Calibrating photon counts from a single image · Calibrating photon counts from a single image...
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Calibrating photon counts from a single image Rainer Heintzmann 1,2* , Peter K. Relich 3 , Robert P.J. Nieuwenhuizen 4 , Keith A. Lidke 3 and Bernd Rieger 4* 1 Leibniz Institute of Photonic Technology, Jena, Germany 2 Institute of Physical Chemistry and Abbe Center of Photonics, Friedrich Schiller University, Jena, Germany 3 Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM, USA 4 Department of Imaging Physics, Delft University of Technology, Delft, The Netherlands * Correspondence to: [email protected] or [email protected] Abstract Current methods for detector gain calibration require acquisition of tens of special calibration images. Here we propose a method that obtains the gain from the actual image for which the photon count is desired by quantifying out-of-band information. We show on simulation and experimental data that our much simpler procedure, which can be retroactively applied to any image, is comparable in precision to traditional gain calibration procedures. Optical recordings consist of detected photons, which typically arrive in an uncorrelated manner at the detector. Therefore the recorded intensity follows a Poisson distribution, where the variance of the photon count is equal to its mean. In many applications images must be further processed based on these statistics and it is therefore of great importance to be able to relate measured values S in analogue-to-digital-units (ADU) to the detected (effective) photon numbers N. The relation between the measured signal S in ADU and the photon count N is given by the linear gain g as S = gN. Only after conversion to photons is it possible to establish the expected precision of intensities in the image, which is essential for single particle localization, maximum-likelihood image deconvolution or denoising [Ober2004, Smith2010, Afanasyev2015, Strohl2015]. The photon count must be established via gain calibration, as most image capturing devices do not directly report the number of detected photons, but a value proportional to the photoelectron charge produced in a photomultiplier tube or collected in a camera pixel. For this calibration typically tens of calibration images are recorded and the linear relationship between mean intensity and its variance is exploited [vanVliet1998]. In current microscopy practise a detector calibration to photon counts is often not done but cannot be performed in retrospect. It thus would be extremely helpful, if that can be determined from analysing the acquisition itself – a single image. A number of algorithms have been published for Gaussian type noise [Donoho1995, Immerkaer1996] and Poissonian type noise [Foi2008, Colom2014, Azzari2014, Pyatykh2014]. However, all these routines use assumed image properties to extract the information rather than just the properties of the acquisition process as in our presented algorithm. This has major implications for their performance on microscopy images governed by photon statistics (see Supplementary Information for a comparison with implementations from Pyatykh et al. [Pyatykh2014] and Azzari et al. [Azzari2014] which performed more than an order of magnitude worse than our method).
Transcript of Calibrating photon counts from a single image · Calibrating photon counts from a single image...
Calibratingphotoncountsfromasingleimage
RainerHeintzmann1,2*,PeterK.Relich3,RobertP.J.Nieuwenhuizen4,KeithA.Lidke3andBerndRieger4*
1LeibnizInstituteofPhotonicTechnology,Jena,Germany2InstituteofPhysicalChemistryandAbbeCenterofPhotonics,FriedrichSchillerUniversity,Jena,Germany3DepartmentofPhysicsandAstronomy,UniversityofNewMexico,Albuquerque,NM,USA4DepartmentofImagingPhysics,DelftUniversityofTechnology,Delft,TheNetherlands*Correspondenceto:[email protected]@tudelft.nl
AbstractCurrentmethodsfordetectorgaincalibrationrequireacquisitionoftensofspecialcalibrationimages.Hereweproposeamethodthatobtainsthegainfromtheactualimageforwhichthephotoncountisdesiredbyquantifyingout-of-bandinformation.Weshowonsimulationandexperimentaldatathatourmuchsimplerprocedure,whichcanberetroactivelyappliedtoanyimage,iscomparableinprecisiontotraditionalgaincalibrationprocedures.
Opticalrecordingsconsistofdetectedphotons,whichtypicallyarriveinanuncorrelatedmanneratthedetector.ThereforetherecordedintensityfollowsaPoissondistribution,wherethevarianceofthephotoncountisequaltoitsmean.InmanyapplicationsimagesmustbefurtherprocessedbasedonthesestatisticsanditisthereforeofgreatimportancetobeabletorelatemeasuredvaluesSinanalogue-to-digital-units(ADU)tothedetected(effective)photonnumbersN.TherelationbetweenthemeasuredsignalSinADUandthephotoncountNisgivenbythelineargaingasS=gN.Onlyafterconversiontophotonsisitpossibletoestablishtheexpectedprecisionofintensitiesintheimage,whichisessentialforsingleparticlelocalization,maximum-likelihoodimagedeconvolutionordenoising[Ober2004,Smith2010,Afanasyev2015,Strohl2015].Thephotoncountmustbeestablishedviagaincalibration,asmostimagecapturingdevicesdonotdirectlyreportthenumberofdetectedphotons,butavalueproportionaltothephotoelectronchargeproducedinaphotomultipliertubeorcollectedinacamerapixel.Forthiscalibrationtypicallytensofcalibrationimagesarerecordedandthelinearrelationshipbetweenmeanintensityanditsvarianceisexploited[vanVliet1998].Incurrentmicroscopypractiseadetectorcalibrationtophotoncountsisoftennotdonebutcannotbeperformedinretrospect.Itthuswouldbeextremelyhelpful,ifthatcanbedeterminedfromanalysingtheacquisitionitself–asingleimage.AnumberofalgorithmshavebeenpublishedforGaussiantypenoise[Donoho1995,Immerkaer1996]andPoissoniantypenoise[Foi2008,Colom2014,Azzari2014,Pyatykh2014].However,alltheseroutinesuseassumedimagepropertiestoextracttheinformationratherthanjustthepropertiesoftheacquisitionprocessasinourpresentedalgorithm.Thishasmajorimplicationsfortheirperformanceonmicroscopyimagesgovernedbyphotonstatistics(seeSupplementaryInformationforacomparisonwithimplementationsfromPyatykhetal.[Pyatykh2014]andAzzarietal.[Azzari2014]whichperformedmorethananorderofmagnitudeworsethanourmethod).
Somedevices,suchasavalanchephotodiodes,photomultipliertubes(PMTs)oremCCDcamerascanbeoperatedinasinglephotoncountingmode[Chao2013]wherethegainisknowntobeone.Inmanycases,however,thegainisunknownand/oradevicesetting.Forexample,thegainofPMTscanbecontinuouslycontrolledbychangingthevoltagebetweenthedynodesandthegainofcamerasmaydeviatefromthevaluestatedinthemanual.Tocomplicatematters,devicesnotrunninginphotoncountingmode,useanoffsetOzerotoavoidnegativereadoutvalues,i.e.thedevicewillyieldanon-zeromeanvalueevenifnolightreachesthedetector,S=gN+Ozero.ThisoffsetvalueOzeroissometimeschangingovertime(“offsetdrift”).Traditionally,aseriesofabout20darkimagesand20imagesofasamplewithsmoothlychangingintensityarerecorded[vanVliet1998].Fromtheseimagesthegainiscalculatedasthelinearslopeofthevarianceovertheseimagesversusthemeanintensityg=var(S)/mean(S)(fordetailsseeSupplementaryinformation).InFigure1weshowatypicalcalibrationcurvebyfitting(blueline)theexperimentallyobtaineddata(bluecrosses).Theobtainedgaindoesnotnecessarilycorrespondtotherealgainperdetectedphoton,sinceitincludesmultiplicativenoisesourcessuchasmultiplicativeamplificationnoise,gainfluctuationsortheexcessnoiseofemCCDsandPMTs.Inadditionthereisalsoreadoutnoise,whichincludesthermalnoisebuild-upandclockinducedcharge.Theunknownreadoutnoiseandoffsetmayseematfirstglancedisadvantageousregardinganautomaticquantification.However,asshownbelow,thesedetailsdonotmatterforthepurposeofpredictingthecorrectnoisefromameasuredsignal.
LetusfirstassumethatweknowtheoffsetOzeroandanypotentialreadoutnoisevarianceVread.TheregioninFourierspaceabovethecut-offfrequencyofthesupportoftheopticaltransferfunctiononlycontainsnoiseinanimage[Liu2017],wherebothPoissonandGaussiannoiseareevenlydistributedoverallfrequencies[Chanran1990,Liu2017].BymeasuringthespectralpowerdensityofthenoiseVHFinthishigh-frequencyout-of-bandregionandaccountingfortheareafractionfofthisregioninFourierspace,wecanestimatethetotalvarianceVall=VHF/fofalldetectedphotons.Thenthegaingisthenobtainedas
(1) 𝑔 = !!""!!!"#$(!!!!"#$)
wherewerelatethephoton-noise-onlyvarianceVall-Vreadtothesumoffset-correctedsignaloverallpixelsintheimage(seeOnlineMethods).Thedevicemanufacturersusuallyprovidethereadoutnoiseleavingonlytheoffsetandgaintobedeterminedfromtheimageitselfinpractise.Toalsoestimateboth,theoffsettogetherwiththegain,weneedmoreinformationfromthelinearmean-variancedependencethangivenbyequation(1).Weachievethisbytilingtheinputimage,e.g.into3×3sub-images,andprocesseachofthesesub-imagestogenerateonedatapointinamean-varianceplot.Fromtheseninedatapointsweobtaintheaxisoffset(Ono-noise).Wethenperformthegainestimation(1)onthewholeimageafteroffsetcorrection(SeeOnlineMethodsandSupplementaryInformation).AsseenfromFigure1thelinearregressionofthemean-variancecurvedeterminestheaxisoffsetADUvalueOno-noiseatwhichzeronoisewouldbeexpected.YetwecannotsimultaneouslydeterminebothoffsetOzeroandreadoutnoiseVread.Ifeitherofthemisknownapriori,theothercanbecalculated:Vread=g(Ozero-Ono-noise),whichis,however,notneededtopredictthecorrectnoiselevelforeachbrightnesslevelbasedontheautomaticallydeterminedvalueOno-noise.
Totestthesingle-imagegaincalibration,simulateddatawasgeneratedforarangeofgains(0.1,1,10)withaconstantoffset(100ADU),arangeofreadoutnoise(1,2,10photonRMS)andmaximumexpectedphotoncountsperpixel(10,100,…,105).Offsetandgainwerebothdeterminedfrom
band-limitedsingleimagesoftwodifferentobjects(resolutiontargetandEinstein)withoutsignificantrelativeerrorsintheoffsetorgain(lessthan2%atmorethan10expectedmaximumphotoncounts)usingtheproposedmethod(seeSupplementaryFiguresS1-S3).Figure1quantitativelycomparestheintensitydependentvariancepredictedbyapplyingourmethodindividuallytomanysingleexperimentalin-focusimages(shadedgreenarea)withtheclassicalmethodevaluatingawholeseriesofcalibrationimages(blueline).Notethatoursingle-imagebasednoisedeterminationdoesnotrequireanypriorknowledgeaboutoffsetorreadoutnoise.Figure2showsafewtypicalexampleimagesacquiredwithvariousdetectorstogetherwiththegainandoffsetdeterminedfromeachofthemandthecalibrationvaluesobtainedfromthestandardprocedureforcomparison.Weevaluatedthegeneralapplicabilityofourmethodondatasetsfromdifferentdetectorsandmodesofacquisition(CCD,emCCD,sCMOS,PMT,GAsPandHybridDetector).Figure3quantitativelycomparesexperimentalsingleimagecalibrationwithclassicalcalibration.20individualimageswereeachsubmittedtoouralgorithmandthedeterminedoffsetandgainwascomparedtotheclassicalmethod.Thevarianceofaseparatelyacquireddarkimagewassubmittedtothealgorithmasareadoutnoiseestimate,butalternativelythereadoutnoisespecificationfromthehandbookcouldbeusedorameasuredoffsetatzerointensity.AsseenfromFigure3thesingle-image-basedgaincalibrationasproposedperformsnearlyaswellasthestandardgaincalibrationusing20images.Therelativegainerrorstaysgenerallywellbelow10%andforcamerasbelow2%.The8.5%biasfortheHyDphotoncountingsystemisunusuallyhigh,andwewereunabletofindaclearexplanationforthisdeviationfromtheclassicalcalibration.UsingonlylowerfrequenciestoestimateVHF(kt=0.4)resultedinamuchsmallererrorof2.5%inthesingle-photoncountingcasesuggestingthatdead-timeeffectsofthedetectormighthaveaffectedthehighspatialfrequencies.
Simulationsaswellasexperimentsshowagoodagreementofthedeterminedgainwiththegroundtruthorgoldstandardcalibrationrespectively.Thebiasofthegaindeterminedbythesingle-imageroutinestayedbelow4%(exceptforHyD).Forintensityquantificationanypotentialoffsetmustbesubtractedbeforeconversiontophotoncounts.Ourmethodestimatesthephotoncountverypreciselyoveralargerangeofparameters(relativeerrorbelow2%insimulations).Ourmethodcouldbeappliedtomanydifferentmicroscopymodes(widefieldtransmission,fluorescence,andconfocal)anddetectortypes(CCD,emCCD,sCMOS,PMT,GAsPandHyDphotoncounting),becauseweonlyrequiretheexistenceofanout-of-bandregion,whichpurelycontainsfrequencyindependentnoise.Thisisusuallytrue,iftheimageissampledcorrectly.AsdiscussedintheSupplementaryInformationthecut-offlimitofouralgorithmcaninpractisebesetbelowthetransferlimitandsingle-imagecalibrationcanevenoutperformthestandardcalibrationifmolecularblinkingperturbsthemeasurement
Insummaryweshowedthatsingleimagecalibrationisasimpleandversatiletool.Weexpectourworktoleadtoabetterabilitytoquantifyintensitiesingeneral.Thiswillhavesignificantimpactonallmethodsthatprofitfromknowledgeaboutthenoisedistribution.Examplesrangefromquantificationoflightintensitiesoverparameterselectionforpatch-baseddenoisinganddeconvolutionprocedurestonoiseprediction.
AuthorContribution
RHconceivedtheinitialidea.BRandRHextendeditandwrotetheinitialsoftware.PRperformedexperiments.BRandRHperformedsimulationsandwrotethepaper.PRandRPJNcontributedtosolvetheoffsetestimation.KALgaveresearchadvice.Allauthorscommentedonthepaper.
CompetingfinancialinterestsTheauthorsdeclaretohavenocompetinginterest.
AcknowledgementsWethankSteffenDietzel,CoreFacilityBioimagingLMUMünchen,forprovidingcalibrationdatasets.ColinParkandSimon Denham,CarlZeissU.K.,arethankedfortheirhelpinmeasuringdataontheZeissLSM800.UteNeugebauerandChristinaGroßearethankedforhelpintakingdataonaZeiss800systemusingordinaryPMTs.ManythankstoRüdigerBader(Hamamatsu)forhissupportinthesCMOSmeasurements.GabrielaFiczisthankedfordiscussionininitialstagesofthemanuscript.DiederikFeilzeristhankedforhelpingcodingtheImageJplugin.P.K.R.andK.A.L.weresupportedbyNationalScienceFoundationgrantno.0954836.R.H.acknowledgesDeutscheForschungsgemeinschaftgrantsTRR166TPA4,B5andHE3492/7-1.B.R.acknowledgesEuropeanResearchCouncilgrantno.648580andNationalInstituteofHealthgrantno.1U01EB021238-01.
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Figure1.Gainestimationbyplottingvarianceovermean.Typicaldetectorcalibrationcurveobtainedfromaseriesofstaticimages.Aseriesof20imagesofadefocussededge(leftinset)andabackgroundseriesof20imagesareprocessedintoavarianceandmeanprojectionimage.Fromtheseamean-varianceintensityplotisobtained(only21datapointsaredisplayedasbluecrossesforbettervisualization).Theexperimentallyfoundmeanbackground(Ozero)andthereadoutnoiseVread,thevarianceatzerointensity,areindicated.Thegaingisobtainedastheslopeofthelinearregression(blueline).Theshadedgreenareacorrespondstotheminimumandmaximumvaluesforsingleimagegainandoffsetresultsobtainedfrom20singlein-focusimageswithourmethod(bottomrightinset,scalebar2µm).
Figure2:Gainandoffsetestimationforvariousmodesofacquisition.Estimatedgainandoffsetasdeterminedforeachimageandthecomparisontostandardcalibratedvalues.Panelsa)andb)showmitochondriaMuntJacAF555cells.Panelc)displayslabelledactinfilaments(Alexa568)andpaneld)aDAPI-labelledcellnucleusofembryonicmousefibroblasts.(allscalebars2µm)
a)CCDDetector(lowgain)
EstimatedGain:9.78Calib.:9.81EstimatedOffset:103Calib.:100.0
b)emCCDDetector,highgain(100)
EstimatedGain:0.096Calib.:0.097EstimatedOffset:113.7Calib.:106.4
EstimatedGain:0.0046Calib.:0.0045EstimatedOffset:34.08Calib.:59.4
c)ConfocalGAsPPMT,Zeiss800 d)ConfocalHyD,LeicaSP8,countingmode
EstimatedGain:1.07Calib.:0.98EstimatedOffset:-0.072Calib.:0.00087
Figure3:Performanceofsingleimagegaindeterminationonvariousdetectors.Relativeerrorofthegainestimationfromsingleimagegaindeterminationscomparedwithastandard“groundtruth”calibrationobtainedfrom20smoothand20darkimageseach.Thegainvaluesvariedfrom0.1(lowgainCCD),1.0(confocalHyDcounting)toalmost10(highgainemCCD).Errorbarsindicateonestandarddeviation.SeeSupplementaryFigure5forexampleimagesforallimagingmodes.
OnlineMethods:Calibratingphotoncountsfromasingleimage
RainerHeintzmann1,2*,PeterK.Relich3,RobertP.J.Nieuwenhuizen4,KeithA.Lidke3,BerndRieger4*
1LeibnizInstituteofPhotonicTechnology,Jena,Germany2InstituteofPhysicalChemistryandAbbeCenterofPhotonics,FriedrichSchillerUniversity,Jena,Germany3DepartmentofPhysicsandAstronomy,UniversityofNewMexico,Albuquerque,NM,USA4QuantitativeImagingGroup,DepartmentofImagingPhysics,DelftUniversityofTechnology,Delft,TheNetherlands
Experimentalsetupsandmethods:Figure1:Calibrationimageswereproducedbyilluminatinganopaqueedgemountedonamicroscopeslide(VWR#48312-003)usingahalogenlamp(OlympusU-LH100L-3).Thesamplewasdefocusedtoproduceasmoothgradientand100imagesofthesamepatternwereacquired.Exposuretimesforeachcompletedatasetrangedbetween15.7msto51.2ms.Foreachdataset,aseriesof100backgroundimageswerecollectedwiththemicroscopecamerashutterclosedatthesameexposuretime.Figure2:Forstandardgaincalibration20imageswereusedinallpanels.AcommerciallyavailabletestslideconsistingoffluorescentlylabeledMuntJacskinfibroblastcells(InvitrogenF36925)wasimagedusinganEMCCDcamera(AndoriXon897)withconventional(panelA)orwithvariousEM(panelB)gainsettings:20,50,100,and200andpre-amplifiergainsettings:1×,2.6×,5.2×.ThemicroscopesetupwasanOlympusIX71withanOlympus150×1.45NAobjective,561nmlaser(CoherentSapphireLP)excitation,575nmdichroicanda600/30nmbandpassemissionfilter.Theresultingbackprojectedpixelsizewas106.7nm.TheexposuretimeforallEMCCDdatasets(panelB)was20ms.WhenthecamerawasoperatedwiththeEMgaindisabledtoactasaconventionalCCDdetector(panelA)theexposuretimewas50ms.PanelCshowsanAlexa-phalloidinstainedcell,imagedusingaZeissLSM800confocalmicroscopeusingaPlanApochromat63×1.4OilobjectiveandGAsPdetector.Thepixelpitchwas82.5nm.ImagedintheAF568channel.PanelDdisplaysaDAPIstainedcellnucleusimagedusinga100x1.4OilHCPlanApoCS2objectiveandthephotoncountingmodeoftheHyDdetectorofaLeicaSP8microscope.Thepixelpitchwas79.6nm.Figure3:Thiscomparisonofvariousdetectorsandimagingmodesusedthefollowingsettings:AllCCDdatawastakenusingvarioussettingsofanAndoriXon897(Andor,Belfast,U.K.)emCCDcamera.For“CCDlowgain”,“CCDmedgain”and“CCDhighgain”,thenon-EMreadoutregisterwasusedandthepre-amplifiergainwassetto1,2and3respectively.Thecalibratedgainresultswere9.8,3.9and1.8respectively.“emCCDlowgain”and“emCCDhighgain”usedtheelectronmultiplyingregisterwithEMgainssetto20and100atpre-amplifiergainof2and3respectively.Calibratedgainresultswere0.88and0.097respectively.“sCMOS”correspondstoaHamamatsuFlash4.0V2(calibratedgain:0.44).“confocalGAsP”usedaZeissconfocalLSM800AiryScanwithaGAsPdetectorin16bitmode(calibratedgain:0.0045).“confocalHyDcounting”referstoaLeicaSP8confocalmicroscopeusingtheHyDdetectorinphotoncountingmode(calibratedgain:0.98).Maximalcountswereabout90ADU.“confocalPMT”usedaZeiss800microscopewitha
standardPMTdetector(calibratedgain:1.1).Avalueofkt=0.8wasusedinthesingle-imagebasedgainestimationalgorithmandthek-axescrosswasavoidedasdescribed.
Derivationofthegainestimationfromout-of-bandinformationofasingleimageonly:Wemodeltheimageformationas(2) 𝐼 𝑥 = 𝑔 𝐵 𝑥 + 𝑛! 𝑥 + 𝑛! 𝑥 + 𝑂!"#$Withxdenotingthespatialdetectorcoordinates,Btheexpectedphotoncountatthedetector,thephotonnoisenp,thereadoutnoisenrandtheoffsetoftheADconverter𝑂!"#$.Becausenoiseisassumedtohaveameanofzeroandthenoisesourcescanbeassumedtobestatisticallyindependent,theexpectedvalueofI(x)issimplygB+𝑂!"#$andthevarianceofIovertimeisequalto(3) 𝑣𝑎𝑟 𝐼 = 𝑔!𝑣𝑎𝑟 𝑛! + 𝑣𝑎𝑟 𝑛! = 𝑔!𝐵+𝜎!!whereσristhestandarddeviationofthereadoutnoise.OurkeyideaistoextrapolatetheexpectedvarianceofIfromtheaveragepoweratspatialfrequencieskbeyondathresholdfrequencyktabovethecut-offfrequencykcut(out-of-bandenergy)ofasingleimagebecauseinthisregionallspectralpowerisentirelyduetonoise.Thereforetheaveragespectralpowerofthenoisevar(I)atthosefrequenciesisequaltotheaveragespectralpowerofthenoiseatallspatialfrequencies:(4) !
!𝑣𝑎𝑟 𝐼! = !
!𝑔!𝑛!!! + 𝑛!! =
!!
𝑔!! 𝑛!! + 𝑛! ! = !
!𝐼 !! !!!
whereMisthetotalnumberofpixelsandTisthenumberofpixelsintheFouriertransformofIbeingabovethethresholdfrequencykt,thehatdenotestheFouriertransformationandthesummationsoverxandkdenotethesumoverpixelsinrespectivelythespatialandFourierdomain.Inthesecondstepofequation(4)weusedParseval’srelation(conservationofsignalenergy)toconvertthesumoverallpixelsinthespatialdomainintoasumintheFourierdomain.Bycombiningalltheabove,therelationshipbetweenthevarianceandaveragemeasuredintensitycanbeexpressedas(5) !
!𝐼 !! !!! = 𝑔 !
!𝐼+ (𝜎!! − 𝑔𝑂!"#$
!!
)𝑥
oralternatively
(6) !!
𝐼 !! !!! = 𝑔 !!
𝐼− (𝑂!"#$ −!!!
!!!"!!"#$%
)𝑥
Equations(5-7)areequaltoequation(1)asgiveninthemaintext.Clearlythegaingcanbefoundusingequation(6)iftheimageoffset𝑂!"#$andthereadoutnoiseσrareknown:
(7) 𝑔 =! ! !! !!!
! (𝐼𝑥 −𝑂𝑛𝑜−𝑛𝑜𝑖𝑠𝑒)
withthedefinitionof𝑂!"!!"#$% asstatedinequation(6), beingthetheoreticalintensityvalueatwhichzerovariancewouldbeexpected.
However,theoffsetismostoftennotknownwhereasthedetectormanufacturerspecifiesthereadoutnoise(inmostcases).OuralgorithmfirstestimatesOno-noiseexploitingthelinearityofthemeantovariancerelationshipasdetailedbelowtothencomputethegainalsoincasesofunknownimageoffset.
GainestimationwithanunknownimageoffsetWesubdividetheimageintotiles(e.g.3x3sub-image,thetiles)andplotforalltilestheout-of-bandestimatedmeannoiseenergy!
!𝐼 !! !!! versusthemeasuredmeanintensity
!!
𝐼𝑥 .Thisgives9datapointsonthevarianceversusmeanintensityplotandweestimate𝑂!"!!"#$%byalinearregressiononthedata.Fromthiswethenobtainthegainbycomputingthemeannoisevarianceandmeanintensityfromtheoverallimageasstatedinequation(6).Withthisalgorithmicallydetermined𝑂!"!!"#$%,eitherthedetectoroffset𝑂!"#$orthereadoutnoise𝜎!!canbeobtainediftheotherrespectivevalueisknownapriory.SeeSupplementaryInformationforafurtherdiscussionontheimplementationdetailsandfitting.
SupplementaryInformation:Calibratingphotoncountsfromasingleimage
RainerHeintzmann1,2,PeterK.Relich3,RobertP.J.Nieuwenhuizen4,KeithA.Lidke3,BerndRieger4
1LeibnizInstituteofPhotonicTechnology,Jena,Germany2InstituteofPhysicalChemistryandAbbeCenterofPhotonics,FriedrichSchillerUniversity,Jena,Germany3DepartmentofPhysicsandAstronomy,UniversityofNewMexico,Albuquerque,NM,USA4QuantitativeImagingGroup,DepartmentofImagingPhysics,DelftUniversityofTechnology,Delft,TheNetherlands
SimulationResultsToobtaininformationonthequalityofthesingle-image-basedgainandoffsetcalibrationprocedureaseriesofsimulationhasbeenperformed.Weusedtwodifferenttypesofimagesboth512×512pixels1)aresolutiontargetand2)apictureofAlbertEinstein(compareFigureS1d)andS2d)),atanoffsetof100ADUs,wherethegroundtruthofthegainisknown.
TheseimageswereconvolvedwithaPSFwithitscutofflimitkcutat2/3oftheNyquistlimit.Avarietyofmaximallyexpectedphotons(10,…,105)andanumberofreadoutnoisecontributions(1,2and10photons)weresimulated.Foreachoftheseconditions,100simulationshavebeenperformed.Foreachindividualsimulatedimagethegainandoffsetwheredeterminedbyouralgorithmusingkt=kcutinthealgorithm,andsupplyingthenominalreadoutnoisetoit.Allsimulatedandexperimentaldataisbasedonthe3×3tileoffsetestimationmethodaspresentedbelow.Theresultswerethencomparedtothegroundtruth(nominalgainandphotoncountobtainedviadetectorcalibrationexperiments)andsummarizedinFigureS1,S2andS3.Herethemidpointofa“relativegainerrorinpercent”entryis
computedby !!!!!"#$!%&!""!!!
!!"#$!%&×100%,where𝑔! istheestimatedgainbyour
algorithminsimulationnumberi.Theerrorbarsrefertoplusandminusthecorrespondingrelativevalueofthestandarddeviation.Thereare4differentrelevantspatialfrequenciesforthesimulations:
• kcut Theopticalcut-offfrequencyoftheOTF.• kt Thethresholdfrequencyforcomputationofspectralpowerdensityof
thenoiseVHF.• kNyquistTherequiredNyquistsamplingfrequencygivenkcut,i.e.kNyquist=2kcut.• ksamp Theactualsamplingfrequency.
Ideallywehaveksamp=kNyquist>=2kcut=2kt.Butinrealitytheserelationsmightdifferslightly.Inpracticetheactualsamplingmightoftendifferfromthestrictlynecessaryone.Ouralgorithmcanonlyvaryktastherestisdetermineduponacquisition.However,asdifferentrelativevaluesofthesefrequenciesinfluenceforemostthecorrectestimationofthespectralpowerdensityofthenoiseVHF,we
investigatetheperformanceofthealgorithmwithrespecttoseveralcases.Wecreatedaslightlyoversampledtestimagewithkcut=2/3*½kNyquistandwespecifykcutasthefractiontothematchingNyquistsampling,i.e.hereaskcut=2/3.Wespecifyktalsorelativeto½kNquist,i.e.kt=2/3=kcutwillonlyconsiderpurenoisefortheestimationofVHF.LowervaluesofktwillconsidersignalintocomputingVHFwhichisineffectsimilartospatiallyundersamplingtheimage.Observationsfromsimulationsonthegainestimation:
• Thehigherthereadoutnoisetheworsethegainestimate.• Higherphotoncountsgivebetterresultsincaseofhighreadoutnoise.• Thehigherthegain,thehigherthevarianceintheestimationofthegain.
However,therelativegainerrorwasessentiallyindependentofthegain.Asmallpositiveconstantbiasofabout1%wasobserved,whichstayedbelow(resolutiontarget)orwascomparableto(Einstein)thestandarddeviationoftheestimation.
• Readoutnoiseof10at10maximallyexpectedphotonsgaverisetoverylargevariances.However,themeanestimateisstillwithin10-20%error(compareSFig1c)and2c)).
• kt=kcutyieldsthebestresults.Forkcut=2/3thereisplentyofout-of-bandnoisepresentinthepixelatedimageduetotheoversampling.Evenforsmallundersampling10-20%e.g.0.9kt=kcutwestillhaveacceptableresults.
• Gainestimationerrorswithchoosingkcutbelowthecut-offlimit,i.e.kcut=2/3andkt=0.5,remainsacceptablewitherrorsbelow10%exceptfor105expectedphotonsormore,asmoreandmoresignaliswronglyassumedtobenoiseinthecomputationforVHF.Ifwedecreasektfurtherbelowkcutlowerphotoncounts(<103)alreadyshowanincreasederrorasexpected(datanotshown).
Insteadofinvestigatingdirectlytherelativeoffsetestimationerrorasthisvaluecannotbereliablyobtainedindependentlyfromthemeasuredsignal,weinsteadevaluatetheperformanceofthealgorithmtoestimatetheeffectivephotoncount.Fordataprocessingpurposesthephysicaloffsetisoftenirrelevantastheoffset-correcteddataconvertedtoeffectivephotonsisusedforanyfurtheranalysissuchasmaximumlikelihooddeconvolution.Theterm“effectivephotons”referstoasignalwiththenoisevariancebeingequaltothemeanvalue.ThereforeweinvestigatetheperformanceforeffectivephotoncountN
𝑁 =𝑆 − 𝑂!"!!"#$%
𝑔
whereSistheimagevalueinADU,Ono-noiseistheADUoffset(ADUvalueatzeroeffectivephotons)andgthegain.Bypropagationoferrorwefindthevarianceinthephotoncountestimation
∆𝑁 ! =𝜕𝑁𝜕𝑔
!
(∆𝑔)! +𝜕𝑁𝜕𝑂
!
(∆𝑂!"!!"#$%)! =
=𝑁𝑔
!
(∆𝑔)! +1𝑔! (∆𝑂!"!!!"#$%)
!
HereΔgandΔOno-noisearethestandarddeviationoftheestimateofthegainandoffsetrespectively,whicharedeterminedfromthesimulationsover100runs.Asthesimulatedphotoncountspans5ordersofmagnitudeweplottherelativeerrorΔN/N⋅100%inFigureS3.Thevariationinthegainestimatedominatestheerrorinthephotoncountinmostcases.Asseenfromtheseresults(errorbarsinFigureS1andS2),themeanrelativeerrorinthegainestimationisindependentoftheactualgain,confirmingthatthealgorithmisinvarianttomultiplicativefactors.Onlyinthepresenceofsubstantialreadoutnoiseaninfluenceofamultiplicativefactorisseen.Itisfurtherobservedthatthebiasstaysbelow1%forallcasesbutwhensimulatingwithamaximumnumberof10expectedphotons. Theonlycase,whichleadstoanunacceptablehighvarianceintheestimation,isfor10photonsreadoutnoiseatamaximalsignallevelof10photons.AnexampleimageforthiscaseisshowninFigureS4.Interestinglyeveninthiscasewecanobtainanacceptableresult(standarddeviations~10-20%forthelowergainvalues),ifthethresholdfrequencyktsuppliedtothealgorithmisreducedfrom2/3to0.2.Thereasonisthatatsuchnoiselevelsthesignaliscompletelylostinthenoisealreadyatfrequenciesfarbelowtheactualcut-offlimit.Areducedthresholdlimitktthenhelpstoestimatethenoisemorerobustly.However,atgoodSNR(highphotoncounts),suchaloweredktleadstounwantedbiasesassignalenergyisfalselyaccountedforasnoiseenergy.
FigureS1:Performanceofthesingleimagegainestimationonasimulatedresolutiontargetasshownind).Panelsa,b,c)showtherelativegainestimationerrorforrespectivelydifferentlevelsofreadoutnoisewithstandarddeviationof1,2and10photonsasafunctionofmaximalphotoncount(10,…105)forthreegainlevels.Markersindicatemeanerrorover100runsanderrorbarsonestandarddeviationofthe100runs.Inpanelc)thestandarddeviationoftheresultsfor10expectedphotonsinthemaximumwhichrangedupto600%isnotshown.For100photonsonlyforgain10itisoutsidethedisplayrange.Paneld)showsthesimulatedresolutiontarget(512×512pixels).Parameterskt=kcut=2/3.
d)
FigureS2PerformanceofthesingleimagegainestimationonEinstein(512×512pixels)asshownind).Panelsa,b,c)showtherelativegainestimationerrorfordifferentlevelsofreadoutnoisewithstandarddeviationofrespectively1,2and10photonsasafunctionofmaximalphotoncount(10,…105)forthreegainlevels.Markersindicatemeanerrorover100runsfordifferentsimulatedgainvalues.Parameterskt=kcut=2/3.
d)
FigureS3:Estimatedrelativeerrorinthephotoncountfromthesingle-imageestimationusingtheEinsteinobjectassimulationinput.Panelsa,b,c)showtherelativephotoncountestimationerrorfordifferentlevelsofreadoutnoisewithrespectivestandarddeviationsof1,2and10photonsasafunctionofmaximalphotoncount(10,…105)for7gainlevels.Inpanelc)weomittedtheresultsfor10expectedphotonswhereestimationbreaksdownanderrorsabove100%werefound.
FigureS4:WorstSNRinsimulationsforreadoutnoise10photonsand10photonsexpectedmaximumphotoncount.
a)CCDlowgain b)CCDmediumgain
c)CCDhighgain d)emCCDlowgain
e)emCCDhighgain
f)sCMOS
FigureS5:a)-i)Exampleimagesusedforthesingleimagephotonquantificationonexperimentaldataofvariousinstrumentsanddetectortypes.RespectiveresultsoftheestimationareshowninFigure3.
g)confocalGAsP h)confocalHyDcounting
i)confocalPMT
TraditionalGainestimationfromaseriesofexposuresHerewedescribethestandardgaincalibrationprocedureaccordingto[vanVliet1998,Lidke2005]. Twoseriesofimages,eachtypicallyabout20images,areacquiredadarkserieswithnolightexposureandaserieswithasmoothintensitydistribution(gradientsequenceasexemplifiedinthemaintext,Figure1,leftinset).Theintensitydistributionshouldcapturethefulllineardynamicrangeofthedetectorwithoutsaturation.Fortheupperlimitweadviseabouthalfthemaximumofthelinearworkingrangeofthedevice,thefullwellcapacityofthecameraorthemaximumoftheADUconverterrange.Suchadistributioncanbeobtainedbyfindingabrightedgeorisolatedstructureanddefocussingit.Itisimportanttoensurethatbothseriesareacquiredunderidenticalsettings,temperaturesandexposuretimes. Theseseriesarethenprocessedbyfirstdeterminingtheoffset(Ozero)asthemeanvalueofalldarkframes(seeFig.1).Suchanestimateisusuallyprecisesinceimagescontainmanypixelsandthisisaveragedovermanyframes.Ofcoursesuchabackgroundseriescanalsoproveusefultoestimateoffset-drifteffectsbyplottingthemeanofeachframeasatimeseries,orresidualnoisesourcesbyaveragingthemagnitudeofallindividual2DFouriertransforms,whereevenweakperiodicfluctuationsoftenshowupasclearlyvisiblepeaks.Thenfromthesequenceofsmoothimages,themeanandthevariancesarecalculated.Tothisendtheimageisbinnedintoregionsofintensitybinsdeterminedfromtheaverageimage.Weuseastandardof100intensitybinstoobtainarobustestimateandaccountfordrifteffectsthatcouldinfluencepurepixelbasedestimation.Thegaingisobtainedfromtheslopeofthefitof𝑣𝑎𝑟 𝑆 = g 𝑆 − 𝑂!"!!"#$% totheplotofthemeanvarianceovermeanintensity(seeFig.1).ThesignaloutputSinADUcomesfromthedetectorinresponsetoaninputofNeffectivephotonswhichareobtainedas𝑁 = 𝑆 − 𝑂!"!!"#$% /g.NotethatforPoissonstatisticsitholdsthatvar(N)=N.Thefitisareweightedleastsquaremethodwiththesquareofthevarianceandthenumberofaveragedvariancevaluesineachbinpredictingthevarianceofthevariance.Fromtheslopeofsuchaplot,thegaincanbeestimatedasgiveninFigure1.Withthisknowledgethemeanofthevarianceprojectionofthedarkseriescanbeconvertedtothereadoutnoiseofthedeviceinelectrons.
DescriptionofthesingleimagegainestimationalgorithmThemeanspectralenergyofthenoise(”out-of-bandenergy”)iscomputedasdepictedinFigureS6by!
!𝐼 !! !!! ,whereT,isthenumberofFourier-spacepixels
beyondthealgorithmiccut-offfrequency 𝑘!comparedtotheareaofthefullimage.Thethresholdfrequencyktistypicallyequalorlargerthantheopticalcut-offfrequencykcut.ForadiscussiononpracticalvaluesofktseediscussionontheSimulationResults.TheFouriertransformationiscomputedusingthediscreteFouriertransformation(DFT)asoutlinedintheimplementationdetailsbelow.
FigureS6:Spectralenergyintheout-of-bandregionintheFourierdomainoftheimage.Tisthenumberofpixelsbeyondthealgorithmicthresholdfrequency 𝑘!.
GainestimationandimageoffsetTobeabletorelatemeasuredvaluestonoiseprediction,weneedtodeterminethevarianceoffset(Vo)atzeroreadoutvalue.AsdetailedintheOnlineMethodswetiletheinputimageintoanumberofsub-imagesortiles.Itisnotoftheessencehowmanyorhowlargethetilesarefortheestimate.Wechose3×3tiles,whichworkedwellinsimulationsandwasalsousedforallcomputationsofexperimentaldata.TheworkflowisdepictedinFigureS7.Increasingthenumberoftilesgivesmoredatapointsonthevarianceversussumintensityplot,buteachdatapointhasahigheruncertaintyasitiscomputedfromalowernumberofpixels.Inaddition,thedifferenttilesneedtocoverarangeofmeanintensitiestogiveagoodscatterontheplotandreducingthetilesizetoomuch,doesnotfosterthisrequirement.OncethevarianceoffsetVoisfoundbyreweightedlinearregression(asdescribedbelow)thegainisestimatedasfollows.
FigureS7:Workflowtoenablegainestimationfromasingleimageinthecaseofunknownimageoffset.Theinputimageisdividedinto3×3sub-images.Foreachsub-imagetheFouriertransformationiscomputedandthemeanout-of-bandenergyofthenoiseismeasured.Thesevaluesareplottedagainstthemeanintensitiesinthetiles(includingtheunknownoffset).FromareweightedlinearregressionwefindthevarianceoffsetVo,whichisfinallyusedtocomputethegain.Weplotthissumagainstthevarianceduetonoiseineachtilecorrectedforthefractionofthearea.ThevarianceoffsetatzeroreadoutvalueVoisthenusedtofindtheoverallgain.Tothisendwecomputeagain 𝐼 !! !!! and 𝐼! butnowonthe
wholeinputimageandfinallyobtainthegainas
𝑔 =𝑀 𝐼 2𝑘 >𝑘𝑡 −𝑉𝑜
𝑇 !!.
Torelatemeasuredvaluesnotonlytoanoisepredictionbuttoactualphoton
counts,weneedtosupplyeitherthereadoutnoise(ine-RMS)𝜎!ortheimageoffset(𝑂!"#$inADU)toouralgorithm.Weobtainthevarianceofthereadoutnoiseas𝑉!"#$ = 𝜎!!.Ifthereadoutnoiseissupplied,theimageoffset𝑂!"#$isobtainedafterthegaingisfound,andVofromtheregressionas𝑂!"#$ =
!!"#$!
− !!!.
ToconverttheimagefromADUtoeffectivephotoncountstheoffset𝑂!"#$isfirstsubtractedbeforewedividebythegain.Notethatforhigherphotoncounts,thereadoutnoiseestimatedoesnotneedtobeprecise,stillyieldinggoodestimatesofthegainandoffset.Forimageswithverylowphotoncountswithunknownreadoutnoise,itmaybeadvantageoustoratherestimatetheoffsetfromadarkareaofthatimage.
Weightedlinearregressiononvariancevs.sumintensityTheobtainedmean-variancedataisfittedbyaPoissonianreweightedleastsquareapproachtofindthevarianceoffsetVoasgivinginFigureS7.Theerrorsintheyvaluesaregivenbythesquaresofthevariances,dividedbythenumberofpixelsinthetile.ThevariancesitselfarejustgivenbythemeanyvaluesforPoissonstatistics.Thefitweightsarethentheinverseofthiserrorandtheslopeandoffset(asvector𝛽)arefoundfrom𝛽 = (𝑋!𝑊!!𝑋)!!𝑋!𝑊!!𝑦.WhereWisthecovariancematrixoftheweights(herethemeasurementsareindependentandthematrixisdiagonal)andXisthematrixofxvaluesgivenbysumoftileintensities.
ImplementationdetailsThediscreteFouriertransformationunavoidablyimpliesthattheinputsignaliscyclicwhichgivesrisetoartifacts(falsespatialfrequencies)intheFourierdomain.Toavoidthiswell-knownproblemtypicallyaTuckey,HanningwindoworsimilarisappliedtotheimagebeforeFouriertransformation.However,applyinganysuchwindowalsodisturbsthenoisepowerspectrumandaffectsthesingleimagegainestimation.Thereforewehavechosennottoapplyanywindowbuttousemirroringoftheinputimageandtilingitintoa2×2arraybeforeFouriertransformationasdisplayedinFigureS8.Thisoperationensuresmatchingboundariesandavoidszero-orderdiscontinuitiesofthediscreteFouriertransformationbutleavesthenoisedistributionunperturbed.Somedetectors,suchassCMOScameras,canexhibitfixedpatternnoiseand/orpixeldependentgainextendingintothehigh-frequencyregion[Huang2013].Diffusionandblinkingduring(confocal)scannedimageacquisitionorafterpulsingcanalsoleadstosimilareffects,asforexampleexploitedinRICS[Brown2008].Thesecanleadtopotentialproblemsofourmethod.ThereforeweadditionallychosetoremovethespectralcrossfromtheFourierdomainbeforeestimationthespectralenergyofthenoise.Thatis,wenullouta3pixelwidestrokeontheFourieraxesnexttonullingallspectralcontent|k|<ktasshowninFigureS6andaccountforthisinthefractionf.
FigureS8:SymmetrizinganinputimagebymirroringtoavoidboundarydiscontinuitieswhenapplyingthediscreteFouriertransformation.Note:IftheinputimagedoesnothaveanequalnumberofpixelsalongbothdimensionsandasquarepixelsizethediscreteFouriertransformwillresultinnon-isotropicpixelsinFourierdomainandthephysicalcut-offfrequencywillnotbeacircleintheFouriertransformedimage,butanellipse.Softwarecanbedownloadedfromourwebsitehttp://www.diplib.org/add-onsprovideMatlab(TheMathworks,MA)codeandanImageJ(NIH,VA)plugin.
Comparisonwithpriorartonsingle-imagebasedalgorithmsEstimationofthecombinedGaussianand/orPoissoniannoiseparametersfromsingleimageshasbeenstudiedinthesignalprocessingcommunity[Immerkaer1996,Foi2008,Pyatykh2014].Theestimationstrategiesintheliteraturerelyonfindingthemean-varianceintensitycurvesfromeitherhomogeneousimageparts/blocksorafterhigh-passfilteringandattemptingtoremovesignalcontributionfromtheimage.ForanextensivereviewseeColometal.[Colom2014]andPyatykhetal.[Pyatykh2014].Inotherwordsassumptionsaremadeonthetypicalpropertiesofobjectsintheimage.Weapproachedtheproblemfromtheacquisitionside.Instarkcontrasttothesemethodsweonlyassumethattheimageisacquiredbyaband-limitingprocess(suchasgivenbyanyopticalimagingsystem)andthatitwasatleastroughlysampledcorrectly.Thatmeansourapproachisindependentoftheimagecontentandstructure.ForestimationofthePoissoncomponentofthenoisethenormalapproximationofthePoissondistributionhasbeenappliedinthepriorart(ortheAnscombetransformwasused).Inourapplication,lightmicroscopyandinparticularfluorescenceimaging,thedetectionprocessislargelyshot-noiselimitedduetothelowsignalintensities,butalsotheverylowreadoutnoiseofcurrent(em)CCDandsCMOSdetectors.GainestimationontheexperimentaldatafromFiguresS5withtheavailableimplementationsbyPyatykhetal.[Pyatykh2014]didtypicallyresultinerrorslarger
than100%(andamuchlargerspread)whencomparedtothegoldstandardcalibrationandtheirimplementationgotstuckinendlessloopsforsomeoftheexperimentalandsimulateddatasets(FigureS9).Thealgorithmhas15freeparameterswhereweusedtheonessetinthesoftwareprovidedbytheauthors.Potentiallyfine-tuningtheparameterscouldsolvetheproblemofthealgorithmgettingstuckbutwedidnotpursuethiseffortasourfine-tuning15parametersseemedfruitlesscomparedtoourveryeasyroutine.FortheCCDmed.gainandemCCDhighgainmodethecodedidnotreturnaresultandweskipthesedatapoints.TheperformanceoftheimplementationofAzzari2014etal.[Azzari2014]wasabitbetterwithrelativeerrorsof10-600%andsmallerspread.However,comparedtoourresultswitharelativegainerrorofonaverage1.4%(0.8%median)(Figure3)theAzzari2014methodisstillabouttwoordersofmagnitudeworsethanourapproachonexperimentaldatawithameanrelativeerrorof181%(66%median),asseeninFigureS9.ForthebestestimationsoftheAzzarimethod,namelytheconfocalGAsPcaseandtheconfocalHyDcountingcase,therelativeerrorvaluesare9.6+-1.6%and9.6+-5.1%respectively,whicharealsostillworsethanourestimationsof5.3+-0.7%and8.4+-1.2%(compareFigure3).
FigureS9:PerformanceofalgorithmbyPytatykhetal.andAzzarietal.forgaindeterminationonvariousdetectors.Relativeerrorofthegainestimationfromsingleimagegaindeterminationscomparedwithastandard“groundtruth”calibrationobtainedfrom20smoothand20darkimageseach.Thegainvaluesvariedfrom0.1(lowgainCCD),1.0(confocalHyDcounting)toalmost10(highgainemCCD).Errorbarsindicateonestandarddeviation.SeeFigureS5forexampleimagesforallimagingmodes.Estimationonsimulateddatawasforbothinvestigatedalgorithmsmuchworsethanouralgorithm.OntheimageoftheresolutiontargetweshowinFigureS10therelativegainestimationerrorsover50noiserealizations.ItisevidentthatespeciallythealgorithmbyAzzarietal.resultsinverypoorestimateswith50-100%relative
errorswhereasthealgorithmofPyatythetal.isintherangeof0-50%.Pleasenotethattheresultsaregaindependent.Thestandarddeviationisomitted,asitislargerthantherelativeerrors.Pleasealsonotethatthephotoncountisrestrictedto104photonsasforlargercountsthecodegotstucklikefortheexperimentaldatainanendlessloop.
FigureS10:PerformanceofalgorithmbyPytatykhetal.andAzzarietal.forgaindeterminationonsimulationsforthe“resolutiontarget”image.Meanrelativeerrorofthegainestimationover50noiserealizationsfordifferentamountsofaddedreadoutnoise.SimulationsettingsareidenticaltoFiguresS1andS2.
FigureS11:PerformanceofalgorithmbyPytatykhetal.andAzzarietal.forgaindeterminationonsimulationsforthe“Einstein”image.Meanrelativeerrorofthegainestimationover50noiserealizationsfordifferentamountsofaddedreadoutnoise.SimulationsettingsareidenticaltoFiguresS1andS2.Ourcurrentalgorithmassumesthattheimagedoesnotcontainsaturatedpixels,i.e.novaluesthatareclippedtothemaximumbitdepthoftheA/Dconverter.In(fluorescent)microscopythisissueisirrelevantaslightlevelsareverylowandstudentsaretaughtinentry-levelcoursesalreadyasbestpractisetoneversaturateimagesbyincreasinganyvariablegain.Priorartsuggesttobeabletocopewithsuchundesirableacquisitionproblembutfailedinourhandsalreadyforproperlyacquireddata.
Wearealsolargelyindependentoftheimagecontent/structureaslongastherearedifferentimageintensitiespresentindifferentpartsoftheimageotherwisethetilingapproachmightfail.Thatisalsoincontrasttoearliermethodsthathavedifficultiestodealwithstrongtexturedcontent.
Priorartalsosuggestbeingabletocopewithundersamplingissues.OurexperimentalandsimulateddataareNyquistsampledorbetter,yettheaforementionedpublishedalgorithmsshowpoorperformance.CommercialsoftwareforintegratedsetupscommonlypointsuserstotherequiredNyquistsamplingrateofλ/(4NA).Thisallowstheusertomatchite.g.byadjustingthepixelsizeinaconfocalsystem.Forwidefieldacquisitionsanobjectivewithahighermagnificationoranadditionalopticalrelaysystemcanfixpotentialundersamplingforagivencamerapixelsize.Asindicatedaboveourmethodsalsotoleratessomedegreeofundersampling(10-20%)especiallyforimagesoflowSNRasthemicroscopeOTFfallsofftozerostronglytowardstheopticalcut-offfrequencyand
onlyasmallfractionofthesignalenergyisthenwronglyassumedasnoiseenergy.Furthermorehaveslightlyundersampledimagesstillanout-of-bandregioninthecornersoftheFourier-transformation,whichcanbeusedforouralgorithm(seeFigureS6).References:[Brown2008]C.M.Brown,R.B.Dalal,B.Hebert,M.A.Digman,A.R.Horwitz,andE.Gratton,Rasterimagecorrelationspectroscopy(RICS)formeasuringfastproteindynamicsandconcentrationswithacommerciallaserscanningconfocalmicroscope.JournalofMicroscopy,229:78-91,2008.[Huang2013]F.Huang,T.M.P.Hartwich,F.E.Rivera-Molina,Y.Lin,C.Whitney,J.J.Long,P.D.Uchil,J.R.Myers,M.A.Baird,W.Mothes,M.W.Davidson,D.Toomre,andJ.Bewersdorf.Video-ratenanoscopyusingsCMOScamera-specificsingle-moleculelocalizationalgorithms.NatureMethods,10(7):653-658,2013.
