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1、RobertCollinsRobertCollinsCSE486,PennStateCSE486,PennStateReminder:-0.00310695-0.00256462.96584F=-0.028094-0.0077162156.381313.1905-29.2007-9999.79Lecture20:TheEight-PointAlgorithmReadingsT&V7.3and7.4RobertCollinsRobertCollinsCSE486,PennStateEssential/Fundamen
2、talMatrixCSE486,PennStateE/FMatrixSummaryTheessentialandfundamentalmatricesare3x3matricesLonguet-Higginsequationthat“encode”theepipolargeometryoftwoviews.Motivation:Givenapointinoneimage,multiplyingEpipolarlines:bytheessential/fundamentalmatrixwilltelluswhiche
3、pipolarlinetosearchalonginthesecondview.Epipoles:EvsF:Eworksinfilmcoords(calibratedcameras)Fworksinpixelcoords(uncalibratedcameras)RobertCollinsRobertCollinsCSE486,PennStateComputingFfromPointMatchesCSE486,PennStateComputingF•Assumethatyouhavemcorrespondences•
4、Eachcorrespondencesatisfies:•Fisa3x3matrix(9entries)•SetupaHOMOGENEOUSlinearsystemwith9unknowns1RobertCollinsRobertCollinsCSE486,PennStateComputingFCSE486,PennStateComputingFGivenmpointcorrespondences…Think:howmanypointsdoweneed?RobertCollinsRobertCollinsCSE48
5、6,PennStateHowManyPoints?CSE486,PennStateSolvingHomogeneousSystemsSelf-studyUnlikeahomography,whereeachpointcorrespondenceAssumethatweneedthenontrivialsolutionof:contributestwoconstraints(rowsinthelinearsystemofequations),forestimatingtheessential/fundamentalm
6、atrix,eachpointonlycontributesoneconstraint(row).[becausetheLonguet-Higgins/Epipolarconstraintisascalareqn.]withmequationsandnunknowns,m>=n–1andrank(A)=n-1Thusneedatleast8points.Hence:TheEightPointalgorithm!Sincethenormofxisarbitrary,wewilllookforasolutionwith
7、norm
8、
9、x
10、
11、=1RobertCollinsRobertCollinsCSE486,PennStateLeastSquaresolutionCSE486,PennStateOptimizationwithconstraintsSelf-studySelf-studyWewantAxascloseto0aspossibleand
12、
13、x
14、
15、=1:Definethefollowingcost:ThiscostiscalledtheLAGRANGIANcostandλiscalledtheLAGRANGIANmulti
16、plierTheLagrangianincorporatestheconstraintsintothecostfunctionbyintroducingextravariables.2RobertCollinsRobertCollinsCSE486,PennStateOptimizationwithconstraintsCSE486,PennStateOpt
