ransac算法
高中物理教学反思-论述题格式
2023年2月18日发(作者：易盛极星)RANSAC算法⽂献
RandomsampleconsensusRandomsampleconsensus(RANSACRANSAC)isantoestimateparametersofamathematicalmodelfromasetofobserved
datathatcontains,ore,italsocanbe
on-deterministicalgorithminthesensethatitproducesareasonable
resultonlywithacertainprobability,orithmwasfirst
edRANSACtosolvetheLocationDeterminationProblem(LDP),where
thegoalistodeterminethepointsinthespacethatprojectontoanimageintoasetoflandmarkswithknownlocations.
Abasicassumptionisthatthedataconsistsof"inliers",i.e.,datawhosedistributioncanbeexplainedbysomesetofmodel
parameters,thoughmaybesubjecttonoise,and"outliers"lierscancome,
forexample,fromextremevaluesofthenoiseorfromerroneousmeasurementsorincorrecthypothesesaboutthe
alsoassumesthat,givena(usuallysmall)setofinliers,thereexistsaprocedurewhichcan
estimatetheparametersofamodelthatoptimallyexplainsorfitsthisdata.
Contents
Example[]
ngthatthissetcontainsbothinliers,i.e.,points
whichapproximatelycanbefittedtoaline,andoutliers,pointswhichcannotbefittedtothisline,aforlinefittingwill
sonisthatitisoptimallyfittedtoallpoints,includingtheoutliers.
RANSAC,ontheotherhand,attemptstoexcludetheoutliersandfindalinearmodelthatonlyusestheinliersinits
donebyfittinglinearmodelstoseveralrandomsamplingsofthedataandreturningthemodelthathas
heinlierstendtobemorelinearlyrelatedthanarandommixtureofinliersand
outliers,artice,thereisnoguaranteethata
subsetofinlierswillberandomlysampled,andtheprobabilityofthealgorithmsucceedingdependsontheproportionof
inliersinthedataaswellasthechoiceofseveralalgorithmparameters.
Adatasetwithmanyoutliersforwhichalinehastobefitted.
FittedlinewithRANSAC;outliershavenoinfluenceontheresult.
Overview[]
TheRANSACalgorithmisalearningtechniquetoestimateparametersofamodelbyrandomsamplingofobserveddata.
Givenadatasetwhosedataelementscontainbothinliersandoutliers,RANSACusesthevotingschemetofindtheoptimal
lementationofthisvoting
schemeisbasedontwoassumptions:thatthenoisyfeatureswillnotvoteconsistentlyforanysinglemodel(fewoutliers)
andthereareenoughfeaturestoagreeonagoodmodel(fewmissingdata).TheRANSACalgorithmisessentially
composedoftwostepsthatareiterativelyrepeated:
irststep,asamplesubsetng
modelandthecorrespondingm
cardinalityofthesamplesubsetisthesmallestsufficienttodeterminethemodelparameters.
econdstep,thealgorithmcheckswhichelementsoftheentiredatasetareconsistentwiththemodel
inlementwillbeconsideredasan
outlierifitdoesnotfitthefittingmodelinstantiatedbythesetofestimatedmodelparameterswithinsomeerror
thresholdthatdefinesthemaximumdeviationattributabletotheeffectofnoise.
SACalgorithmwilliterativelyrepeatthe
abovetwostepsuntiltheobtainedconsensussetincertainiterationhasenoughinliers.
TheinputtotheRANSACalgorithmisasetofobserveddatavalues,awayoffittingsomekindofmodeltothe
observations,achievesitsgoalbyrepeatingthefollowingsteps:
issubsetthehypotheticalinliers.
isfittedtothesetofhypotheticalinliers.
ointsthatfittheestimatedmodelwell,accordingto
somemodel-specific,areconsideredaspartoftheconsensusset.
imatedmodelisreasonablygoodifsufficientlymanypointshavebeenclassifiedaspartoftheconsensusset.
ards,themodelmaybeimprovedbyreestimatingitusingallmembersoftheconsensusset.
Thisprocedureisrepeatedafixednumberoftimes,eachtimeproducingeitheramodelwhichisrejectedbecausetoofew
pointsarepartoftheconsensusset,atter
case,wekeeptherefinedmodelifitsconsensussetislargerthanthepreviouslysavedmodel.
RANSAC:InliersandOutliers.
Algorithm[]
ThegenericRANSACalgorithmworksasfollows:
Return:
bestFit–modelparameterswhichbestfitthedata(ornulifnogoodmodelisfound)
iterations=0
bestFit=nul
bestErr=somethingreallylarge
whileiterations maybeInliers=nrandomlyselectedvaluesfromdata maybeModel=modelparametersfittedtomaybeInliers alsoInliers=emptyset foreverypointindatanotinmaybeInliers{ ifpointfitsmaybeModelwithanerrorsmallerthant addpointtoalsoInliers } ifthenumberofelementsinalsoInliersis>d{ %thisimpliesthatwemayhavefoundagoodmodel %nowtesthowgooditis betterModel=modelparametersfittedtoallpointsinmaybeInliersandalsoInliers thisErr=ameasureofhowwellbetterModelfitsthesepoints ifthisErr bestFit=betterModel bestErr=thisErr } } incrementiterations } returnbestFit Matlabimplementation[] Given: data–asetofobservations model–amodeltoexplainobserveddatapoints n–minimumnumberofdatapointsrequiredtoestimatemodelparameters k–maximumnumberofiterationsallowedinthealgorithm t–thresholdvaluetodeterminedatapointsthatarefitwellbymodel d–numberofclosedatapointsrequiredtoassertthatamodelfitswelltodata AMatlabimplementationof2DlinefittingusingtheRANSACalgorithm: %%Plotthedatapoints figure;plot(data(1?,data(2?,‘o’);holdon; number=size(data,2);%Totalnumberofpoints bestInNum=0;%Bestfittinglinewithlargestnumberofinliers bestParameter1=0;bestParameter2=0;%parametersforbestfittingline fori=1:iter %%Randomlyselect2points idx=randperm(number,num);sample=data(:,idx); %%Computethedistancesbetweenallpointswiththefittingline kLine=sample(:,2)-sample(:,1);%twopointsrelativedistance kLineNorm=kLine/norm(kLine); normVector=[-kLineNorm(2),kLineNorm(1)];%Ax+By+C=0A=-kLineNorm(2),B=kLineNorm(1) distance=normVector(data-repmat(sample(:,1),1,number)); %%Computetheinlierswithdistancessmallerthanthethreshold inlierIdx=find(abs(distance)<=threshDist); inlierNum=length(inlierIdx); %%Updatethenumberofinliersandfittingmodelifbettermodelisfound ifinlierNum>=round(inlierRationumber)&&inlierNum>bestInNum bestInNum=inlierNum; parameter1=(sample(2,2)-sample(2,1))/(sample(1,2)-sample(1,1)); parameter2=sample(2,1)-parameter1*sample(1,1); bestParameter1=parameter1;bestParameter2=parameter2; end end %%Plotthebestfittingline xAxis=-number/2:number/2; yAxis=bestParameter1*xAxis+bestParameter2; plot(xAxis,yAxis,‘r-’,‘LineWidth’,2); end %%Generaterandomdatafortest data=150(2rand(2,100)-1);data=data.*rand(2,100); ransac_demo(data,2,100,10,0.1); Parameters[] Thethresholdvaluetodeterminewhenadatapointfitsamodelt,andthenumberofclosedatapointsrequiredtoassert thatamodelfitswelltodatadaredeterminedbasedonspecificrequirementsoftheapplicationandthedataset,and berofiterationsk,however,canbedeterminedasafunctionofthe thedesiredprobabilitythattheRANSACalgorithm returnsasuccessfulresultifinsomeiterationitselectsonlyinliersfrom theinputdatase{displaystylew}be theprobabilityofchoosinganinliereachtimeasinglepointisselected,thatis, function[bestParameter1,bestParameter2]=ransac_demo(data,num,iter,threshDist,inlierRatio) %data:a2xndatasetwith#ndatapoints %num:efittingproblem,num=2 %iter:thenumberofiterations %threshDist:thethresholdofthedistancesbetweenpointsandthefittingline %inlierRatio:thethresholdofthenumberofinliers w{displaystylew}=numberofinliersindata/numberofpointsindata Acommoncaseisthatw{displaystylew}isnotwellknownbeforehand,ng thatthenpointsneededforestimatingamodelareselectedindependently,wn{displaystylew^{n}}isthe probabilitythatallnpointsareinliersand1−wn{displaystyle1-w^{n}}istheprobabilitythatatleastoneofthen pointsisanoutlier,obabilitytothepower ofkistheprobabilitythatthealgorithmneverselectsasetofnpointswhichallareinliersandthismustbethesameas1 −p{displaystyle1-p}.Consequently, 1−p=(1−wn)k{displaystyle1-p=(1-w^{n})^{k},} which,aftertakingthelogarithmofbothsides,leadsto k=log(1−p)log(1−wn){displaystylek={frac{log(1-p)}{log(1-w^{n})}}} Thisresultassumesthatthendatapointsareselectedindependently,thatis,apointwhichhasbeenselectedonceis oftennotareasonableapproachandthederivedvaluefork shouldbetakenasmple,inthecaseof findingalinewhichfitsthedatasetillustratedintheabovefigure,theRANSACalgorithmtypicallychoosestwopointsin eachiterationandcomputesmaybe_modelasthelinebetweenthepointsanditisthencriticalthatthetwopointsaredistinct. Togainadditionalconfidence,ndarddeviationofkisdefinedas SD(k)=1−wnwn{displaystyleoperatorname{SD}(k)={frac{sqrt{1-w^{n}}}{w^{n}}}} Advantagesanddisadvantages[] Thcedmaterialmaybechallengedandremoved.(September2014)() AnadvantageofRANSACisitsabilitytodoofthemodelparameters,i.e.,itcanestimatetheparameterswithahighdegree vantageofRANSACisthatthereisno upperboundonthetimeittakestocomputetheseparameters(exceptexhaustion).Whenthenumberofiterations computedislimitedthesolutionobtainedmaynotbeoptimal, thiswayRANSACoffersatrade-off;bycomputingagreaternumberofiterationstheprobabilityofareasonablemodel er,RANSACisnotalwaysabletofindtheoptimalsetevenformoderately contaminatedsetsanditusuallyperformsbadlywhenthenumberofinliersislessthan50%.OptimalRANSACwas proposedtohandleboththeseproblemsandiscapableoffindingtheoptimalsetforheavilycontaminatedsets,evenforan inlierratiounder5%.AnotherdisadvantageofRANSACisthatitrequiresthesettingofproblem-specificthresholds. nyone-modelapproachwhentwo(ormore)model instancesexist,nealternativerobustestimationtechniquethatmaybeuseful rapproachformultimodelfittingisknownasPEARL,which combinesmodelsamplingfromdatapointsasinRANSACwithiterativere-estimationofinliersandthemulti-modelfitting beingformulatedasanoptimizationproblemwithaglobalenergyfunctionaldescribingthequalityoftheoverallsolution. Applications[] TheRANSACalgorithmisoftenusedin,e.g.,tosimultaneouslysolvetheandestimatetherelatedtoapairofstereo cameras. Developmentandimprovements[] Since19812006,forthe25th anniversaryofthealgorithm,aworkshopwasorganizedattheInternational(CVPR)tosummarizethemostrecent contributionsandvariationstotheoriginalalgorithm,mostlymeanttoimprovethespeedofthealgorithm,therobustness andaccuracyoftheestimatedsolutionandtodecreasethedependencyfromuserdefinedconstants. RANSACcanbesensitivetothechoiceofthecorrectnoisethresholdthatdefineswhichdatapointsfitamodelinstantiated thresholdistoolarge,thenallthehypothesestendtoberankedequally(good).On theotherhand,whenthenoisethresholdistoosmall,theestimatedparameterstendtobeunstable(lyadding orremovingadatumtothesetofinliers,theestimateoftheparametersmayfluctuate).Topartiallycompensateforthis undesirableeffect,edtwomodificationofRANSACcalledMSAC(M-estimatorSAmpleandConsensus) andMLESAC(MaximumLikelihoodEstimationSAmpleandConsensus).Themainideaistoevaluatethequalityofthe consensusset(athatfitamodelandacertainsetofparameters)calculatingitslikelihood(whereasinthe originalformulationbyFischlerandBollestherankwasthecardinalityofsuchset).AnextensiontoMLESACwhichtakes intoaccountthepultingalgorithmis imilarlines,Chumproposedtoguidethesamplingprocedureifsomeaprioriinformation regardingtheinputdataisknown,posedapproachiscalled PROSAC,PROgressiveSAmpleConsensus. oposedarandomizedversionofRANSACcalledR-RANSACtoreducethecomputationalburdento icideaistoinitiallyevaluatethegoodnessofthecurrentlyinstantiatedmodelusingonlya strategywilltellwithhighconfidencewhenitisthecaseto evaluateasonabletothinkthattheimpact ofthiseofstrategyproposedbyChum érproposedaparadigmcalledPreemptiveRANSACthatallowsrealtimerobust eideaoftheapproachconsistsin generatingafixednumberofhypothesissothatthecomparisonhappenswithrespecttothequalityofthegenerated hypothesisratherthanagainstsomeabsolutequalitymetric. Otherresearcherstriedtocopewithdifficultsituationswherethenoisescaleisnotknownand/ormultiplemodelinstances enteachdatumwiththe ltiplemodelsarerevealedasclusterswhich steringalgorithm,calledJ-linkage,doesnotrequirepriorspecification ofthenumberofmodels,nordoesitnecessitatemanualparameterstuning. RANSAChasalsobeentailoredforrecursivestateestimationapplications,wheretheinputmeasurementsarecorruptedby outliersandKalmanfilterapproaches,whichrelyonaGaussiandistributionofthemeasurementerror,aredoomedtofail. SuchanapproachisdubbedKALMANSAC. Relatedmethods[] (MaximumLikelihoodEstimateSAmpleConsensus)–thatthedatawasgeneratedfromthesample-fittedmodel,e.g.a ofinliersandoutliers (MaximumAPosteriorSAmpleConsensus)–extendsMLESACtoincorporateaoftheparameterstobefittedand maximizesthe –ofthestateofa Notes[] ttingandUncertainty,,SpringerVieweg(2ndedition,2016) Statistics,,Wiley,1981(republishedinpaperback,2004),page1. 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