英文文献及翻译级联迭代傅里叶变换算法在光学安全中的应用.docx

英文文献及翻译级联迭代傅里叶变换算法在光学安全中的应用.docx

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英文文献及翻译级联迭代傅里叶变换算法在光学安全中的应用

AcascadediterativeFouriertransformalgorithm

foropticalsecurityapplications

Abstract:

AcascadediterativeFouriertransform（CIFT）algorithmispresentedforopticalsecurityapplications.Twophase-masksaredesignedandlocatedintheinputandtheFourierdomainsofa4-fcorrelatorrespectively,inordertoimplementtheopticalencryptionorauthenticityverification.Comparedwithpreviousmethods,theproposedalgorithmemploysanimprovedsearchingstrategy:

modifyingthephase-distributionsofbothmaskssynchronouslyaswellasenlargingthesearchingspace.Computersimulationsshowthatthealgorithmresultsinmuchfasterconvergenceandbetterimagequalityfortherecoveredimage.Eachofthesemasksisassignedtodifferentperson.Therefore,thedecryptedimagecanbeobtainedonlywhenallthesemasksareunderauthorization.Thiskey-assignmentstrategymayreducetheriskofbeingintruded.

Keywords:

Opticalsecurity–opticalencryption–cascadediterativeFouriertransformalgorithm

1.Introduction

Opticaltechniqueshaveshowngreatpotentialinthefieldofinformationsecurityapplications.RecentlyRéfrégierandJavidiproposedanoveldouble-random-phaseencodingtechnique,whichencodesaprimaryimageintoastationarywhitenoise.ThistechniquewasalsousedtoencryptinformationinthefractionalFourierdomainandtostoreencryptedinformationholographically.Phaseencodingtechniqueswerealsoproposedforopticalauthenticityverification.

WangetalandLietalproposedanothermethodforopticalencryptionandauthenticityverification.Unlikethetechniquesmentionedabove,thismethodencryptsinformationcompletelyintoaphasemask,whichislocatedineithertheinputortheFourierdomainofa4-fcorrelator.Forinstance,giventhepredefinitionsofasignificantimagef（x,y）asthedesiredoutputandaphase-distributionexp{jb（u,v）}intheFourierdomain,it’seasytooptimizetheotherphasefunctionexp{jp（x,y）}withamodifiedprojectionontoconstraintsets

（POCS）algorithm[10].Thereforetheimagef（x,y）isencodedsuccessfullyintoexp{jp（x,y）}

withtheaidofexp{jb（u,v）}.Inotherwords,thefixedphaseexp{jb（u,v）}servesasthelockwhiletheretrievedphaseexp{jp（x,y）}servesasthekeyofthesecuritysystem.Toreconstructtheoriginalinformation,thephasefunctionsexp{jp（x,y）}andexp{jb（u,v）}mustmatchandbelocatedintheinputandtheFourierplanerespectively.Abookasisetalimplementedthisschemewithajointtransformcorrelatorforopticalverification.

However,becausethekeyexp{jp（x,y）}containsinformationoftheimagef（x,y）andthe

lockexp{jb（u,v）},andthe4-fcorrelatorhasacharacteroflinearity,itispossiblefortheintrudertofindoutthephase-distributionofthelockfunctionbystatisticallyanalyzingtherandomcharactersofthekeysifthesystemusesonlyonelockfordifferentimage.Inordertoincreasethesecurelevelofsuchsystem,oneapproachistousedifferentlockfunctionfordifferentimage.Enlargingthekeyspaceisanotherapproachtoincreasethesecurelevel.ItcanbeachievedbyencryptingimagesinthefractionalFourierdomain;asaresult,thescalefactorsandthetransformorderofferadditionalkeys.Ontheotherhand,notethatthephase-maskservesasthekeyofthesystem,enlargingthekeyspacecanbeachievedbyencodingthetargetimageintotwoormorephasemaskswithamodifiedPOCSalgorithm.Changetalhaveproposedamultiple-phasesretrievalalgorithmanddemonstratedthatanopticalsecuritysystembasedonithashigherlevelofsecurityandhigherqualityforthedecryptedimage.However,thisalgorithmretrievesonlyonephase-distributionwithaphaseconstraintineachiteration.Asaresult,themasksarenotsoconsistentandmayaffectthequalityoftherecoveredimage.

Inthepresentpaper,weproposeamodifiedPOCSalgorithmthatadjuststhedistributionsofbothphase-maskssynchronouslyineachiteration.Asaresult,theconvergentspeedoftheiterationprocessisexpectedtosignificantlyincrease.Andthetargetimagewithmuchhigherqualityisexpectedtorecoverbecauseoftheco-adjustingofthetwomasksduringtheiterationprocess.Whentheiterationprocessisfinished,thetargetimageisencodedintothephase-maskssuccessfully.Eachofthesemasksseversasthekeyofthesecuritysystemandpartoftheencryptedimageitselfaswell.Moreover,thealgorithmcanbeextendedtogeneratemultiplephase-masksforarbitrarystagescorrelator.Toacquirethemaximumsecurity,eachkeyisassignedtodifferentauthoritysothatthedecryptioncannotbeperformedbutbeingauthorizedbyallofthem.Thiskey-assignmentschemeisespeciallyusefulformilitaryandgovernmentapplications.

ThealgorithmdescriptionispresentedinSection2.ComputersimulationofthisalgorithmandthecorrespondingdiscussarepresentedinSection3.

2.CascadedIterativeFourierTransform（CIFT）Algorithm

Considertheoperationoftheencryptionsystemwiththehelpofa4-fcorrelatorasshowninFig.1,thephasemasksplacedintheinputandtheFourierplanesaredenotedas

and

respectively,where（x,y）and（u,v）representthespaceandthefrequencycoordinate,respectively.Oncethesystemisilluminatedwithamonochromaticplanewave,atargetimagef（x,y）（animagetobedecryptedorverified）isexpectedtoobtainattheoutputplane.Thephase-masks

and

containtheinformationoff（x,y）,thatis,f（x,y）isencodedintothesephase-masks.Theencodingprocessistheoptimizationofthetwophase-distributions.Itissomewhatsimilarwiththeproblemsoftheimagereconstructionandthephaseretrieval,whichcanbesolvedwiththePOCSalgorithm.However,thepresentproblemcomesdowntothephaseretrievalinthree（ormore,ingeneral）planesalongthepropagationdirection.SotheconventionalPOCSalgorithmshouldbemodifiedforthisapplication.

ThecascadediterationFouriertransform（CIFT）algorithmbeginswiththeinitializationofthephase-distributionsofthemasks.Supposetheiterationprocessreachesthekthiteration（k=1,2,3,…）,andthephase-distributionsintheinputandtheFourierplanearerepresentedas

and

respectively.Thenanestimationofthetargetimageisobtainedattheoutputofthecorrelatordefinedby

whereFTandIFTdenotetheFouriertransformandtheinverseFouriertransform,respectively.Iffk（x,y）satisfiestheconvergentcriterion,theiterationprocessstops,and

and

aretheoptimizeddistributions.Otherwise,thefk（x,y）ismodifiedtosatisfythetargetimageconstraintasfollows

Thenthemodifiedfunctionistransformedbackwardtogeneratebothofthephase-distributionsasfollows

whereang{·}denotesthephaseextractionfunction.Thenkisreplacedbyk+1forthenextiteration.ItisshowninEqs.3（a）and3（b）thatbothofthephase-distributionsaremodifiedineveryiteration,accordedtotheestimationofthetargetimageinthepresentiteration.Itensureshealgorithmconvergeswithmuchfasterspeedandmoreconsistentforthephase-masks.

Ingeneral,theconvergentcriterioncanbetheMSEorthecorrelationcoefficientbetweentheiteratedandthetargetimage,whicharedefinedby

whereM*Nisthesizeoftheimage,andE[·]denotesthemeanoftheimage.TheconvergentbehaviorofthisalgorithmissimilartothatoftheconventionalPOCS.Thatis,theMSEreducesrapidlyintheforemostfewiterations,thenitkeepsreducingslowlytillitreachestheminimum.Correspondingly,thecorrelationcoefficientisexpectedtoincreaserapidlyatfirstandkeepincreasingslowlytillthestoppingcriterionissatisfied.

Indecryption,thedeterminedphase-masks

and

（thekeysoressentially,theencryptedimages）areplacedintheinputandtheFourierplane,respectively,andthentransformedintotheoutputplanethroughthecorrelationdefinedbyEq.

（1）.Themodulusoftheoutputisthedecryptedimage.TheCIFTalgorithmretainsthepropertyoftheconventionaliterationalgorithm,thatis,thefinalphase-distributionsofthemasksaredeterminedbytheinitializationsofthem.Thereforedifferentinitializationswillresultindifferentdistributionsof

and

.Thetargetimagecannotbedecryptedifthekeysmismatch（thatis,thekeysweregeneratedfromthedifferentiterationprocess）.

Inpracticalsystem,thephasesofthemasksarequantizedtofinitelevels,whichmightreducethesolutionspaceandintroducenoisetotherecoveredimage.Tocompensatethelossofthequality,thetargetimagecanbeencodedintomorephase-maskstoprovideadditionalfreedomforsolutionssearching,whichmeanstoencrypttheimagewithamulti-stages（cascaded）correlator.Fromthepointofviewofsecurity,thisstrategysignificantlyenlargesthekeyspace（becausemorekeysweregenerated）,andmakestheintrusionmoredifficult.Generally,thet-stagescorrelationisdefinedas

fortiseven,or

fortisodd,wherethematrixI（x,y）representstheinputplanewave,andthesuperscripti（i=1,2,…,t）denotestheserialnumberofthemasksinthesystem.Thephase-distributionsofthesemasksmaybededucedbyanalogousanalysisforEq.3.

3.Computersimulation

Inthissectionwenumericallydemonstrateourgeneralconcept.Ajetplaneimageofthesize128´128with256grayscaleisusedasthetargetimageasshowninFig.2.Thesizesofbothphase-masksaresameasthetargetimage.Andwesupposetheopticalsystemisilluminatedbyaplanewavewiththeamplitudeequatingto1.Thealgorithmstartswiththerandominitializationofthetwophase-masks.Thenthephasefunctionsaretransformedforwardandbackwardalternativelythroughth