Fuzzy Clustering to the Detection of Defects from Nondestructive Testing

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1 SETIT rd Internatona Conferene: Senes of Eetron, Tehnooges of Informaton and Teeommunatons Marh 7-3, 005 TUNISIA Fuzzy Custerng to the Deteton of Defets from Nondestrutve Testng Hasanzadeh R.P.R. *, Morad M.H. ** and Sadegh S.H.H. * * Eetra Engneerng Department, Amrkabr Unversty of Tehnoogy, Hafez Ave., Tehran, 594, IRAN. hasanzadeh@aut.a.r sadegh@aut.a.r ** Bomeda Engneerng Department, Amrkabr Unversty of Tehnoogy, Hafez Ave., Tehran, 594, IRAN. Abstrat: In NDT mages, defet, generay ontans aggregaton of spea brghtness eve n ts poston. In onsderabe ases, these brghtness eves have no notabe ontrast reatve to bakground of mage. Hene deteton and reonstruton of defet usng automat or hande threshod eves ause to msft the resut. In order to sove ths probem, we used densty and sze of defet for assfaton. In ths paper, Gath-Geva fuzzy usterng agorthm s obtaned for usterng and reonstruton of defets reatve to nose and bakground of mages and ts resut ompared wth two most popuar fuzzy usterng agorthms: fuzzy -means and fuzzy Gustafson-Kesse agorthms. Key words: Custerng, Fuzzy, NDT, Defet.. Introduton Today, dgta sgna proessng s an mportant souton to detet and reonstrut of defets n NDT appatons. Severa knd of appaton n neura networks, fuzzy og and mage proessng for usterng and segmentng of NDT mages have been more attenton than the other knd of dgta sgna proessng methods. Upon havng knowedge howsoever tte of grabbed data, we an use neura network. For nstane, pereptron and bakpropagaton s utzed n order to uster of defet features. In ths method, preproessng stage s apped for reduton of data dmensons usng mage proessng and sef organzng map neura network. Aso pattern reognton tehnque s apped to assfy two knds of defets: rak and deamnaton, n utrason mages (Km et a., 00. Severa knds of asses of eddy urrent data are deteted usng modeed nput sgna traned by a neura network, but anayta nverse modes are possbe ony f strong smpfyng assumptons are onsdered (udpa et a., 99. In addton fuzzy og appatons are expanded n NDT ategory. Wed mage of x-ray ne by ne s proessed whh three features, wdth, peak ntensty and mean square error between the objet and ts Gaussan are extrated. Extrated features were assfed by fuzzy K-NN agorthm (Lao et a., 997. On the other hand they used tra and error urve fttng for nose reduton and feature extraton. Wedng faw n x-ray mages was deteted by fuzzy -means usterng agorthm (Lao et a., 999. Fuzzy nferene s proposed to enhane ontrast of NDT mages. Ths method based on knowedge of ntensty and propagaton deay n sanned data whh s obtaned from puse-radar (Matsumoto et a., 000. Athough waveet-based texture segmentaton s apped for utrason mages, utmatey perentage of damage s omputed by segmentaton usng ISODATA agorthm whh s rsp knd of fuzzy -means agorthm (Km et a., 998. A of the above methods extrated a number of features or apped methods that were based on knowedge howsoever tte of nput sgnas. A gane revew of above methods shows that ony mprovng of agorthms an not hep us to nreasng of effeny. Searhng and fndng new features or nreasng of knowedge of features about test and defet an hep us to enhanng of effeny. In nosy mages and ow ontrast mages, densty and sze of defet are adequate for assfaton. By usng densty and sze of usters, defet, speke nose and bakground, varous asses an be separated.. The Gath-Geva Fuzzy Custerng Gath-Geva (GG fuzzy usterng agorthm s an

2 extenson of the Gustafson-Kesse (GK fuzzy usterng agorthm that aso takes the sze and densty of the usters nto aount (Hoppner et a., 999. GK agorthm s produed by repang the Eudean dstane n fuzzy -means usng a postve defnte and symmetr matrx that epsoda usters oud aso be reognzed, nstead of ony sphera ones. Aso n GG agorthm, the knd of usterng suh as GK agorthm oud aso be reognzed n epsoda usters. We dsuss about GK and -means agorthms ater. Probabst nterpretaton of GG usterng s shown by equaton (: p(x η p(x, η p(x η p( η ( GG agorthm, as a prevous dsussed, aso take the densty and sze of usters for assfaton. Hene, t has better behavour for rreguar features. Gath and Geva assumed that the norma dstrbuton N wth expeted vaue ν and ovarane matrx A s hosen for generatng a datum wth a pror probabty P (Gath & Geva 989. P p(x η.exp[ (X ν A (X ν ] T j P j j ( π det(a ( X {x,x,...,x N}, s the data vetor, the p p ovarane matrx s A?, p s the number of dmensons of data and s the number of asses. The dstane funton, d (X,Cfor GG agorthm s j hosen to be ndrety proportona to equaton ( whh s posteror probabty (kehood funton. A sma dstane means a hgh probabty and a arge dstane means a ow probabty for membershp. GG agorthm s based on mnmzaton of the sum of weghted squared dstanes between the data X and the uster enters of objetve funton n equaton (3. N m,k,k (3 j J(X,U,V u d Condtons for probabst uster partton are: N m [, ; U? ; U,j [0,],, j; N u,k, j; 0 < u,j < N, j (4 In order to approxmatey mnmze J wth respet to a probabst uster parttons ( ν,a,p and gven membershps u j,, the parameters ( ν,a,p of the norma dstrbuton n equaton ( shoud be hosen as foows ( s the number of teratons: N m (u j,j X j ν N m (u j,j N m T (u,j (X j j ν(x j ν A N m (u j,j N m (u j,j X j p N m (u j k k,j (5 (6 (7 We teratvey ompute the parameters n (5 and apped them for auaton the dstane funton and membershps as foows equatons (8, (9: p ( π det (A T d,k(xj, ν exp[ (X j ν (A (X j ν] p u (8,k (9 /(m (d k,j(x j, ν/d k,j(x j, νk For reduton the nose, we an use membershps whh are resut of Dave s objetve funton (Dave 99: u,k /( m /( m (d,j(x j, ν k /d k,j(x, j νk + (d,j(x j, ν / δ (0 The membershp degree of a datum to the nose uster s defned as one mnus the sum of the membershp degrees to a other usters. In the other hand we produe an ambguty rejeton regon whh nreases the number of usters to +. The addtona uster orresponds to the nose uster ν nose for whh d nose,k(xj, νnose δ. 4. The Fuzzy C-means Agorthm Hard -means or hard ISODATA agorthm was deveoped by Duda and Hart (Duda & Hart 973 then Dunn (Dunn 974 and Bezdek (Bezdek 973 ntrodued a fuzzy verson of ths agorthm. The resutng of fuzzy -means agorthm reognzes sphera usters n p-dmensona spae. The usters are assumed to be approxmatey the same sze and eah uster s represented by ts enter. Aso ths agorthm teratvey tres to mnmze the equaton (3 but the dstane funton obtans from equaton (:,k υ j υ ( d (Xj, X - The equatons (6 and (7 have no effet n fuzzy - means agorthm.

3 5. The Gustafson-Kesse Agorthm Ths agorthm desgned by Gustafson and Kesse (Gustafson & Kesse 979. In omparson wth the fuzzy -means agorthm, n addton to uster enters eah uster s haraterzed by a symmetr and postve defnte matrx A. thus, ony the uster shapes are varabe now, but not the uster s szes. Aso n ths agorthm, the equaton (3 shoud be teratvey mnmzed. Dstane matrx obtans from equaton (: T,k ν j ν j ν ( d (Xj, (X A(X The ovarane matrx an be omputed from equaton (3 and (4: p A det(s(s (3 N m T S (u (X ν (X ν,j j j (4 j 6. Expermenta Resuts Fgure (-a shows the orgna utrason mage whh was obtaned from NDE (Moysan et a The mage s a three dmensona data whh s normazed n ts x and y axes and n ts ntensty. Usng any knd of fuzzy usterng just for extraton the ntensty of mage eads to severa threshods, the number of whh s equa to the number of usters. Irregardess of fuzzy usterng tehnques, defets tend to the usters, the threshod eves of whh are smar to ts ntensty. Hene, the defets wth dfferent ntensty may be ed to dfferent asses. In order to avod of above probems, we use densty and sze of usters whh are n x and y axes of mage. Now we evauate resut of the Gath-Geva agorthm. In prmary stage, we apped ths agorthm for two-ass ase: defet-bakground and n seondary stage, we apped t for three-ass ase: defet-speke nose-bakground. Fgures (- a, b show the resut of two-ass GG fuzzy usterng agorthm usng equaton (9. Image of defet s shown n fgure (-a and mage of bakground s shown n fgure (-b. On the other hand n fgures (- a, b and we used three-ass GG agorthm usng equaton (9. We ompare the resuts of two ast stages. Threeass ase separates speke nose from defet whh ause better ontrast of defet s mage of three-ass ase reatve to defet s mage of two-ass ase, but omputng tme of three-ass ase s more than of two-ass ase. Fgures (3- a, b and show the resut of two-ass GG agorthm usng equaton (0 wth δ 3. Ths nos e usterng method uses an ambguty rejeton regon to separate speke nose from defet and bakground. The resut of defet s mage n fgure (3-b s better than defet s mage n fgure (-b and a few ess than defet s mage n fgure (-b but the omputng tme s smar to twoass GG agorthm. Fgures (4- a, b and show that nreasng ofδ redues the speke nose n defet s mage but t moves part of defet to ambguty rejeton ass. Tabe ( shows the abty of eah method. In a fgures, the mages of bakground are approxmatey smar. Usuay the veoty of threeass GG agorthm s 67% of two-ass GG agorthm. We apped nos e usterng suh as two-ass GG agorthm but ambguty rejeton auses a thrd ass wth same veoty. Fgures (5- a, b show the resut of two-ass -means agorthm. Ths agorthm an not orrety assfy the defets. Ths probem repeated for GK agorthm n fgures (6- a, b, the resut of whh s same. Fgures (7- a, b and are for threeass -means agorthm and fgures (8- a, b and are for three-ass GK agorthm. Both of them suh as prevous an not orrety assfy the defets. Tabe Compare the abtes of methods Method Abty Computng Tme Deteton of Defet Nose Reduton Cass GG 3 Cass GG Cass wth GG Nose Custerng Low Medum Low Low Hgh Medum Medum Hgh Hgh 7. Conuson C-means and GK agorthms oudn t orrety assfy defets as a resut of negetng the sze and densty. We used Gath-Geva fuzzy usterng agorthm that was senstve to densty and sze of usters whh apped on three dmensona of mage ontanng area and ntensty. By usng an utrason mage of defets, we assfed the speke nose, defet and bakground n separated usters. Quaty and tme of usterng depends to the number of asses. Nos e usterng redues the number of asses and the tme of usterng. Ths method apped wthout any mage proessng tehnque suh as removng parte, morphoogy and et. but ths method oney oud remove nose and segment defets of mage. We obtan t on utrason mage but t an be apped on eah NDT mage. Referenes J.C. Bezdek, Fuzzy Mathemats n Pattern Cassfaton Ph.D., Thess, Apped Math, Center, Corne Unversty, Ithaa, 973. R. N. Dave, Charaterzaton and Deteton of Nose n Custerng Pattern Reognton Letters, vo., pp. 657~664, 99. R. Duda, P. Hart, Pattern Cassfaton and sene Anayss Wey, New York, 973. J.C. Dunn, A Fuzzy Reatve of the ISODATA Proess and ts Use n Detetng Compat We separated Custers, Journ. Cybern., vo. 3, pp. 95~04, 974. I. Gath, A.B. Geva, Unsupervsed Optma Fuzzy Custerng IEEE, Trans. Pattern Anayss and

Mahne Integene, vo., pp. 773~78, 989. E.E. Gustafson, W.C.

F. Hoppner, F. Kawonn, R. Kruse, T. Runker, Fuzzy Custer Anayss John Wey and Sons, 999. J.Y.

Km, Adaptaton of Neura Network and Appaton of Dgta Utrason Image Proessng for the Pattern

Booth, V. Amn, J. Lm, S. Udpa, Utrason Image Proessng for Tendon Injury Evauaton IEEE, pp.

Systems, vo. 08, pp. 45~58, 999. T. W. Lao, D.

89~303, 997. T. Matsumoto, K Ohta, Y. Hata, K.

4 Mahne Integene, vo., pp. 773~78, 989. E.E. Gustafson, W.C. Kesse, Fuzzy Custerng wth a Fuzzy Covarane Matrx IEEE CDC, San Dego, Caforna, pp. 76~766, 979. F. Hoppner, F. Kawonn, R. Kruse, T. Runker, Fuzzy Custer Anayss John Wey and Sons, 999. J.Y. Km, H.J. Jeong, H.C. Km, C.H. Km, Adaptaton of Neura Network and Appaton of Dgta Utrason Image Proessng for the Pattern Reognton of Defets n Semondutor IEEE, Int Sym. on Eetron Materas and Pakagng, pp.74~79, 00. N.D. Km, L. Booth, V. Amn, J. Lm, S. Udpa, Utrason Image Proessng for Tendon Injury Evauaton IEEE, pp. 4~44, 998. T.W. Lao, D.M. L, Y.M. L, Deteton of Wedng Faws from Radograph Images wth Fuzzy Custerng Methods Fuzzy Sets and Systems, vo. 08, pp. 45~58, 999. T. W. Lao, D. L, Two Manufaturng Appaton of the Fuzzy K-NN agorthm Fuzzy Sets and Systems, vo. 9, pp. 89~303, 997. T. Matsumoto, K Ohta, Y. Hata, K. Tanguh, Three- Dmensona Construton for Non-Destrutve Testng Aded by Fuzzy Log IEEE, pp.603~606, 000. J. Moysan, G. Corneoup, F. Gueraut, O. Roy, Deveopment of a New Method for the Inspeton of the 3D Voume of Utrason Data QNDE, Vo. 5, pp. 875~88,996. L. Udpa, S.S. Udpa, Neura Network for the Cassfaton of Nondestrutve Evauaton Sgnas IEE Pro, vo. 38, No., Feb. 99. Fg..GG usterng usng two asses, a Orgna Images, b defet, bakground Fg.. GG usterng usng three asses, a nose, b defet, bakground Fg.3. GG nos e usterng usng two asses ( δ 3, a ambguty rejeton, b defet, bakground Fg.4. GG nos e usterng usng two asses ( δ 5, a ambguty rejeton, b defet, bakground

5 (a (b Fg.5. fuzzy -means usterng usng two asses (a (b Fg.6. GK usterng usng two asses Fg.7. fuzzy -means usterng usng three asses Fg.8. GK usterng usng three asses

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