A new technique to reconstruct the defect shape from Lock-in thermography phase images

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Clifford Parks
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1 Ju -5 8 Kakw - Pan A nw tchniqu t cnstuct th fct shap fm Lck-in thmgaph phas imags b C. Zöck* A. Langmi* R. Stöß * an W. An** *EADS Innvatin Wks Munich Gman ** Faunhf-Institut f Nn-Dstuctiv Tsting Saabückn Gman psnt ass: Dpatmnt f Matias Saaan Univsit Saabückn Gman Abstact W psnt a nw mth f cnstucting th shap f fcts in th imnsins fm ptica ck-in thmgaph phas imags with imag pcssing agithms. Th pint-spa functin which scibs th buing ffct f thma imags iv fm ptica ck-in thmgaph is cmput. It is shwn that th pth an th shap f a pana fct can b tiv. Kws: ptica ck-in thmgaph thma tmgaph quantitativ vauatin PSF invs pbms cmpsit aicaft matia.. Intuctin Ta th quimnts abut th quait n pats in th aspac inust a ising. Dfcts such as aminatins impacts amags cacks hav t b fun in quait cnts aft manufactu uing insvic inspctins. Rsachs a king f nw fast an saf mths that aw n t btt tct an chaactis th fcts. Nn-stuctiv tsting (nt) incus a vait tsting mths that psnt th avantag that th pat un tst tains its intgit. Thmgaph sstms a much us in nt. Th hav th avantag t b mbi s that aipans can b inspct ict in th inspctin ha. But unti nw n imags can b gist an n quantitativ infmatin is avaiab n th tct fcts. In this wk w wi fcus n th ptica ck-in thmgaph mth which pvis th avantag t cmput imags with a high signa-t-nis ati (SNR). In th fwing imag pcssing mths a cmbin in t pfm quantitativ vauatin f atificia fcts in a cabn infc pm matia. In thmgaph th istubanc f th iffusin f hat is th igin f th cntast t bsv fcts. This thma iffusin pcss infuncs th appant shap f th fcts. Th fcts appa gat as th a. This pcss can b m as an invs pbm. W us th th f Manis [] f piic thma wav fis t cmput th pint-spa functin (PSF) ict in th Fui main. Th pat t b inspct is assum t b an infinit pat. T gt an appimatin f th PSF th Bn appimatin is us. With this PSF th phas imags a cnvv in t gt tu fct shap an siz []. Aft a thtica sciptin f th appi mths masumnts n th spcimn with atificia fcts a scib an th suts f th imag pcssing an cnvutin a iscuss.. Optica ck-in thmgaph In ptica ck-in thmgaph th absptin f a ight bam is us t psit hat int th matia t b amin. Th bam is sinusia muat an ampitu an phas imags a cmput fm imag squncs utiizing a Fast Fui tansfm. B vaing th citatin fqunc it is pssib t tct fcts in iffnt pths bcaus th thma pntatin pth α π f µ () pns n th fqunc f f th thma citatin wav [3].

Fig.. Eamp f ptica ck-in phas imags btain f iffnt fquncis On main avantag f ptica ck-in thmgaph is in th fact that th ng f th signa is cncntat int a sing fqunc.

2 Fig.. Eamp f ptica ck-in phas imags btain f iffnt fquncis On main avantag f ptica ck-in thmgaph is in th fact that th ng f th signa is cncntat int a sing fqunc. Cnsqunt th ampitu an phas imags hav a high SNR. In t bcm a pth sv mth sva fquncis hav t b us (figu ). Th is a atin btwn th s-ca bin fqunc i.. th fqunc at which th fct appas f th fist tim an a phas vsin ccus an th pntatin pth (Eq. ()). Thus th fqunc has t b chsn sma nugh in t b ab t tct p fcts. Th masumnt tim is ng spcia whn wking with w fquncis an can asi b minuts. B knwing th bin fqunc th pth f a fct can b tiv. 3. Thtica sciptin f th PSF W utiiz th th f piic thma wav fis as iv b Manis [] t sv th hat quatin with piic hating [4]. Th sutin is sught in Fui spac an th sutin is intpt as th stapiic spns at a sing fqunc. Th hat quatin in th Fui spac withut vum suc is θ θ () ( ) ( ) ( ) H ( i) α wh [ m s ] α is th thma iffusivit. B cnsiing th Gn functin G which satisfis Eq. () th sutin f th thma wav-fi with n vum sucs is [] θ ) α [ G ) θ ) θ ) G ) ] S. (3) S H w assum that th si f th pat is ng nugh (infinit pat) that n vum suc ists an that th sufac is hmgnus iuminat in t hav Numann cnitin at th sufac ( OG ) S z an that th tim hamnic fu pscib v th sufac at z is s it ~ s ( t) F ( ) F( ) F δ ( ) ( ) F. (4) Th suting thma wav-fi can b intpt as th sum f th thma wavfi withut fct an th thma wav-fi f th fct. Fm Eq. (3) w gt θ ) θ ( ) [ ( ) ( ) ( ) ( α G θ θ G )] S S θ ( ) ( G ) θ )) S z S α (5) Th tmpatu an th Gn functin at pth z can b wittn as θ G ) R ( θ ) θ )) ) R ( G ) G( )) (6) wh R is th fctin cfficint f th fct. Fwing th fist Bn appimatin w ngct th scn tm in th intga tm an gt ) θ ) α R ( G ) θ )) z S S θ (7) Th tmpatu istibutin can b pss as th cnvutin f th PSF with th fct shap

3 NDE ( ) ( ) θ θ ( ) ( ) ( ) ( ) S sh G z R θ α (8) Th functin ( ) ( ) S S sh ) ( (9) scibs th shap f th fct an is qua n n th fct sufac an z utsi th fct. With ( ) G πα 4 an ( ) ( ) z k F δ θ th functin h can b cmput an uc t ( ) 4 R k F h π () ( ) ( ) Phas PSF As shwn in [5] th pint-spa functin f th phas imag can b appimat b th imagina pat f th cmp pint-spa functin: ( ) Phaspsf ( ) ( ) ( ) 4 Im π R k F () wh. In figu th vutin f th PSF in spct f th fqunc is shwn f iffnt fct pth. A phas vs ccus at a imit fqunc ca bin fqunc. Fig.. Thtica vutin f th PSF f iffnt fct pths

Cmpaisn btwn masumnts an simuatins T tmin th vau f th thma cnuctivit th thtica PSF has bn fitt t th vutin f th phas with th fqunc (figu 3). Sinc th pth f th fct is knwn th vau f α can b tiv.

4 Cmpaisn btwn masumnts an simuatins T tmin th vau f th thma cnuctivit th thtica PSF has bn fitt t th vutin f th phas with th fqunc (figu 3). Sinc th pth f th fct is knwn th vau f α can b tiv. A vau f α was fun an takn f th simuatin..mm s Fig. 3. Fit f th thtica PSF t th masu ata in t tiv th thma cnuctivit Thn a ttu pattn f cmpsit matia is us an th pfi f th fct is vai n th pattn. Figu 4 shws a cmpaisn btwn an tact fct an th simuatin f th fct fmatin pcss in th cas f th ict pbm f iffnt ck-in fquncis. In figu 5 th imag fmatin pcss f a fct in iffnt pth is cmpa t masumnts. Th simuatins ag with th masumnts. Fig. 4. Cmpaisn btwn masumnt (a) an simuatin (b) f a fct in mm pth f iffnt fquncis btwn. t. Hz

5 NDE Fig. 5. Evutin f th phas imag with th pth in mm f a ck-in fqunc f. Hz; masumnt (a); simuatin (b) 4. Epimnts F th pimnts w us a spcimn f cmpsit matia with i hs f vaius sizs an pth psnting atificia fcts (figu 6). P-pcssing f th thma imags Fig. 6. Schmatic awing f th pat f cmpsit matia A st f imag pcssing agithms a fist appi in t btt tct abs an fcts an t caibat th imags. In t iminish th ttu ffct f cmpsit matia a backgun subtactin agithm is appi. W us th mth scib in [6] wh th man vau f a nighbuing aa T su(ij) (t) is subtact fm ach pi vaut pi (ijt) fwing T i j t T i j t T ( () ( ) ( ) ) pi su( i j) t Thn a sca is intuc in th imags using phtgammtic mths [7]. F this task spcia tspctiv abs a us. Th a appi t th pat an a asi tct in th imags. An ips tctin agithm [8] is us t tct autmatica th abs an tmin th cnt f th ab. Th ight a i i...8 f th hmgaph tansfmatin can b cmput with th psitins f fu fnc paamts { } pints a b c a b X 3 3 a b c a b Y (3) 3 wh X Y a th cinats f th fnc pints in th cct imag S (in m) a th cinats f th fnc pints in th igina imag S p (in pis). Futhm this appach aws n t aign pcis th iffnt phas imags vn whn th cama psitins a sight shift uing th pimnt. 3

Dtminatin f th fct pth F ach imag pint w tmin thugh th vutin f th phas with th fqunc th bin fqunc. Sinc th thma cnuctivit is knwn th atin btwn th pth an th bin fqunc f Eq.

6 Dtminatin f th fct pth F ach imag pint w tmin thugh th vutin f th phas with th fqunc th bin fqunc. Sinc th thma cnuctivit is knwn th atin btwn th pth an th bin fqunc f Eq. () aws us t tmin th pth f a fct. Thb thugh th anasis f th histgams an upp imit is tmin an us f th tctin f fcts. A Gaussian cuv is fitt t th histgam f ach phas imag an th tctin imit (6 B) is cmput thugh th man an with f th Gaussian cuv. It must b ntic that th phas imags a sca an cntain th infmatin in a sca wa. F ach pi a tctin vct i is buit an us t tmin th bin fqunc in a isct wa. Whn th pi vau in th i-th phas imag is v th tctin imit th vau f i is st t : i. S th tctin vct cntains inict infmatin abut th pth f th atificia fcts. Th in f th bin fqunc cspns t th atst vau in th vct. Th tw-imnsina map f th bin fqunc f an aa f th phas imags is shwn in figu 7. In this wa a map f th fct pths was cacuat. Th a maining atfacts in th map nam th ata ffcts f th iffusin f th hat. W appi a custing agithm th s-ca kmans [9] in t fm fct cust an th pth f th cnt f ach cust is assign t th cust. Dtminatin f th fct siz Th bu phas imags can b pss as th cnvutin btwn th a shap f th fct an th phas PSF. Th thtica fm f th PSF f Eq. () is us in t tiv th shap f th fcts. Th pi siz is knwn fm th caibatin stp. Th tact fcts a cnvv using th LucRichasn invsin fit []. Figu 8 shws th sut f th cnvutin f a fct. Th fct appas shap. 5. Discussin Th pcss f hat iffusin can b m an th simuatin f ptica ck-in phas imags shw g agmnt with th masumnts. Th ffct is invs in t tiv th a shap an siz f pana fcts. A wk f autmatin f th pcss has t b n f amp t impv th custing. Th invs pbm f cnvutin ucs th appant siz f th fct an shu in futu aw cmputing th siz f th fcts.

[] Vaviv V. Maagu X. Pica J. Thmas R.L. an Fav L.D.

7 NDE Fig. 7. Dtctin vct which cntains th infmatin n th pth (a) with ata ffcts (b) withut ata ffct Fig. 8. Dcnvutin f th tact fct an pt f th pfi REFERENCES [] Manis A. Diffusin-Wav Fis-Mathmatica Mths an Gn Functins Sping Vag Nw- Yk (). [] Vaviv V. Maagu X. Pica J. Thmas R.L. an Fav L.D. Dnamic thma tmgaph - Nw NDE tchniqu t cnstuct inn sis stuctu using mutip IR imag pcssing in Chimnti D.E. an Thmpsn D.O. its Rv. Pg. Quant. Nnst. Eva. A (99) Pnum Pss [3] Buss G. Optacustic phas ang masumnt f pbing a mta App. Phs. Ltt. 35 (979)

8 [4] Casaw H. S. an Jag J.C. Cnuctin f hat in sis. Of Univsit Pss (959). [5] Fiich K. Haupt K. Si U. an Wath H. Dfinitin sutin an cntast in phtthma imaging J. App. Phs. 7 (99) [6] Oma M. Hassan M. Sait K. & A R. IR sf-fncing thmgaph f tctin Infa Phsics & Tchng 46 (5) [7] Luhmann T. Nahbichsphtgammti: Gunagn Mthn un Anwnung. Aufag Wichmann Vag Hibg (3). [8] Eips tctin agithm: [9] MacQun J. B. Sm Mths f cassificatin an Anasis f Mutivaiat Obsvatins in L Cam L. M. an Nman J. its Pc. 5th Bk Smp. Math.. Statistics an Pbabiit Univ. f Ca. Pss (967) [] Luc An itativ tchniqu f th ctificatin f bsv istibutins Astn. J. 79 (974)

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