STATISTICAL PROCESSING OF STRAIGHT EQUISPACED FRINGE PATTERNS

Transcription

1 Journl of Optoelectronics nd dvnced Mterils Vol. 6 o. 3 September 4 p SSCL PROCESSG OF SRGH EQUSPCED FRGE PERS V. scov *. Dobroiu D. postol V. Dmin tionl nstitute for Lser Plsm nd Rdition Phsics PO Bo MG-36 R-775 Buchrest Romni he pper is deling with sttisticl processing of digitll recorded stright equispced fringe ptterns. t is determined the highest degree of ccurc tht cn be chieved in estimting fringe prmeters b sttisticl processing under given sttisticl fluctution conditions ffecting the recorded imge. Received ugust 3; ccepted June 4) Kewords: Fringes Fringe pttern Digitl imge Sttistics Lest squres fitting. ntroduction Stright equispced fringe ptterns occur in mn interference cses. hese re the simplest fringe ptterns but hve gret importnce becuse n other fringe ptterns of n compleit m be decomposed in terms of these simple ptterns. Furthermore processing of these fringe ptterns is ver importnt in opticl metrolog [3]. here re mn methods for fringe processing. he most modern is considered the phse shifting interferometr method tht hs to process t lest three phse shifted fringe ptterns to retrieve the phse mp of n object wvefront reltive to reference wvefront. he sttisticl properties of the errors of this method re well described in Refs Since the fringe ptterns tht we concern ourselves with hve n priori known phse mp liner rmp) we don t hve to determine the phse mp but onl it s grdient nd phse offset which must be constnt prmeters. For this purpose we use lest squres fitting LSF) to compute the bsic prmeters of the fringe ptterns s sttisticl quntities [9]. LSF method widel used for dt processing is well described in the ppendi using mtri lnguge well suited for computer progrmming. We lso hve implemented this method in computer progrms tht we used to process fringe ptterns nd to stud the relted errors. We do not give detils on these computer progrms since the min purpose of this pper is to evlute the sttisticl errors for the LSF computed fringe prmeters. Becuse these fringe ptterns re simple periodic) the Fourier trnsform method [] is optiml to process them nd m be fster thn LSF which is ver generl method. However we consider tht LSF is ver importnt s clssicl method nd the min point of the present work resides in the fct tht we could obtin nlticl epressions for the sttisticl errors of LSF. We cn s tht the errors found on LSF m serve s reference errors to compre the errors of n other methods used to process fringe ptterns. Moreover we think tht LSF is one of the most ccurte sttisticl methods.. Fringe pttern processing.. Periodic hrmonic fringes he simplest fringe pttern hs hrmonic distribution of intensit long direction or over plne ): * Corresponding uthor: nv@emil.ro

2 84 V. scov. Dobroiu D. postol V. Dmin D fringe pttern: ) + cosν + ) ) ϕ D fringe pttern: ) + cos ν + ν ) + ϕ) b) is the bckground intensit is the intensit of the fundmentl hrmonic is the sptil frequenc of the fringes hving two components for D fringe pttern nd ϕ is phse prmeter tht specifies the globl positioning of the fringe pttern. Such fringe ptterns occur b the interference of two uniform plne wvefronts. Fringe pttern cquisition b CCD leds to sequence of smpled nd digitized vlues of the intensit distribution: D: k k ) + cos f k + ϕ; f ν k ) D: mn f ) cos f f mδ n δ + m + n + ϕ; ν δ ν δ m n f b) where nd denote the smpling steps which re the spcing between two djcent CCD piels on two orthogonl directions. nsted of the sptil frequenc we prefer further using the dimensionl prmeter f tht is the number of fringes in the fringe pttern. t hs lso two components in the cse of D fringe pttern. We should mention tht ctull the smpling performed on the fringe pttern b the CCD is not just the simple smpling of the functions ) becuse the CCD piels hve finite size nd the output the verge light intensit over their ctive re. We describe this effect for the D cse ssuming liner dt cquisition sstem. ccording to the theor of liner sstems the output J) cn be written s convolution of the input ) with the impulse response function g) of the cquisition sstem: J ) g) ) ) g ) d 3) Let s consider rectngulr impulse response function: δ g ) ε δ > ε δ 4) tht signifies light verging over the ctive re of CCD piel whose size equls frction of the spcing between two djcent piels. With this impulse response function we obtin the following sequence of smples: J J k J k ) J + J cos f k + ϕ; ε J ε sinc ε ) f f k ν 5) herefore the contrst in the cquired dt of the fringe pttern is reduced but ver slightl if the CCD resolution is lrge enough. Of course beside light verging due to the finite size of piels there re mn other effects such s light diffrction. ll effects superpose nd give more complicted impulse response function but for n liner cquisition sstem the cquired dt remins hrmonic like the fringe pttern nd the prmeters f ϕ) remins unffected in spite of the ctul impulse response function. Further we shll use the formule ) to describe the fringe ptterns tking into ccount tht the fringe contrst never reches lws < ).

3 Sttisticl processing of stright equispced fringe ptterns 843 Let s show how LSF is pplied to determine the set of 4 prmeters f ϕ) in ) respectivel 5 prmeters f f ϕ) in b). hese prmeters re sttisticl quntities nd the LSF computer progrm for fringe processing gives their sttisticl mens nd vrinces. heir vrinces re due to the fluctutions of the cquired dt tht we suppose to hve Gussin distribution with men nd σ vrince. n fct onl mking this ssumption is the LSF correctl pplicble. he LSF pplicble on ) to determine is nonliner see ppendi) becuse the reltions ) cnnot be written in the form of liner combintion of independent functions hving the prmeters s coefficients s the eqution ) shows. onliner LSF needs set of pproimte strting vlues for to itertivel compute high precision finl results. t s es to get the strting vlues using course precision methods for emple ppling Fst Fourier rnsform to subset of the sequence ). % precision hs been proved to be enough to successfull strt the itertions. Usull 5- itertions led to the finl result. he formul 7) llows us to clculte the covrince mtri of the prmeters tht is their sttisticl errors. his formul is pplicble s well for nonliner LSF since it epresses the first order pproimtion of the error propgtion lw. For the D fringe pttern we clculte firstl 4-component vector of functions tht re the prtil derivtives of the function ) with respect to ech of the prmeters: ) ) ) ) ) D f ϕ 6) hen we hve to compute the 4.4 mtri of the sttisticl weights P : 34 ) ) ) ) j i dk k k j i k k j k i ij D D D D P δ δ σ σ 7) ssuming ver lrge number of smples ) we hve pproimted the 6 sums bove with integrls which we could clculte nlticll. However some of them re still complicted but using the limit of ver lrge fringe number f the simplif ver much. et one needs onl inversing the P mtri in order to get the covrince mtri σ. he covrince mtri for the prmeters of D fringe ptterns re clculted similrl ecept tht the sums nd integrls become double ccording to the ) coordintes nd the mount of computtion is much greter. s described bove we hve found nlticl epressions for the sttisticl errors of the prmeters using n smptotic pproimtion vlid for lrge number of fringes f nd lrge number of smples : D: f 8/ / / σ σ ϕ 8) D: f f 4/ / / σ σ ϕ 8b)

4 844 V. scov. Dobroiu D. postol V. Dmin he correltions between f ϕ) re irrelevnt for the sttisticl errors ssessment onl the digonl elements of the covrince mtri re sufficient. ll the prmeters hve vrince inversel proportionl to consequent to n universl sttisticl lw. t is now es to find out the probbilit densit function of the sttisticl fluctutions of providing tht the intensit fluctutions of ll the piels in the fringe pttern re sttisticll independent nd ech of them hs the Gussin distribution function f G of vrible nd µ prmeters: f δ ) e σ δ σ f δ ; σ ) We see tht ccording to reltion 5) there is liner reltionship between the vritions of the prmeters nd tht of the intensit : G 9) B k B ki i k 345) i ) in reltion 9) should not be mistken for in reltion ). he former quntit is sclr representing the intensit vrition of piel in the fringe pttern while the ltter denotes the intensit vritions ll over the fringe pttern. t is known tht the probbilit densit function of the sum of two independent rndom quntities equls the convolution of their probbilit densit functions. Hence ech of the prmeter in the set of hs probbilit densit function tht is the convolution of Gussin distribution function of different widths : f k ) fg δ ; Bk σ )* fg δ ; Bk σ )* * fg δ ; Bk σ ) ) t is lso known tht convolving two Gussin functions leds to nother Gussin function ccording to the following rule: fg ; µ σ)* fg ; µ σ ) fg ; µ + µ σ + σ ) ) herefore ech component k hs the distribution f G k; kk) nd moreover the fluctutions of the whole set of hve the multivrite norml distribution []: f ) r ) det )) e r 45) 3) r being the number of prmeters equl to 4 or 5 whether the fringe pttern is D or D. o obtin the covrince mtri with more precision one needs to perform ver difficult nlticl clcultion which leds to ver complicted results. ht s wh we restrict ourselves to the simple pproimte results 8). n order to check the level of ccurc of these nlticl formule we hve processed lrge number of computer simulted fringe pttern nd we hve performed sttisticl nlsis on the results to compre the ctul errors with the theoreticll predicted ones in Eq. 8). We simulte D fringe ptterns of the form ) ech of them with 56 smples. For ll the fringe ptterns in the set the prmeters ) were ssigned the vlues.5.4) which give good fringe contrst nd clibrte the intensit ) within the rnge tht we suppose to be the dt cquisition rnge we tke into ccount the previous mention tht lws < ). he prmeter f ws given uniform rndom vlues in the intervl of 3 3 fringes. We chose this intervl becuse there is no sense to process fringe ptterns with too few fringes f < 3) nd we limit the upper limit of fringe number to 3 to void lising ssocited with undersmpling we onl consider well smpled fringes with smpling rte >8 smples/fringe spcing). he prmeter ϕ ws given uniform rndom vlues in the intervl -. We dd Gussin noise of men nd.5 stndrd devition. he noise due to usul 8-bit quntistion is ver smll but the simultion took it into

5 Sttisticl processing of stright equispced fringe ptterns 845 ccount too. Sttisticl mthemthics shows tht b uniform quntistion with step q the quntistion error is uniform in the intervl -q/ q/ nd hs the vrince σ q q / ). n these conditions the reltions 8) give for the devitions of the prmeters the vlues: theor σ ) equl to the squred root of the digonl elements in the covrince mtri. Let s denote with the prmeters given to the simulted fringe ptterns while nd σ denote their sttisticl men vlues nd stndrd devitions provided b the computer progrm ppling 7) to the dt ). We define the displcement between the ect vlues nd the computed vlues of the fringe prmeters: δ nd we do sttistics on the reduced prmeters δ / σ over the whole set of the simulted fringe ptterns. Here re the results for verge nd stndrd devition: ) σ ). he men vlues show tht the estimtions of the prmeters re bised but ecept the first component these bises re quite smll considering tht the re epressed s frctions of σ. Fig.. Results on processing set of hrmonic fringe ptterns. σφ/σφ theor) σφ/σφ theor) σf/σf theor) σ f/σ f theor) σ/σ theor) σ /σ theor) σ /σ theor) σ /σ theor)

6 846 V. scov. Dobroiu D. postol V. Dmin he.63 bis of the intensit is minl due to quntiztion. Quntiztion error is ctull not sttisticl error. t m be regrded s sttisticl onl when the imge hs lrge vritions. he DC component of n imge is sstemticll bised b quntiztion. his bis is not greter thn ± / σ being the number of quntiztion levels. f we repet the whole procedure of q q sttistics once gin without quntizing the fringe ptterns we obtin the mens ) nd the stndrd devitions σ ). his time ll bises re ver smll nd we cn regrd them s being onl due to sttisticl fluctutions: < σ / where p is the number of processed fringe ptterns. p ll the devitions σ re close to proving the fct tht the computed vlues of the sttisticl errors on single fringe pttern processing estimte ver well the ctul fluctutions tht hve been observed over the set of simultions. Moreover we hve proved tht the histogrms of the sttisticl quntit fit well with Gussin distribution functions of prmeters σ ) ver close to stndrd norml distribution functions which hve ) prmeters. he Fig. shows the result of the simultions in form of grphs tht show in wht mnner the number of fringes f nd the phse prmeter ϕ influence the numericll computed stndrd devitions of the prmeters. he top left plot hs the nrrower rnge from 6 to 6 fringes for the bsciss to give more detils. hus it is es to see tht the errors become minim when the fringe number f tkes integer vlues. ht s becuse the hrmonic functions re orthogonl over rnges of integer periods. he covrince mtri 8) is clculted just ssuming ect orthogonlit between the functions sine nd cosine involved in the sums 7). We conclude tht the formuls 8) estimte ver well the ctul sttisticl errors even if the number of fringes is not lrge nd there re onl 56 smpling points. We should mention tht for f 3 fringes the difference between the theoreticl nd computed evlution of the devition for the prmeter f is onl 3%... on-hrmonic periodic fringes Often fringe ptterns m be periodic but not hrmonic so tht their intensit profile cn be written s Fourier series: ) + lk cosk + ϕk ) k ν 4) here re two min resons wh the fringe ptterns to be processed hve high-order hrmonics: - he non-hrmonic profile is specific to certin interference conditions for emple in multiple bem interference. - B cquiring dt with non-liner device the output signl becomes higher order hrmonics. Often we m need to process fringe ptterns with unknown profile nd determine the set of prmeters f ϕ) s if the fringe were hrmonic. Let s nlze if there is n sense in doing this. f the number of fringes is ver lrge then the hrmonic functions cos k + ϕ) build up n orthogonl sstem over the domin of the fringe pttern. f the Fourier series 4) were finite then due to the orthogonlit of the hrmonics the ccurc of this series would increses onl b dding new terms in the series not b djusting the former ones. hus the prmeters ) computed with the lgorithm for hrmonic fringes will ectl mtch the th nd the first orders of hrmonic intensities in the non-hrmonic fringe pttern. lso the lst two prmeters f ϕ) estimte correctl the number of fringes nd the phse offset prmeter in the non-hrmonic fringe pttern nd the hve the sme ccurc s the fringes would hve hrmonic profile with the intensit.

7 Sttisticl processing of stright equispced fringe ptterns 847 f the fringe number is not ver lrge then the orthogonlit of the hrmonic functions cos k + ϕ) is distorted nd the prmeters ) do not mtch ectl the orders nd of the hrmonic intensities in the non-hrmonic fringe pttern nor do the lst two prmeters f ϕ) estimte correctl the number of fringes nd the phse prmeter. ll prmeters f ϕ) computed with the ssumption of hrmonic fringes re bised estimtions of the corresponding prmeters in the non-hrmonic fringe pttern nd the tend to the ect vlues s the number of fringes increses surel the smpling rte must be incresed correspondingl to void lising). o see the conditions suitble for good results on processing non-hrmonic fringes with the computer lgorithm for hrmonic fringes we use sttisticl nlsis on the results of processing lrge set ) of computer simulted fringe ptterns. We simulte D Fbr Perot fringe ptterns ech of them with 56 smples. For ll the fringe ptterns in the set the prmeters ) were ssigned the vlues.5.5) so tht the intensit is confined within the rnge. Fig. displs the ctul fringe profile of the simulted fringes. he prmeter f ws given uniform rndom vlues in the intervl of 3 3 fringes nd the prmeter ϕ ws given uniform rndom vlues in the intervl -. We dd Gussin noise with men nd.5 stndrd devition. n these conditions the reltions 8) gives for the devitions of the prmeters the vlues: σ ). theor Fig.. Periodic Fbr -Pérot fringes. he Fig. 3 shows the result of the simultions in form of grphs tht indicte how the number of fringes f nd the phse prmeter ϕ influence the displcements of the prmeters. We see tht these influence becomes insignificnt if f >8. his time we preferred to displ the theor displcements δ / σ insted the normlized stndrd devitions δ / σ s we did for hrmonic fringes becuse we epected greter vritions in respect to prmeter f so tht the histogrms of the displcements which re the ctul sttisticl errors) don t fit with Gussin distribution functions if the number of fringes is not lrge enough. et we repet sttistic over new set of computer-simulted fringes with the sme prmeters ecept f tht ws given uniform rndom vlues in the rnge 8 fringes. We hve chnged the rnge of f thinking tht so the errors δ / σ will become distribution closer to the stndrd norml distribution one. his time for the reduced prmeter δ / σ we obtin the following results for verge nd stndrd devition: ) σ ). n comprison with the hrmonic fringes the estimtions of the prmeters re more bised especill hs displcement of 33σ. he prmeters f ϕ) hve smll displcements. he devitions σ re greter thn showing tht the computed sttistic errors re underestimted. he histogrm of the sttisticl quntit for ech component fits well with Gussin distribution.

8 848 V. scov. Dobroiu D. postol V. Dmin Fig. 3. Results on processing set of Fbr-Pérot fringe ptterns. δφ/σφ theor) δφ/σφ theor) δ f/σ f theor) δ f/σ f theor) δ /σ theor) δ/σ theor) δ /σ theor) δ /σ theor) We conclude tht the LSF lgorithm mde for hrmonic fringe ptterns cn be used to process strongl non-hrmonic fringe ptterns s well being ble to compute ccurtel the number of fringes nd the phse offset f ϕ) if t lest well smpled fringes re present in the fringe pttern to be processed. t s obvious one cn mke LSF routine tht tkes into ccount more prmeters relted to high order hrmonics but the mount of computtion increses s the squred number of prmeters..3. Modulted fringes Most fringe ptterns do not hve uniform contrst the re modulted so tht their intensit profile m be written with vrible coefficients ) nd the fringes re sid to be qusiperiodic:

9 Sttisticl processing of stright equispced fringe ptterns 849 ϕ ) ) + ) cosν + ) 5) Obviousl the most generl fringe profile m be written s Fourier series 4) with vrible intensities k. Often we m use to process vrious qusiperiodic fringe ptterns nd determine the set of prmeters f ϕ) s if the fringes were hrmonic. Let s nlze if there is reson for doing this. n the Fourier spectrum of the function 5) the component of frequenc is widened its spectrl profile being the spectrum of the function ) due to the convolution of the function ) with the hrmonic function). Similrl the component of frequenc tkes the spectrum of the function ). B lest squres fitting of the function 5) with the hrmonic function ) the computed prmeters f ϕ) hve the following menings: become the verge of the functions ) ) respectivel while f nd ϕ estimte the number of fringes nd the phse prmeter in the formul 5) but ll these vlues re bised estimtions nd the tend to be ect s the number of fringes increses. o see the conditions suitble for good results on processing qusiperiodic fringes with the computer lgorithm for hrmonic fringes we mde sttistics on the results of processing of lrge set ) of computer simulted fringe ptterns. We simulte D modulted fringe ptterns ech of them with 56 smples with such profile s tht of the interference pttern of two Gussin bems. Fig. 4 displs the ctul fringe profiles used for the simulted fringes. For ll the fringe ptterns in the set the prmeters ) were ssigned the vlues.4.334) so tht the intensit lies in the rnge. he prmeter f ws given uniform rndom vlues in the intervl of 3 4 fringes nd the prmeter ϕ ws given uniform rndom vlues in the intervl -. We dd Gussin noise of men nd.5 stndrd devition. n these conditions the reltions 8) give for the devitions of the prmeters the vlues: theor σ ). Fig. 4. Modulted fringes. he Fig. 5 shows the result of the simultions in form of grphs tht show how the number of fringes f nd the phse prmeter ϕ influence the displcements of the prmeters. We see tht these influence becomes insignificnt if f >.

10 85 V. scov. Dobroiu D. postol V. Dmin Fig. 5. Results on processing set of modulted fringe ptterns. δφ/σφ theor) δφ/σφ theor) δ f/σ f theor) δ f/σ f theor) δ /σ theor) δ/σ theor) δ /σ theor) δ/σ theor) We further mde sttisticl nlsis over new set of computer-simulted fringes with the sme prmeters ecept for f tht ws given uniform rndom vlues in the rnge 4 fringes so tht the distribution of the errors δ / σ becomes closer to the stndrd norml distribution. his time for the reduced prmeter verge nd devition: ) δ / σ we obtin the following results for the σ ). n comprison with the hrmonic fringes the estimtions of the prmeters re more bised especill the prmeters f ϕ) hve displcements of bout. he devitions σ re greter thn thus the numericll computed sttisticl errors re underestimted up to 8% in the cse of prmeter f. he histogrm of the sttisticl quntit for ech component fits well the Gussin distribution. We conclude tht the LSF lgorithm for hrmonic fringe ptterns is lso suitble for rocessing qusiperiodic fringe ptterns to compute ccurtel the number of fringes nd the phse offset f ϕ) if t lest well smpled fringes re provided in the fringe pttern to be processed.

11 Sttisticl processing of stright equispced fringe ptterns Conclusions We develop nlticl epressions for the sttisticl errors concerning the mesurement of the prmeters f ϕ) for the simplest fringe pttern b). hese theoreticl results obtined using n smptotic pproimtion vlid for lrge number of fringes) give the upper limit of ccurc the fringe prmeters cn be determined with this method. Performing sttistics on set of simulted fringe ptterns we hve proved tht the nlticl formule 8) estimte correctl the sttisticl errors for hrmonic fringes if the fringe pttern hs t lest 3 fringes which re well smpled the smpling rte considered ws >8 smple/period). We stted the vlid conditions for processing non-hrmonic fringes with the LSF lgorithm for hrmonic fringes. Doing sttistics on the results of processing set of simulted Fbr Perot fringes we proved tht we get the fringe prmeters with high ccurc if there re t lest 8 fringes in the fringe pttern. We should mention tht lthough the numericll computed sttisticl errors re underestimted nd not lws relible) the theoreticl sttisticl errors 8) lws estimte ver well the ctul sttisticl errors if the number of fringes re greter thn 8. n the sme mnner we treted the processing of qusiperiodic fringe with the LSF lgorithm mde for hrmonic fringes. B sttisticl processing of set of computer simulted fringe ptterns we proved tht we cn clculte the fringe prmeters with good ccurc if there re t lest fringes in the fringe pttern. lso in this cse the numericll computed sttisticl errors re underestimted but the theoreticl results 8) lws estimte ver well the ctul sttisticl errors if the number of fringes re greter thn. he generl conclusion is tht the fringes must be hrmonic with mimum contrst to obtin mimum precision in mesuring its prmeters. However even for strongl non-hrmonic fringes the estimtion of the prmeters f ϕ) is ver good if the fringe profile is periodic nd if there re t lest - fringes vilble. Specil cre must be tken for processing of qusiperiodic fringes fringes with non-uniform contrst or modulted fringes). n this cse especill the prmeters f ϕ) cn hve lrge displcements nd fluctutions if there re not t lest fringes in the fringe pttern to be processed. he results of this pper re pplicble for emple to predict the precision tht one cn chieve when mesuring wvelengths lengths nd displcements using high precision interferometers with stbilized lsers. he give the nswer to the following questions: Wht resolution number of piels) must hve the CCD used to cquire the fringe pttern to mesure wvelengths lengths or displcements with given ccurc? How do these errors depend on the number of fringes? t hs been shown tht these errors do not depend on the number of fringes if this number is lrge enough. he depend onl on the number of smples nd the fringe contrst. 4. ppendi 4.. Lest squres fitting for liner combintions of functions Let be two quntities which hve functionl reltionship tht we pproimte b liner combintion of r independent functions: ) f ) + f ) + + r f r ) f ) f ) ) ) f f r f r ) )

12 85 V. scov. Dobroiu D. postol V. Dmin he smbol represents the opertion of mtri trnsposition. Let us consider tht ) is sclr function nd m be single vrible or set of independent vribles. Let be set of eperimentl results sttisticl dt) mesured with vrince σ to which set of dt is ssocited. We suppose the dt belonging to the vribles hve zero vrince. ) ) ) ) ) ) ) n the most generl cse σ is the covrince mtri of the vlues hs the dimension nd contins the vrinces on the min digonl while the other elements re the covritions between the vrious dt i) i... n mn prcticl situtions ll the vlues re mesured with the sme ccurc nd re sttisticll independent so tht the quntit σ becomes sclr. lterntivel we use s well nother equivlent quntit clled sttisticl weight defined b [ ] P. We re ttempting to compute the coefficients from reltion ) so tht this reltion best pproimte the eperimentl dt. For this purpose we use the lest squres fitting method from sttisticl mthemtics widel used for dt processing. Firstl we define column vector whose elements re the difference between the dt nd the ssumed function ): ) ) f ) D f ) f f f D ) ) ) ) f ) fr ) ) ) ) ) f ) f ) ) ) ) ) f ) f ) r r D f ) 3) s globl ssessment of ll these devitions one builds up sclr function ) which represents the sum of the squred normlized devitions of the eperimentl dt from the ssumed function ): ij ) i) j) ) δ ) δ ) ) P ) χ P 4) i j Lest squres fitting consists in finding the vlues of the prmeters for which the function ) reches its minimum. he vlues of the prmeters for which the function ) is minimum re obtined b solving n eqution sstem consisting of r liner equtions estblished b equling to zero the prtil derivtives of the function ) reltive to ech of the prmeters. his is clled the norml eqution sstem : χ ) D P [ D ] [ D P D] [ D P ] 5) he solution of this sstem is: [ D P D] [ D P ] B B 6) t is interesting to notice tht there is liner reltionship between the dt nd the computed prmeters. he prmeters re sttisticl quntities. Reltion 6) gives their

13 Sttisticl processing of stright equispced fringe ptterns 853 sttisticl men. Using the error propgtion theorem we obtin the covrince mtri of these prmeters tht contins on the min digonl their vrinces: [ D P D] [ P ] B B 7) We find out tht the function vector : ) is n r-dimensionl prbol in the components of the χ ) [ D ] P [ D ] [ D ) + D ] P [ D ) + D ] [ ] P [ ] + χ χ [ D ] P [ D ] const 8) Fig. 3: section of this prbol to one component s is simple D prbol shown in ) ) χ σ χ + 9) hus the prbol vrinces σ. ) reches its minimum in nd hs n etent corresponding to the χ ) σ χ Fig. 6. For liner LSF the function is prbol. Using the criterion of minimizing the function ) for computing the prmeters tht give the best fit of the eperimentl dt b the formul ) is bsed on the mimum plusibilit principle from sttisticl mthemtics. t is right to use this criterion onl with the ssumption tht the eperimentl dt devites from the formul ) onl due to norml distributed fluctutions with men nd σ vrinces. Otherwise different criterion is necessr. f the devitions δ ) re normll distributed then the minimum of the function nmel the vlue ) χ ) χ which is lso sttisticl quntit hs distribution with degrees of freedom. he vlue χ is useful to check if the reltion ) fits well the eperimentl dt. We distinguish two min cses:

14 854 V. scov. Dobroiu D. postol V. Dmin - f χ < r it is highl probble tht the reltion ) pproimtes ver well the eperimentl dt which devite from the reltion ) onl due to norml distributed sttisticl fluctution with men nd vrinces. - f χ < r the reltion ) is not suitble for fitting the eperimentl dt or the ssumption tht the devitions δ ) hve norml distribution with men nd vlid. σ vrinces is not χ ) σ χ Fig. 7. For nonliner LSF the function χ hs qudrtic form round it s globl minimum. 4.. Lest squres fitting for nonliner functions Let s ssume certin nonliner reltionship between two quntities nd which cnnot be epressed s liner combintion of independent functions: f ) ) he function f ) hs known behvior but depends on the set of prmeters which re to be determined in order to find the best fit of n eperimentl dt set ) b this function in the sense of minimizing the function ). We use the ssumption tht the dt is ffected b norml distributed fluctutions with men nd σ vrinces. he norml equtions for determining the set of prmeters re generll not liner. he cnnot be lws solved nlticll nd the numericl solution cn often prove to be difficult. he problem cn be tckled itertivel s shown further. We strt from n rbitrr initil pproimtion ) of the prmeter set obtined b some method or other with n rbitrr ccurc. he nonliner function f ) is developed s lor series round the vlues ) nd the liner term is sved: f ) f ) f ) D ) + D ) ) ) ) f ) ) f ) ) f ) ) f ) ) f ) ) f ) ) f ) r ) f ) r ) f ) r ) )

15 Sttisticl processing of stright equispced fringe ptterns 855 he -element vector tht contins the devitions of the function f ) from the eperimentl vlues is: ) ) ) f ) D ) f ) ) ) nd the sum of the squred nd normlized devitions is: ) ) ) ) P ) [ D ) ] P [ D ) ] χ 3) he norml eqution sstem hs the sme form s 5) but insted of we hve - ) nd is replced with δ f ) ): χ ) ) [ D ) ] [ D P D] ) [ D P ] ) D P 4) Obviousl its solution hs the sme form s 6): ) B [ D P D] [ D P ] B 5) B now we hve the result of the first itertion of series tht must be continued. he result thus obtined is input for new itertion insted of the zero-order pproimtion ) nd the procedure is repeted until the difference between the results of two successive itertions becomes negligible. Generll the lgorithm hs rpid convergence but not lws. fter the finl result ws obtined with sufficient ccurc its vrinces cn be clculted using the sme formul s in the liner cse 7). Let s nlze the form of the function ). f we develop the nonliner function f ) in first order lor series round the vlues : f ) f ) f ) + D ) D 6) then we obtin n r-dimensionl prbol form for the function n etent tht depends on the vrinces σ : ) hving the minimum in nd χ ) [ f ) ] P [ f ) ] [ D ) + f ) ] P [ D ) + f ) ] [ ] P [ ] + χ χ [ f ) ] P [ f ] const ) 7) Outside the vicinit of the function ) is no longer prbolic s shown in figure 4 tht presents one-dimensionl section of this r-vrible function. he bsic ide of this lgorithm consists in the fct tht the function ) which depends on the set of r prmeters hs n pproimtel r-dimensionl qudrtic shpe round no mtter if the function f ) is liner or not) nd its minimum cn be determined s solution of liner eqution sstem. n the non-liner cse the result obtined fter ech itertion progressivel drws ner the solution ecept in some inpproprite singulrit cses when the convergence is not ttined. s generl rule to ensure convergent itertions the first pproimtion ) must be close enough to the ect vlues otherwise the itertions m diverge or the m led to locl minimum. t is remrkble tht ll the opertions described for this lgorithm re crried out using onl liner lgebr.

16 856 V. scov. Dobroiu D. postol V. Dmin References [] K. R. V. Murt pplied Optics ). [] D. postol. postol.. Mih ilescu nfrred Phsics ). [3] D. postol V. Dmin St. Cerc. Fiz ). [4] J. Schwider R. Burow K. E. Elssner J. Grzn R. Spolczk K. Merkel pp. Opt ). [5] C. Broph J. Opt. Soc. m ). [6] G. Li. Ytgi J. Opt. Soc. m ). [7] C. Rthjen J. Opt. Soc. m ). [8] B. Zho pp. Opt ). [9] D. W. Robinson pplied Optics ). [] D. W. Robinson G.. Reid nterferogrm nlsis nstitute of Phsics Publishing Bristol nd Phildelphi 993. [] G. Li. Ytgi pp. Opt ). [] bout multivrite norml distribution.