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SceneplanesandhomographiesHomographiesgiventheplaneandviceversaProofofresult12.1Example12.2Acalibrated

stereorigAcalibratedstereorig2Acalibratedstereorig3ThehomographyinducedbyaplaneFig12.1LegendHomographiescompatiblewithepipolarGeometryTwosetsof4arbitrarypointsfrom2imagesEpipolargeometrydefineconditionsonhomographiesCountingdegreesoffreedomCompatibilityconstraintsFig.12.2a

e’=HeCompatibilityconstraints2

Fig.12.2bHTle’=le

Compatibilityconstraints3

Fig.12.2cFig12.2CompatibilityconstraintsResult12.3HomographiesarecompatiblewithfundamentalmatrixCorollary12.4Result12.513.6PlaneinducedhomographiesgivenFandimagecorrespondences:

(a)3points,(b)alineandapoint

12.2.1ThreepointsThreepointsThefirst(explicit)methodispreferredDegenerategeometryforanimplicitcomputationofthehomographyFig.12.3Fig.12.3LegendDeterminingthepointsXiisnotnecessaryinfirstmethodAllthatisimportantResult12.6ProofProof2ConsistencyconditionsConsistencyconditions2Algorithms12.1Theoptimalestimateofhomographyinducedbyaplanedefinedby3points12.2.2ApointandlineAoneparameterfamilyofhomographiesFig12.4(a),(b)Fig12.4LegendResult12.7Proofofresult12.7Proofofresult12.7(2)Proofofresult12.7(3)Result12.8Result12.82GeometricinterpretationofthepointmapH(m)ExploretheAhomographybetweencorrespondinglineimagesFig.12.5Fig.12.5LegendDegeneratehomographiesDegeneratehomographies2AdegeneratehomographyFig.12.6Fig.12.6Legend12.3ComputingFgiventhehomographyinducedbyaplanePlaneinducedparallaxPlaneinducedparallaxFig.12.7Fig.12.7LegendPlaneinducedparallax2Fig.12.8Fig.12.8LegendPlaneinducedparallax2Algorithm12.2ComputingFgiventhecorrespondenceof6points,4ofwhicharecoplanarFundamentalmatrixfrom6pointsofwhich4arecoplanarFig.12.9Fig.12.9LegendProjectiveDepthExample12.9Binaryspacepartition:leftandrightimagesFig.12.10a,b(c)Pointswithknowncorrespondence

(d)Atripletofpointsselectedfrom(c)andthistripletdefinesaplaneFig.12.10c,d(e)Pointsononesideoftheplane

(f)Pointsontheotherside

Fig.12.10e,fFig12.10LegendTwoplanesTwoplanes2TheactionofthemapH=H2-1H1onx

Fig.12.11Fig.12.11LegendTwoplanes3Uptothispoints,theresultsofthischapterhavebeenentirelyprojective12.4TheinfinitehomographyHinfTheinfinitehomographyHinf2TheinfinitehomographyHinf3VanishingpointsandlinesTheinfinitehomographyHinfmapsvanishingpointsbetweenimagesFig.12.12AffineandmetricreconstructionAffineandme

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