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1、THEFASTCALCULATIONOFFORMFACTORSUSINGLOWDISCREPANCYSEQUENCESALEXANDERKELLERDEPARTMENTOFCOMPUTERSCIENCE,KAISERSLAUTERNUNIVERSITY,POSTFACH3049,D-67653KAISERSLAUTERN,GERMANYABSTRACT.Thecalculationofformfactorsisanimportantproblemincomputingtheglobalilluminationintheradiositysetting.Closedforms
2、olutionsoftenareonlyavailableforobjectswithoutobstructionandareveryhardtocalculate.UsingMonteCarlointegra-tionandraytracingprovidesafastandeleganttoolfortheestimationoftheformfactors.Inthispaperweshow,thatusingdeterministiclowdiscrepancysamplepointsissuperiortorandomsampling,resultinginana
3、ccelerationofmorethanhalfanorderofmagnitude.1.INTRODUCTIONIncomputergraphicsmostintegralshaveadiscontinuouskernelandassucharehardtobesolvedanalytically.Inadditionvisibilityhastobechecked,whichisanexpensiveoperation.Fortheevaluationofsuchintegrals,MonteCarlomethodsprovideaefficientandelegant
4、tool.OnacomputertherandomsamplesusedforMonteCarlointegration,areapproximatedbymeansofpseudo-randomnumbers.Butthereexistdeterministicpointsetsespeciallydesignedforintegration,whichpromiseaconvergencefasterthantheMonteCarlorateofO(√1),whereNisthenumberofsamplesdrawn.NThepapernowinvestigatest
5、heapplicationofso-calledlowdiscrepancypointsfortheformfactorintegralandcomparesittorandomsampling.Thereforeweintroducethequasi-MonteCarlomethod(foraprofoundintroductiontoquasi-MonteCarlointegrationandlowdiscrepancypoints,see[Nie92b])inthenextsection.Thenweexplainthealgo-rithmusedforthecalc
6、ulationoftheformfactors.Afterdiscussingthenumericalevidenceofsomeexperiments,wedrawtheconclusions.2.MONTECARLOANDQUASI-MONTECARLOINTEGRATIONIncomputergraphicsweoftenhaveintegrandswithdiscontinuities,thatarenotaxis-aligned.Sousualquadraturerulesdonotworkatmaximumefficiencyandwethereforeuseth
7、eMonteCarlomethodtoapproximateanintegralontheunitcubebyZNX−11g(x)dx≈g(xi)IsNi=0wherePN={x0,...,xN−1}isauniformlydistributedsequenceofpointsinthes-dimensionalunitcube[0,1)s=Is.FortheclassicalMonteCarlomethod,thesepointsarechosenran-domly(onacomputertypicallymod
