Poisson-BasedContinuousSurfaceGenerationforGoal-BasedYonghaoYueColumbi - PDF document

Poisson-BasedContinuousSurfaceGenerationforGoal-BasedYonghaoYueColumbiaUniversityKeiIwasakiWakayamaUniversityandUEIResearchBing-YuChenNationalTaiwanUniversityandUEIResearchYoshinoriDobashiHokkaidoUniversityandUEIResearchTomoyukiNishitaUEIResearchandHiroshimaShudoUniversityWepresentatechniqueforcomputingtheshapeofatransparentobjectthatcangenerateuser-deÞnedcausticpatterns.Thesurfaceoftheobjectgeneratedusingourmethodissmooth.Thankstothisproperty,theresult- Y.Yueetal. Fig.1.Thefabricatedobject(right)anditscaustics(left)onthescreen.PleaseseeFigure7fortheshapeoftherefractivesurfaceoftheobject. Light sourceNearlyparallel light e.g. sunlight, projector Transparent objectScreenCaustic pattern Incident surface: plane Refraction surface Screen: plane Fig.2.Anillustrationoftheproposeddisplaysystem.2.PREVIOUSAPPROACHESTheapproachestakeninthepreviousmethods[Finckhetal.2010;Papasetal.2011;Yueetal.2012]canbeclassiÞedintothefol-lowingtwotypes.TheÞrstapproach,takenbyFinckhetal.[2010],randomlyperturbstherefractivesurfacetoÞndthesolution.Thesecondapproach,takenbyPapasetal.[2011]andYueetal.[2012],Þrstdeterminestherelationshipbetweenthelightontheincidentsurfaceandthatonthescreen,andthencomputestheshapeoftherefractivesurface.IntheÞrstapproach,thesolutionspaceiscomplex,andtherecanbemanysolutionsthatareoptimallocally.Papasetal.[2011]pointedoutthattheresultgeneratedusingthisapproachcanconvergetoasolutionthatdoesnotspanthefullcon-trastrangeofacomplextargetimage(criterion(i)inSection1).Inthesecondapproach,thekeyelementisthewayinwhichtherelationshipbetweenthelightontheincidentsurfaceandthatonthescreenisfound.Withpreviousmethodsonlyadiscreterelation-shipcouldbefound.Inthosepreviousmethods,theincidentsur-face(ortherefractivesurface)andthescreenaresubdividedintosetsofsmallregions,andtherelationshipsbetweenthesesmallre-gionsareestablished.Papasetal.Õsmethod[2011]subdividestheincidentsurfaceintoagridandthecausticsintoasetofGaussiankernels,andusessimulatedannealingtoÞndtherelationship.InYueetal.Õsmethod[2012],theobjectcanberearrangedusingasetofdiscretesticks.Inthesemethods[Papasetal.2011;Yueetal.2012],thelightincidentonneighboringregionsgenerallyarrivesatdistantregionsonthescreen.Duetothisdiscontinuity,thefollowingproblemsoc- Transparent objectScreen xyzCoordinate system p(x,y) ’ Fig.3.Anillustrationofthecomputationoftherelationship.cur.First,eachsmallregionusuallyneedstobeofacertainsize,whichrestrictstheresolutionofthecausticpattern(criterion(ii)inSection1).Second,thedynamicrangeofthecausticsissomewhatrestricted(criterion(iii)inSection1).Thisisbecausetheinten-sityofthecausticsisbasicallydeterminedbythenumberofover-lappingregionsonthescreen,andsinceeachsmallregionontheincidentsurfaceisofasimilarsize,theintensityofthecausticsbe-comesdiscrete.Furthermore,therangeoverwhichthecausticsareinfocusisshort(criterion(iv)inSection1),andiftheobjectisplacedoutoffocus,theresultingcausticpatternwouldbeasetofdiscretespots(criterion(v)inSection1).Ourgoalistoeliminatediscretizationoftheresultingimageandtoenhancethestabilityoftheprojectedimagetoperturbationsinthephysicaldisplaysetting.Inthisarticle,weproposeanewmethodforcomputingtheshapeoftheobject.OurmethodcanbeclassiÞedasthesecondapproachdescribedpreviously.Thedistinctaspectofourmethodisthatwecomputeacontinuousrelationship.Owingtothisproperty,there-fractivesurfacebecomescontinuous,andthedistributionofthelightleavingtherefractivesurfaceisalsocontinuous.Theprob-lemsdescribedearliercanberesolvedasaresultofthis.3.OURMETHODOurmethodconsistsoftwosteps.IntheÞrststep,wecomputetherelationshipbetweenthelightreachingtheincidentsurfaceandthatreachingthescreen.Inthesecondstep,wecomputetheshapeoftherefractivesurfacefromtherelationship.3.1ComputingtheRelationshipAsshowninFigure3,the-axisisalignedwiththedirectionoftheincidentparallellight(thelightiscomingfromtoward),andtheincidentsurfaceisaplaneperpendiculartothe-axis.PointsontheincidentsurfacecanberepresentedusingAsthelightisperpendiculartotheincidentsurface,itenterstheobjectwithoutrefractionandreachestherefractivesurface.Thecoordinateofanypointontherefractivesurfacecanberepresentedasasingle-valuedfunctionof.Thelightrefractsattherefractivesurfaceandthenreachesthescreen.Todistinguishthecoordinatesonthescreenfromthoseontheincidentsurface,wetodescribethecoordinatesonthescreen.Forsimplicity,weusex,yu,vanddropthecoordinatetodescribethepointsontheincidentsurfaceandthescreen.Thelightpathcanbedescribedbytherelationshipbetweenax,yontheincidentsurfaceandapointu,vonthescreen.Inourmethod,weconsiderthecasewhenthisrelationshipisone-to-one,andrepresentthecausticsbychangingthedensityofthelightreachingthescreen.Thus,weneedtoÞndfromwhichregionsoftheincidentsurfaceweneedtogatherlightinordertoformtheintensitydistributionofthedesiredcausticpattern.ThisproblemcanbeformulatedastheproblemofÞndingamappingu,vx,ysatisfyingthefollowingtwoconditions.ACMTransactionsonGraphics,Vol.31,No.3,Article31,Publicationdate:May2014. Poisson-BasedContinuousSurfaceGenerationforGoal-BasedCaustics (a)(b)(c)Fig.4.(a)Thedifference(redandblueregionsindicatepositiveandnegativeportions);(b)pressureÞeld;(c)vectorÞeldTheÞrstconditionisthatthemappingiscontinuousovertheentiredomain.Ifthismappingiscontinuous,wecanobtainacon-tinuousrefractivesurface.Thesecondconditionisthatthismap-pingpreservesthelightenergy.ImaginethelightbeamshowninFigure3.IfthelightincidentonaninÞnitesimalregionx,yontheincidentsurfacehasintensityx,y(letusforsimplicity),andtheintensityofthislightonaninÞnitesi-malregionaroundpointu,vonthescreenisu,v(letusforsimplicity),then.TheaimofourmethodistoÞndthismapping.SinceitisdifÞculttoanalyticallycomputethismapping,weuseageometricßowapproach,asdescribednext.Themappingfromu,vx,ycanberegardedasparam-eterizationof.Thatis,westartfromaninitial,andthenmaintainthecontinu-ityoftheparameterizationwhilemodifyingitsothatitincreasinglysatisÞestheconservationoflightenergy.Undertheparameterizationof,theratioisequiv-alenttothedeterminantoftheJacobiangivenby
u
y
u
x
v
y .Iftheparameterization,orthemapping,satisÞesatanarbitrarypoint,whereistheintensityofthecaus-ticsatthecorrespondingpointonthescreen,thenthatmappingisthesolutiontotheproblem.Duringcomputation,isnotalwayssatisÞed,andthereisadifference.Letthisdifferencex,y(again,weusesimplicity)beWereducethisdifference(seeFigure4(a))byupdatingthepa-rameterizationtocontinuouslymodifythepointx,yontheincidentsurface,whileÞxingthecorrespondingpointu,vthescreen.Althoughwecouldconsiderthegradient
)andmodifythepointsas isavirtualtimetorepresentthecomputationprocess),isusuallynotcontinuousandmaytakeaninÞnitevalue.ThusitisdifÞculttostablyperformthecomputationusingInstead,inspiredbyaconceptincomputationalßuiddynamics,wemodifytheparameterizationasfollows(ourmethodisrelatedtoonethatgeneratesarea-preservingparameterization[Zouetal.2011]).First,thereareregionswherethedifferenceispositiveornegative.Analogoustothoseincomputationalßuiddynamics,weregardthepositiveandnegativeregionsassourceandsink,respec-tively.Torelaxthesourceandsink,wesolvePoissonÕsequationinordertocomputethepressureÞeld(seeFigure4(b)),andup-datethepointsx,yalongthegradientof.Thatis,welet Unlike(seeFigure4(c))isamuchsmoothervectorÞeld,thuswecanstablyupdatethepointsx,y.Inourmethod,werepeatedlyupdatetheparameterizationbycomputingthedifferenceusingEq.(2),solvingthePoissonÕsequation(3),andupdatingaccordingtoEq.(4).Aphysicalinterpretationoftheprecedingtechniqueisasfol-lows.Ifwewantaregiononthescreentobebrighter/darker,thecorrespondinglightbeamneedstocollectlightfromawider/narrowerregion(i.e.,shouldbelarger/smaller).Thisim-pliesthatwehavetolocallyÒexpandÓorÔshrinkÓ.BysolvingEq.(3)andcomputing,wegetaÒcompressibleÓßowÞeldthatrealizesthisexpansionorshrinkage.Byiterativelyupdatingthepa-rameterization,thecontrastoftheresultingcausticsincreasesandapproachesthecontrastoftheinputpattern.Inourimplementation,weuseatriangularmeshtorepresenttheparameterization.Tocomputethelightenergy,insteadofcomput-ingtheJacobian,wecomputeamediandualmesh(wherewehaveafaceforeachvertexinthetriangularmesh,eachofwhichisgener-atedbyconnectingthemidpointsoftheedgesandthecentroidsofthetrianglesadjacenttothecorrespondingvertexinthetriangularmesh).Wethenintegratetheintensitiesineachfaceinthemediandualmeshtocomputethelightenergy.PoissonÕsequations(3)and(7)aresolvedusingaÞrst-orderGalerkinÞnite-elementmethod.Toupdatetheverticesofthetriangularmesh,wediscretizeEq.(4)usingtheequation,where-thcomputationstep,andisdeterminedsothattherewillbenoßippedtriangles.WeobtainbyÞrstregardingafunctionof.Next,foreachtriangle,wecomputethemaximumthatthe(-th)trianglecantakesuchthattheareaassignedtoitremainsnonnegative.Then,wecomputetheminimumvalue=min).Finally,weset3.2ComputingtheRefractiveSurfaceAswehaveassumedtherefractivesurfacetobeasingle-valuedfunction,itscoordinatecanberepresentedasx,ynormalvectortotherefractivesurfacecanberepresentedby x isnormalizedsothatitscoordinateis.Consideringonlythecomponents,wehaveisavectorcomposedoftheponentsof.TakingthedivergenceofbothsidesofEq.(6),weobtainanotherPoissonÕsequation(similartoYuetal.[2004])Thus,ifthenormalvectoratanarbitrarypointontherefractivesurfaceisknown,wecanuseEq.(7)tocomputetheshapeoftherefractivesurface.Tocomputethenormalvectors,weuseSnellÕslawbasedontheparameterizationobtainedfromSection3.1,and ACMTransactionsonGraphics,Vol.31,No.3,Article31,Publicationdate:May2014.