activated. The existing pro-grams, like “Rough Set Exploration System” (http://logic.mimuw.edu.pl/~rses/get.htm), can be helpful tools for clas-sification experiments.

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Table 1 The exemplaryinformation system

A → a1 a2 a3 DU ↓u1 0 0 0 D1

u2 0 1 0 D2

u3 0 0 1 D3

u4 1 1 0 D2

u5 1 1 1 D4

u6 1 0 1 D5

By rejection of redundant attributes we obtain “reduct”,i.e. minimal subset of attributes that enables the same clas-sification of elements of the universe Uas the whole set ofattributes. Usually one obtains set of reducts. Every reduct(set of attributes) contains certain subset of attributes whichis identical for all reducts. It is called “core”. If sets IND(B)=IND(A), then CORE(B) =∩ RED(A), where RED(A)—set of all reducts classifying elements of U identically likewith use of all attributes of A. The parameters of discretiza-tion influence substantially the sizes of sets (2),(3),(4) as wellas forms and number of reducts. In general, the problem ofdetermining of reducts for big sizes of Decision Tabels isconsidered as non-deterministic polinomial-time hard (NP-hard) one. That is why genetic algorithms or other heuristicsare used for determining of reducts. Fortunately, the conclu-sions and advices given in many papers (see previous section)contain statements that weld defect classification should bedone with use of not to big number of relevant parameters.

Let us illustrate the “philosophy” of idea of creating ofclassification rules by elementary example. For Informa-tion System defined by Table 1 we can create DiscernibilityMatrix (Table 2) and Functions, or other equivalent repre-sentations (without redundant data like in symmetric Dis-cernibility Matrix) which are more convenient for computerprocessing. The Discernibility Matrix shows these attributeswhich obtains different values for elements belonging to cou-ple defined by raw and column labels. Thus, one obtainsinformation which attributes are necessary in order to dis-tinguish objects of U , or classes of abstractions ui , if uirepresents class of objects fulfilling the given indiscernibil-ity relation. Let us note, that SI in Table 1 is deterministicone, because it does not contain identical sets of values of

conditional attributes which generate different decisions. Thelack of such property makes SI non-deterministic.

For example, object (or class) u2 differs from u5becauseof values of attributes a1, a3.The discernibility function foru2 is:

f 2 = a2 ∧ (a2 ∨ a3) ∧ a1 ∧ (a1 ∨ a3) ∧ (a1 ∨ a2 ∨ a3)

= ... = (a1 ∨ a2) ∧ (a2 ∨ a3)

This function shows attributes which differ u2 from otherobjects of U. For u4 is:

f 4 = (a1 ∨ a2) ∧ a1 ∧ (a1 ∨ a2 ∨ a3) ∧ a3 ∧ (a2 ∨ a3)

= ... = (a1 ∨ a3) ∧ (a2 ∨ a3)

Hence the decision rule classifying objects to decisionclassD = D2 is:

I F (a1 = 0 AND a2 = 1)OR(a2 = 1 AND a3 = 0)

OR (a1 = 1 AND a3 = 0) T HEN D = D2

Let us note that CORE of SI in Table 1 is composed ofattributes {a1, a2, a3} because of known property, thatCOREcontains all these attributes which exist in “cells” of discerni-bility matrix as single attribute. This means, that number ofattributes can not be reduced (all attributes are necessary) ifthe same classification like in Table 1 has to be kept.

The procedure of data processing for weld defect classi-fication is shown in Fig. 1. The algorithms LERS (Learn-ing from Examples based on Rough Sets) similarly toother approaches based on “learning from examples” whichprocess Decision Tables of big sizes (numerous objects,attributes and ranges of attributes) yield the satisfying clas-sification rules after execution of iterative process whereparameters for discretization of attributes, choice of subsetsof accessible attributes, choice of objects to learning and test-ing sets are changed many times. The welds defects used forlearning of classifier have to be assigned to standard classes ofdefects by top experts. The research process of classificationwith the usage of rough sets theory consists of the six steps,those represented in Fig. 1 as well as by following stages:

Table 2 The discernibilitymatrix

u1 u2 u3 u4 u5 u6

u1 – a2 a3 a1, a2 a1, a2, a3 a1, a3

u2 a2 – a2, a3 a1 a1, a3 a1, a2, a3

u3 a3 a2, a3 – a1, a2, a3 a1, a2 a1

u4 a1, a2 a1 a1, a2, a3 – a3 a2, a3

u5 a1, a2, a3 a1, a3 a1, a2 a3 – a2

u6 a1, a3 a1, a2, a3 a1 a2, a3 a2 –

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1. Loading and preparation of data 2. Discretization of

data

3. Important features selection and

dimension reduction - estimation of reducts

4. Division of data into testing and learning

sets

5a. Creating of learning set

6. Rules estimation and extraction

7. Gaining rules accuracy - confusion matrix

5b. Creation of testing set

Fig. 1 Procedure for weld imperfections identification

• data retrieval from calculated parameters on recognizedweld imperfections,

• discretization of attributes,• computation of attribute importance and decision aimed

at dimension reduction,• dividing of information table into learning and testing

sets,• process of extracting rule sets,• computation of confusion matrix and other qualitative

coefficients.

The commonly used measures of classification accuracycan be applied to results generated by procedures basedon rough sets approach. Thus, one can calculate “basic”evaluating parameters, like true positive TP, false positiveFP, true negative TN, false negative FN, positive Pos =TP + FP, negative Neg = TN + FN, true True = TP +TN, false False = FP + FN. Depending on needs one canalso determine the “derived” parameters: sensitivity (recall)Se = T P/(T P+ FN ), specificity Sp = T N/(T N + FP),precision P = T P/(T P + FP), measure Fb = [(b2 +1)T P]/[(b2 + 1)T ]P + FP + b2FN , accuracy Acc =(T P + T N )/(Pos + Neg), the balanced accuracy BAcc =c1(T P(Pos)1 + T N (Neg)−1)) and some others.

4 The Classification of Weld Defects on the Basis ofReal Data

The main problem discussed in current paper concerns thepossibilities of using of algorithms drawn from rough sets

Fig. 2 The cropped weld radiograph

Fig. 3 B-Spline reconstructed weld image

Fig. 4 Absolute difference image

theory to classification of weld defects. Nevertheless, assign-ment of defects to class defined by standards can be treatedas the “terminal” stage which is executed after several oper-ations processing primary radiographs image. Thus, somebrief information on preparing data to design of weld defectclassifiers is necessary. On the beginning all radiographs werenormalized. This operation fits the distribution of illumina-tion of processed image to distribution represented by knownparameters (mean, standard deviation). Then algorithm ofweld detection and its shape extraction is applied. There arefollowing operations included to this algorithm: low pass fil-tering of image (reduction of noise and texture details), localtresholding of filtered image with use of sliding window inaccordance to Nibalac’s method, removing of morphologicalnoise by closing operation, filtering of obtained binary imageon the basis of pixel area criteria (removing of small objectswith acceptable sizes in relation to defects sizes accepted bystandard requirements), hole filling (creation of weld mask),calculation and extraction of rectangular bounding box witha certain margins from the detected weld. The exemplarycropped and normalized image of the radiograph is shownin Fig. 2. The feature extraction is the next set of operations.The cropped radiograph image is “passed” through medianfilter (removing of “paper and salt” type of disturbances).Next, the weld is divided into N -pixel wide windows. Foreach window an average profile is computed. Then averagedwindowed profiles are fitted by using the B-Spline functions(reconstruction of weld profile). The reconstructed image isshown in Fig. 3. Afterwards the calculation of absolute dif-ference between the reconstructed and original image is done(Fig. 4). Finally, the simple thresholding (threshold level isdetermined on the basis of image mean and STD values)generates the binary images which can be treated asfinal result of stage under consideration. During the nextstage the final filtering by morphological closing reduces theunwanted objects in the image. Next the area based particle

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eliminates one-pixel artefacts obtained due to thresholdingprocess. To erase artifacts outside the image the maskingtechnique is applied. Finally, the objects still existing in theimage are labeled and “exported” for further processing bymeans of identification procedures—see Fig. 5. The exem-plary image of weld containing two types of defects (crack,pores) is shown in Fig. 6. Result of processing of this imageaccording to above mentioned procedure is shown in Fig. 7.

Fig. 5 Detected weld flaws

Fig. 6 Weld containing two types of flaws (crack and pores)

Fig. 7 Detected weld flaws (crack and pores)

The database used later in machine learning process wascreated on the basis of 640 original radiographs of thethin-walled welded joints of aircraft structures. All the radio-graphs were achieved from digital radiography system. Theflaws have been marked manually by an expert with use ofdedicated software application. For each defect its binarymask was created to facilitate computation of the flaw’s geo-metrical attributes. In certain number of cases generatingof binary masks was supported by manual correction. Theexemplary images of the flaw and its mask for each defecttype are presented in Fig. 8. Finally, the images of the flawscut from the original radiograph along with their masks weresaved and the full information about each defect for everyradiograph was gathered in resulting .xml file.

The statistics of flaw’s imperfections in database repre-senting certain process of welding in aircraft industry ispresented in Table 3. One can observe strongly differingnumbers of examples in consecutive classes of imperfec-

Table 4 The statistics of reduced database dedicated for classificationrules induction and extraction

No. offlaws

No. of cracks(class 1)

No. of porosities(class 2)

No. of undercuts(class 5)

152 39 109 4

Fig. 8 Examples of flaws andtheir binary masks: a lack offusion, b undercut, c porosity,d tungstem inclusion, e group ofpores, f crack

Table 3 The statistics ofprocessed data

No. offlaws

No. of cracks(class 1)

No. of porosities(class 2)

No. of inclusions(class 3)

No. of lacks offusion (class 4)

No. of undercuts(class 5)

499 39 453 1 1 4

123

J Nondestruct Eval (2017) 36 :40 Page 9 of 12 40

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40 Page 10 of 12 J Nondestruct Eval (2017) 36 :40

tions which can be result of applying of restrictive, repeatableprocedures during welding. For purpose of design of clas-sifier, the table has to be changed to balance differencesbetween numbers of different type of flaws which is nec-essary for creation of identification algorithm. The numberof porosities has been reduced to 109 and classes referring toinclusion and lack of fusion have been eliminated, becausefarther researches does not make sense for single examplesin these classes (compare the contents in Tables 3, 4). Thementioned above introductory processing of defect imagesyielded binary in grayscale forms representing defects (seeFigs. 5, 7, 8). That is why 21 attributes characterizing shapeof collected flaws have been taken into account. The familyof attributes was initially composed of the seven momentsof inertia I , the seven normalized moments of inertia N andmoments of Hu (H) of orders 1÷ 7, expressed in form offunctions of normalized moments of inertia (compare to setof attributes in Table 5).

The researches were aimed at obtaining of acceptable clas-sifiers based on possible small number of attributes. Thespecificity and scrupulosity of welding process under consid-eration make that set of forms of defects contains “relativelysimilar” members belonging to given class of defects. Thisallowed to diminish substantially the number of relevantattributes up to 2÷3 while 21 attributes were initially takeninto account. Furthermore, due to mentioned above rigors theacceptable accuracies of classification were obtained even forchecking of values of single attribute. These extremely sim-ple classifiers based on single attribute yielded decision afterchecking the value Ixx or H1, where Ixx - moment of inertia(shift invariant moment about the center of mass of defectimage)

Ixx =∑

xx − (∑

xx)2

A, (5)

H1-invariant to shift, scale, rotation first order moment of Hu:

H1 = Nxx + Nyy = IxxA2 + Iyy

A2 . (6)

The A in (6) denotes the area of defect representation (num-ber of pixels), Nxx , Nyy represent normalized moments ofinertia, x, y are the coordinates of pixels of defect in relationto axis x or y respectively. According to procedure in Fig.1 the data has been randomly divided into learning data set(50% of examples) and testing set (50% of examples). Theexemplary “piece of input data” covering six examples forextraction of classifying rules is shown in Table 5.

The accuracy of classification obtained with rules gener-ated by means of RSES program (http://logic.mimuw.edu.pl/~rses/get.htm) is described by figures gathered in the con-fusion matrixes shown in Tables 6, 7. The figures withoutbrackets refer to results of applying of classification rules to

Table 6 The results of classification (confusion matrix for reductbased on Ixx )

Table 7 The confusion matrix for classification by means of reductbased on H1

reduced population of examples, these containing 152 mem-bers (see Table 4). The numbers in brackets represent resultsof applying of the same rules (i.e. drawn from 152 examples)to classification of the entire population of examples (i.e. for499 examples).

5 Conclusions

Let us note, that results for class 5 have been obtained forprocessing of data referring to 4 examples. Thus, they haveto be treated in category of „curiosity”, especially in casewhere all or almost all examples were included to learningset (see figures without brackets for class 5 in Table 6). Thecomparison of qualities of results obtained for classes 1 and 2(accuracy, coverage) to those obtained by use of other meth-ods (see rough overview in Sect. 2) inclines to conclusion,that approach based on rough sets theory is competitive inrelation to other methods. The specificity of processing ofunbalanced data covering in fact 2 classes of defects does notempowers to draw the general conclusions. Nevertheless, oneought to take into account, that presented good results distin-guishing 3 classes of weld imperfections were obtained withuse of extremely “low-size” reducts. This inclines to supposi-tion that classifiers of weld defects “adjusted” to specific andrigorous welding process can be based on small number ofattributes. Thus, the obtained initial results have to be treatedas promising. The final stage of radiograph processing, i.e.classification of weld defects by use of small number of con-

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ditional attributes, cannot be treated as very complicated task.Nevertheless, one should keep in mind the complicated wayof processing of radiographs to the forms enabling calcu-lation of conditional attributes, like these given by (5) and(6). Presently Authors of current paper try to supplementthe data base with more examples referring to imperfectionclasses 3,4,5. The application of described methodology toprocessing of big size, balanced data base covering all classesof weld imperfections should lead us successfully to compre-hensive evaluation of concepts associated with use of roughsets theory to weld defect classification and in fine to designand implementation of automatic system classifying defectsfor chosen operations of welding in aircraft industry.

Acknowledgements The research obtained the financial support ofThe National Centre for Research and Development, Project NoINNOLOT/I/9/NCBR/2013 CASELOT entitled “Advanced methodsfor manufacturing of aero engine case module components”.

Open Access This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate creditto the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.

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