HYBRID OPTIMUM SIGNAL PROCESSING FOR A STRAIN SENSOR

Transcription

1 HYBRID OPTIMUM SIGNAL PROCESSING FOR A STRAIN SENSOR C. Kargel and B. Zagar Deparmen of Elecrical Measuremen and Measuremen Signal Processing Kopernikusgasse 4/4, A-800 Graz Graz Universiy of Technology, Ausria Absrac: Recenly here has been a shif in maerial sciences owards he use of non conacing laser opical mehods o characerize maerial behaviour. This developmen has been driven by he need o es specimens a relaively high emperaures. We presen a hybrid sysem for racking laser speckles consising of a Fourier-opical preprocessor followed by a digial signal processing uni ha is able o deermine engineering srain wihin specimen a viewing disances of several hundred millimeers. We will show in his paper ha advanced digial signal processing using a polariy coincidence correlaion mehod allows for an increase of he processing speed wihou undue degrading of sysem performance. We will furher show ha an opimum analog o digial conversion characerisic exiss which if applied resuls in displacemen esimaes wih almos ideal variances. Keywords: opical and digial signal processing, opimum analog o digial converer, laser speckle, srain sensor. INTRODUCTION Sandard maerial esing procedures of modern high emperaure maerials like ceramic marix compounds which operae a emperaures beyond 000 C demand for non conacing displacemen, displacemen gradien and srain measuremen mehods. Several opical mehods are discussed in he lieraure [, ] mos of which ry o rack imaged pars of he specimen s surface during mechanical esing. Engineering srain can be easily deermined by racking wo separaed pars of he specimen's surface, calculaing he change in separaion and dividing by he iniial value. Subjecive laser speckle paerns [3] being characerisic for paricular surface elemens [4] allow for high resoluion racking even under he influence of high levels of hermal background radiaion [5]. This racking has o be performed a leas in wo dimensions (D) in space even if only displacemen componens in he srain direcion are of ineres because of imperfecions in he bearings and grips of commercial sress esing equipmen. The racking can be done by digial signal processing (e.g. by esimaing he crosscorrelaion funcion of surface elemens before and afer he deformaion process []). Because he numerical load of wo dimensional signal processing algorihms is very high he racking rae aainable is oo slow for some sandard maerial esing procedures. Raes of only abou a few per second can be achieved even if numerically very efficien algorihms and high power PCs or dedicaed signal processors are used. Real-ime applicaions like load dynamic maerial esing or cyclic esing are no feasible and so all available mehods are no able o cope wih he relaively high srain rae necessary for high emperaures maerials o avoid creep effecs. In recen publicaions [6, 7, 8] we demonsraed ha exremely fas operaing Fourier-opical signal processing [9, 0, ] wihin so called opical 4f-arrangemens can be uilized o perform a leas some pars of he necessary processing by paricular manipulaions of he laser speckles opical power specra. We will poin ou in his paper how for he D problem a hand he (elecrical) daa acquisiion and he subsequen digial signal processing can be performed one-dimensionally (secion ). As a resul of he D processing we are able o speed up he processing subsanially. In secion 3 we will show a paricular unbiased cross-covariance esimaor o furher increase he measuring rae and compare his esimaor s variance o ha of he mos ofen used sandard cross-correlaion esimaor. Furhermore we will prove in secions 4 and 5 ha by implemening an appropriaely designed opimum analog-o-digial converer he variance as a measure of random error of he suggesed esimaor can be kep nearly as low as for he sandard esimaor. This way boh real high speed and high qualiy measuremens become feasible for he firs ime.

2 FOURIER-OPTICAL SIGNAL PROCESSING OF LASER SPECKLES Fourier ransforming properies of lenses [9] offer a very smar way o perform coheren opical signal processing by (spaial) Fourier-filering. I is shown [8] in grea deail how appropriae spaial filering of laser speckles produced by reflecive coheren wave-scaering from objecs wih opical rough surfaces wihin well designed opical 4f-arrangmens can be carried ou and how desired power specra and as a resul paricular laser speckle paerns can be shaped using Fourier-plane filer masks. I is well known [3] ha he power specral densiy (psd) for subjecive (bu also in a slighly modified way for objecive) laser speckle paerns consiss of a δ-funcion componen a zero spaial frequency plus an exended componen ha akes he shape of he auocorrelaion funcion of he inensiy ransmiance of he lens pupil if no spaial filer mask is presen in he opical signal processing sysem. For a circular lens pupil of diameer D he psd of he speckle paern in he image plane (e.g. see Fig., righ side) is given by Eqn. (): ëz 4 ( ) ( ) ëz ëz ëz psd f x,fy = E[I] ä fx,fy + arccos f f f () D ð D D D for f < (D/λz) and zero oherwise, wih E[...] denoes expecaion or mean value f x, f y spaial frequencies in x- and y-direcion respecively f= (f x + f y ) / spaial frequency magniude λ wavelengh of he illuminaing laser ligh z disance beween he lens pupil and he image plane. We had shown in recen papers [6, 7] ha well dimensioned, inexpensive binary ampliude filer masks are bes suied o be applied in pracical se-ups when commercial sress esers are performing he loading process. Inensiy paerns like he one shown in Fig. (lef) which are basically sensiive only o displacemen- and srain componens in loading direcion (here he horizonal direcion) can be produced in he image plane of an opical signal processing sysem opimized for he measuremen of D displacemens and srains. Figure. Image plane inensiy paerns of an appropriaely designed opical signal processing sysem wih (lef) and wihou (righ) a suiable Fourier-plane filering mask This kind of elongaed speckle paern lends iself o D signal acquisiion wih a properly oriened line scan camera followed by only D digial signal processing. I can easily be seen ha unwaned displacemens in he sysem's non-sensiive verical direcion will cause on he spaially sable line scan camera only small signal decorrelaions compared wih he signal aken a he reference posiion whereas displacemens in he horizonal direcion can be measured wih high sensiiviy. I is proven [8] in grea deail ha he properies of his measuremen sysem are excellen when sough afer in-plane displacemens in horizonal direcion have o be deermined even if bohersome displacemens in verical in-plane direcion and ou-of-plane direcion are presen a he same ime. If no spaial filering is applied, more familiar speckle paerns similar o ha depiced in Fig. (righ) are formed in he image plane. In his case he line scan camera signals will srongly decorrelae [4] even if he specimen under es experiences only small displacemen componens in verical direcion which resul in erroneous measuremen values.

3 3 DIGITAL SIGNAL PROCESSING One of he principal applicaions of correlaion in digial signal and image processing is he area of emplae maching. The closes mach beween signals can be found by selecing hose signals ha yield he correlaion funcion wih he larges value. Correlaion is also widely used for he deerminaion of linear displacemens. Acceping more complicaed mahemaics robus algorihms can be designed even if roaion and scaling during he displacemen cause problems. Because hese effecs normally do no provoke undesired side effecs in sandard maerial esing procedures we are allowed o keep he mahemaics easy and focus on he basic new concep wihou loss of generaliy. The mehod shown in he following can be adaped o all robusness mainaining algorihms. Correlaion funcions as a second order saisical propery of speckle signals are defined over an ensemble-average of rough surfaces. Even if ergodiciy can be assumed only esimaes of he rue correlaion funcion can be calculaed because he capured signals have o be resriced o finie exens in all pracical cases. Therefore a finie se N of daa is assumed o be used in esimaing he correlaion funcions. 3. Polariy Coincidence Correlaion Funcion Esimaor Because appropriae, exremely fas operaing, opical signal processing reduces he D feaure racking problem o a D one, all furher digial processing can be performed much faser in D. Alhough he daa which are fed ino a convenional opimum linear, nonparameric cross-correlaion esimaor [0] is only D he compuaional load remains sill high. I should be menioned ha using he convoluion heorem and implemening i hrough he fas Fourier ransform (FFT, []) can offer a numerically more effecive calculaion possibiliy. Measuring raes up o 00 per second are achievable for his D digial correlaion mehod. Moreover we are able o furher decrease he compuaion ime by implemenion of dedicaed correlaion funcion esimaors. The simplificaion of he classical or convenional cross-correlaion esimaor yields an esimae which is called polariy coincidence correlaor PCC (Eqn. ): N π RPCC ( i) = sin sign( I( n) E[ I] ) sign( I ( n i) E[ I] ), () N- i n= i+ wih: i space lag argumen, -(N-) i (N-) N oal number of samples (pixels of he CCD camera) Ι (n) recorded inensiy signal before he specimen s loading process Ι (n) recorded inensiy signal afer he specimen s loading process E[...] denoes expecaion or mean value This kind of correlaion funcion is deermined by he signals zero crossings and does only use he signum (sign) of he signals. I is obvious ha he compuaional complexiy have o be subsanially reduced compared o he convenional correlaor. Therefore he measuring raes can be grealy increased up o several hundreds or even housands per second depending on he ype of he calculaing hardware. The displacemen of he specimen in he sysem s sensiive direcion can be derived from he locaion of he peak of R PCC (i) by aking he opical magnificaion m (=image size/objec size) of he signal processing sysem and he pixel pich p of he uilized CCD line-scan camera ino accoun: p = arg max[ R ( i) ] PCC (3) m i Engineering srain ε wihin he specimen can be calculaed wih he difference of he displacemens and of wo racked objec regions separaed by he base lengh l 0 : Ä Ä ε = (4) l0 3. Variances of Correlaion Funcion Esimaors The variances - as a measure of qualiy and performance - of he wo esimaors menioned and defined in secion 3. are shown in Fig., where N is again he lengh of he digial signal sequences. Specificaions of he somewha unhandy and complicaed mahemaical expressions of he approximaed esimaors' variances [3, 4] shall no be given here, insead we decided o show he variances in a much more explaining graphical way.

4 Because he sough afer displacemen is derived from he posiion of he peak of he correlaion funcion as explained in secion 3. and only small signal decorrelaions occur when he specimen is displaced, he region in which he (normalized) rue correlaion funcion approaches is of greaes ineres: I can be seen ha he lower compuaional cos of he polariy coincidence correlaor is paid for by a larger variance which is sill small, however, in his region. To decrease he variance of he esimaes he oal number N of pixels mus be as high as possible. If similar variance magniudes for boh esimaors are required he signal sequence lenghs N have o be somewha increased when using he PCC mehod. Figure. Variances of esimaors as a funcion of he normalized rue correlaion funcion for large (uncorrelaed) signal sequence lenghs N 4 OPTIMUM ANALOG TO DIGITAL CONVERTERS (ADC) Analog o digial conversions of D inensiy signals as in our case and of images in general consis of boh (spaial) sampling and gray level quanizaion. Whereas he sampling is defined by he pixel geomery of he CCD-sensor he quanizaion of image inensiies is deermined by he uilized ADC. Each quanizer maps a coninuous variable I ino a discree one Î, which akes values from a finie se {r,...r L } of numbers. This mapping is a saircase funcion and he quanizaion rule deermines a se of increasing ransiion levels { k, k=,... L+} wih and k+ as he minimum and he maximum values of I respecively. I is considered o be mapped o r k if i lies in he half-open inerval [ k, k+ ). The quanizer design problem is o selec he bes i and r i for a paricular opimizaion crierion and a given coninuous probabiliy densiy funcion p I (I) of he scalar random variable I. Each quanizaion process is irreversible, he inpu values canno be deermined uniquely from he oupu values and so he quanizer inroduces disorion which should be aemped o be minimized. The opimizaion crierion used here is he minimizaion of he mean-square quanizaion error (MSE). Furhermore a realizable esimaor Î can be obained by minimizing he MSE because Î is an unbiased esimae of I: L+ L i+ ( I Iˆ ) pi ( I) di = ( I ri ) pi ( I) MSE = E[( I Iˆ) ] = di (5) i = i The condiions necessary for he minimizaion are obained by differeniaing he MSE given in Eqn. (5) wih respec o k and r k and equaing he resuls o zero (noe ha I corresponds o k and Î o r k ). Afer a few mahemaical seps he resuls are obained and can be inerpreed as follows: The opimum reconsrucion levels lie a he cener of mass of he probabiliy densiy funcion over he specified ransiion levels, which in urn are halfway beween he opimum reconsrucion levels. The expressions for k and r k are combined so ha hey have o be solved simulaneously by any kind of ieraive scheme. An approximae soluion for he ransiion levels given in Eqn. (6) can be obained by modelling p(i) as a piecewise consan funcion [5]:

5 ( L+ ) k ( L L+ ) + ( I) ] 3 k + + L+ [ p ( I) ] 3 I di [ p I di wih k =,..., L. This mehod requires ha he dynamic range which is defined by he quaniies and L+ is finie and has o be known before deermining he opimum ransiion levels k. The opimum reconsrucion levels r k can be deermined easily by averaging k and k+. The corresponding quanizer is ofen called a Lloyd-Max quanizer [6]. To pu in he ideal probabiliy densiy funcion p I (I) of laser speckle inensiy paerns [3] I p I ( I) = exp (7) E [ I] E [ I] in Eqn. (6) yields he opimum quanizaion funcion which is shown in Fig. for an ADC wih 5 bi gray level resoluion. I can be seen clearly ha he ransiion and reconsrucion inervals, respecively are much smaller a small inensiies han a larger ones which is caused by he negaive exponenial saisics defined in Eqn. (7). (6) Figure 3. Opimum quanizaion funcion (5 bi gray level resoluion) if laser speckle signals are o be analog o digial convered For uniform disribuions he equaions governing he quanizer become linear, giving equal inervals beween he ransiion and reconsrucion levels. Quanizers of his kind, called linear quanizers, are implemened in mos commercially available ADCs. One way o produce he desired opimum quanizer funcion wih a sandard linear quanizer is o implemen code conversion from a high resoluion code (e.g. 8 bi) o a lower one (e.g. 5 bi or lower). This was done for he invesigaions in his paper by appropriae designed look-up-ables (LUTs). LUTs were also used o produce linear quanizers wih differen gray level resoluions. 5 EXPERIMENTAL AND MEASUREMENT RESULTS Our aim is o invesigae he performance of differen quanizers wihin he enire srain measuremen sysem specified so far which yields opical preprocessed images shown in Fig. on he lef side. Because speckle signals from which he "displacemen of he specimen" is esimaed are random variables, he qualiy of differen esimaors and quanizers, respecively can only be expressed in a saisical way. Therefore he following resuls are drawn from 8 signal realizaions each aken exacly a he same horizonal displacemen x. The specimen is displaced in he sysem's sensiive horizonal direcion by using a hyseresis compensaed piezo acor having an absolue posiioning accuracy of 30 nm. A several differen posiions he acually grabbed line signals are correlaed wih he corresponding reference signals and he displacemens are deermined. The resoluion of he

6 digial signal processing echnique was enhanced by heoreical ideal inerpolaions of he correlaion funcions by a facor of 6 where /6 pixel corresponds o 0.45 µm on he specimen s surface. Invesigaions were made for gray level resoluions of he uilized linear quanizers and opimum Lloyd-Max quanizers from 8 bi down o bi. The resuls show ha only minor differences occur if resoluions from 8 bi down o 4 bi for eiher quanizer ypes are implemened. Lower gray level resoluions han 4 bi cause differen resuls of measuremen which are discussed in he following: 6 Esimaed mean value Esimaed displacemen in in pixel bi bi (Lloyd-Max) 4 bi 4 bi(llyod-max) 8 bi Conrol volage driving he piezo acor in Vol Figure 4. Mean value of esimaed displacemens when using differen ypes of quanizers (linear, Lloyd-Max) and differen gray level resoluions Esimaed sandard deviaion in pixel Esimaed sandard deviaion bi bi (Lloyd-Max) 4 bi 4 bi(llyod-max) 8 bi Conrol volage driving he piezo acor in Vol Figure 5. Sandard deviaion of esimaed displacemens when using differen ypes of quanizers (linear, Lloyd-Max) and differen gray level resoluions Fig. 4 shows esimaes of he displacemens' mean values which do no differ significanly when differen gray level quanizers (linear, Lloyd-Max) are implemened. I can also be seen ha he relaionship beween he conrol volage driving he piezo acor and he displacemen of he specimen is no highly linear. This fac does no influence he resuls of he invesigaions in any way because relaive comparisons are of ineres. Fig. 5 depics unbiased esimaes of he sandard deviaions σ x as a measure of random errors and uncerainy of measuremen. Each individual daa poin shown in he diagrams is esimaed as menioned before from 8 samples aken under consan condiions exacly a he same horizonal posiion. Periodical deviaions of σ x a very low gray level resoluions of he ADC are caused by he pixel fill facor of he uilized CCD camera which is around 50 %. Maximum values of σ x occur a displacemens which are odd-numbered muliples of half of he pixel pich (see

7 curves for bi linear and bi Lloyd-Max quanizers in Fig. 5). The opimum quanizer in an ADC wih bi gray level resoluion reduces σ x by a facor of approximaely 4 compared wih he bi linear quanizer. Under wors case condiions, ha is a disinc displacemens, σ x can be a maximum wice as large as for high gray level resoluions. If he geomerical properies of he uilized camera are properly chosen, σ x will no show periodical deviaions and will reach values as in he high resoluion (8 bi or even higher) case. Therefore he polariy coincidence correlaion esimaor defined in Eqn. () can be implemened wihou increase of σ x if he bi opimum quanizer derived in his paper is used. 6 CONCLUSIONS Maerial esing procedures for elevaed emperaures demand very high processing speed of he measuremen equipmen. In his paper we showed ha an increase in he measuremen rae over convenional laser speckle exensiomeers can easily be achieved if besides using Fourier-opical means o preprocess speckle images he image acquisiion process as well as he image processing iself can be performed on opimally quanized daa wihou grealy degrading sysem performance. Fuure work will be devoed ino incorporaing he repored research ino programmable logic devices. REFERENCES [] H. Manser, H. Weiss, B. Zagar, High Resoluion Elecronic Speckle Paern Phoography for Srain Measuremen a High Temperaures, Proc. 8 h Inernl. Symposium on Arificial Inelligence Based Measuremen and Conrol (AIMaC 9),99, pp 9-4. [] R. Jones, C. Wykes, Holographic and Speckle Inerferomery, Second Ediion, Cambridge Universiy Press, 989. [3] J.W. Goodman, Saisical Properies of Laser Speckle Paerns in: J.C. Dainy Laser Speckle and Relaed Phenomena, Springer, Berlin, 984. [4] I. Yamaguchi, Speckle Displacemen and Decorrelaion in he Diffracion and Image Fields for Small Objec Deformaion, Opica Aca, Vol. 8, 98, pp [5] M. Anwander, B. Weiss, B. Zagar, H. Weiss, A laser speckle correlaion mehod for srain measuremens a elevaed emperaures, Proc. Inernl. Symposium on Local Srain and Temperaure Measuremen in Non-uniform Fields a Elevaed Temperaures, Berlin, March 996, Woodhead Pub. Limied, Cambridge, UK, 996. [6] Ch. Kargel, B. Zagar, Opical preprocessing in a laser-speckle correlaion measuremen echnique for he deerminaion of engineering srain wihin specimen, 43h Annual SPIE meeing "Algorihms, Devices and Sysems for Opical Informaion Processing", San Diego, CA, 998, pp [7] B. Zagar, Ch. Kargel: A laser-based srain sensor wih opical preprocessing, IEEE Transacions on Insrumenaion and Measuremen, Vol. 48, No., 999, pp [8] Ch. Kargel: Opical and digial signal processing in a laser speckle measuremen echnique, Ph.D. hesis, Technical Universiy of Graz, 999. [9] J.W. Goodman, Inroducion o Fourier Opics, McGraw-Hill, Second Ediion, 988. [0] W. Lauerborn, T.Kurz, M. Wiesenfeld, Coheren Opics, Springer, Berlin, 995. [] L.J. Curona, E,N. Leih, C.J. Palermo, L.J. Porcello, Opical Daa Processing and Filering Sysems, IRE Transacions on Informaion Theory, Vol. IT-6(3), 960, pp [] A.V. Oppenheim, R.W. Schafer, Discree Time Signal Processing, Prenice Hall Inc. Englewood Cliffs, 989. [3] H.C. Cramer, Mahemaical Mehods of Saisics, Princeon Universiy Press, N.J., 974. [4] M.G. Bellanger, Adapive Digial Filers and Signal Analysis, Marcel Dekker, Inc., N.Y., 987. [5] A. K. Jain, Fundamenals of Digial Image Processing, Prenice Hall Inc., Englewood Cliffs, 989. [6] R.C. Gonzalez, R.E. Woods, Digial Image Processing, Addison-Wesley Publishing Company, Massachuses, 993. AUTHORS: Chrisian KARGEL and Bernhard ZAGAR, Deparmen of Elecrical Measuremen and Measuremen Signal Processing, Kopernikusgasse 4/4, A-800 Graz, Graz Universiy of Technology, Ausria