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Wave-frontestimationfromwave-frontslopemeasurementsW.H.SouthwellRockwellInternationalCorporation,RocketdyneDivision,CanogaPark,California91304(Received28August1979;revised21February1980)Theproblemofwave-frontestimationfromwave-frontslopemeasurementshasbeenexaminedfromaleast-squarescurvefittingmodelpointofview.Itisshownthattheslopemeasurementsamplinggeometryinfluencesthemodelselectionforthephaseestimation.Successiveover-relax-ation(SOR)isemployedtonumericallysolvetheexactzonalphaseestimationproblem.Anewzonalphasegradientmodelisintroducedanditserrorpropagator,whichrelatesthemean-squarewave-fronterrortothenoisyslopemeasurements,hasbeencomparedwithtwopreviouslyusedmodels.Atechniquefortherapidextractionofphaseaperturefunctionsispresented.Errorpropagationprop-ertiesformodalestimationareevaluatedandcomparedwithzonalestimationresults.INTRODUCTIONInthispaperweconsidertheproblemofestimatingwave-frontphasefromasetofdiscretephaseslopemeasurements.Thisreconstructionprocessisnecessarytoobtainphaseprofileestimatesfromcertaintypesofsensors,suchasashearinginterferometeroraHartmannsensor.ItisalsonecessaryinimagerestorationusingtheKnox-Thompson'speckleimagingtechnique.Avarietyofapproacheshavebeenproposedtoaccomplishthisphasereconstruction.Basically,eachestimationap-proachmaybecategorizedasbeingeitherzonalormodaldependingonwhethertheestimateisaphasevalueinalocalzoneoracoefficientofanaperturefunction.Ineithercase,least-squaresestimationisusedforthephasereconstruc-tion.Differencesinthemethodscanbedistinguishedbythedifferentmodelsusedtofitthelocalslopemeasure-ments.Someimportantconsiderationsinselectingamodelare(i)compatibility;doesthemodelfitthegeometryoftheslopemeasurementsasgivenbythesensor?,(ii)numericalcom-plexity;arethereconvergenceproblems,storage,andcom-putationspeedrequirements?,and(iii)errorpropagation;whateffectdoesnoiseintheslopemeasurementshaveonthephaseestimates?Thereconstructionproblemwasfirstaddressedinthelit-eraturebyRimmer,2andsincethenbyanumberofauthors.34Mostofthesepapersuseaphasedifferenceslopemodelforzonalreconstructionthatischieflyapplicabletoashearinginterferometricsensor,wherethex-slopemeasurementpo-sitionsarenotcoincidentwiththey-slopemeasurementpo-sitions.Withsomeapproximations,Hudgin3calculatedthenoisepropagatorforthismodel.Usingamodelthatassumedcoincidentx-andy-slopemeasurementsandsomeapproxi-mations,Fried4alsocalculatedthenoisepropagationpa-rameter.Recently,Herrmann8constructedanexactsolutionforthenoisepropagatorforthemodelsofbothFriedandHudgin.Herrmann'sresultsessentiallyconfirmedthepre-viousresults.Inthispaperwepresentanewphasereconstructionmodelwhich,likethatofFriedisapplicabletocoincidentx-andy-slopemeasurements(andthussuitableforHartmannsensors).TheerrorpropagationpropertiesofthisnewmodelhavebeenshowntobesuperiortothemodelsofbothHudginandFried,partlybecauseitprovidesamoreuniformsamplingoftheaperture.Wealsodescribe,andillustratewithexam-ples,amatrixiterativetechniquethatsolvestheexactleast-squareszonalreconstructionproblem,iseasytoimplement,andexhibitsrapidconvergence.Modalestimationisalsodiscussed.Somenumericalmethodsfortherapidextractionofthelow-orderorthogonalmodesaredescribed.Finally,noisepropagationeffectsusingmodalestimationarecomparedwiththoseusingzonalesti-mation.ZONALESTIMATIONThreegridconfigurationsthatillustratethegridpositionsforthex-andy-slopemeasurementsandthereconstructedphasesareshowninFig.1.Itisimportanttopointoutthattheformulationoftheproblemdependsonthegridpatternused,which,inturn,dependsonwheretheactualslopemea-surementsaretaken.Figure1(A)wouldbeapplicable,forexample,foraHartmannsensor,whereeachsmalllensmea-998J.Opt.Soc.Am.,Vol.70,No.8,August19800030-3941/80/080998-09$00.50(01980OpticalSocietyofAmerica-998Syj=(bij+1-oij)/hi=1,N1Ji=1,N-1FIG.1.Slopemeasurementsamplinggeometryandwave-frontmeshpoints.Thehorizontaldashesindicatepositionsofx-slopesampling.Theverticaldashesarethey-slopesamplingpositions.Thedotsaretheesti-matedphasepoints.Configura-tionBhasbeenconsideredpre-viouslybyHudginandconfigura-tionCbyFried.suresboththexandyslopesatthesamepoint.Figure1(B)couldbeapplicable,forexample,inashearinginterferometerwherethebeamisfirstdividedinordertoapply,separately,thexandyshears.Figure1(C)showscoincidentxandyslopes,butwithadisplacedphasegrid.Theproblemistoleast-squares-fittheslopemeasurementdatatosomefunctionormodelofthephasegivenatitsgridpoints.ConsiderfirstthedataconfigurationgiveninFig.1(B).InthatconfigurationthereareN2phasepointsand2N(N-1)slopemeasurementpoints.Letusassumethatthephasedependencebetweengridpointsinthexdirectionisrepresentedbythepolynomialt1=Co+clx2.(1)TheslopeisSx=c1+2c2x.(2)Butsincewehaveonlyoneslopemeasurement,whichisgivenatthemidpointoftheinterval,wecanonlydeterminec1.Theconstantc0isthevalueofthephaseattheleftsideofthein-terval.Thus,allthatthedatacansupportisalinearphasemodel,q5=+STx.(3)Thephaseattherightoftheinterval(x=h)iski+.Applyingthisboundaryfactleadstotheslopemodel,S-T=(0i+1-,i)lhi=1,N-1,(4)whereNisthenumberofphasesamplepointsinthexdirec-tion.Generalizing