Central Panoramic Cameras Geometry and Design

CzechTechnicalUniversity,FacultyofElectricalEngineeringDepartmentofControlEngineering,CenterforMachinePerception12135Prague2,Karlovonamest13,CzechRepublicfax+420224357385,phone[phone],:[e-mail]335/97/147December5,1997AcknowledgementThisresearchwassupportedbytheCzechMinistryofEducationgrantVS96049,theinternalgrantoftheCzechTechnicalUniver-sity3097472/335,theGrantAgencyoftheCzechRepublic,grants102/95/1378,102/97/0480,102/97/0855andEuropeanUniongrantCopernicusCP941068.AbstractThisreportgivesthefoundationsofpanoramicstereovisionbypresentingtheanalysisofepipolargeometryforpanoramiccameras.Weshowthatthepanoramiccameraswithconvexhyperbolicorparabolicmirrors,socalledcentralpanoramiccameras,allowthesameepipolargeometryasperspectivecameras.Weworkoutthemodelofimageformationbyacentralpanoramiccamera.Itisshownthattheepipolarcurvesincentralpanoramicimagesareconicsandtheirequationisderived.Theapproachhowtodesignaperspectivecamera{hyperbolicmirrorsystemisalsopresented.Weproposeasimpleadjustmentprocedureofaperspectivecameraandahyperbolicmirrorinordertoformapropercentralpanoramiccamera.Thetheoryisdemonstratedbyasimulatedexperimentwhichcorroboratestheconclusionsdrawnfromthetheory.Keywords:computervision,omnidirectionalvision,epipolargeometry,panoramiccameras,hyperbolicmirror,stereo.Contents1Introduction21.1Motivation............................21.2Theproblemstobesolved....................31.3Workrelatedtopanoramicstereo................32Designofapanoramiccamera52.1Previousworks..........................52.2Shapeofthemirror........................52.3Modelofapanoramiccamerawithahyperbolicmirror....72.4Designofarealmirror......................92.4.1Ourdesign........................102.4.2Settingandcalibratingcentralpanoramiccameras..123Geometry143.1Epipolargeometryforperspectivecameras...........143.2Epipolargeometryofcentralpanoramiccameras.......153.2.1Formulaofepipolarconics................173.3Usingtheepipolargeometry...................183.3.1Ego-motionestimation..................184Experiments195Summary221Chapter1IntroductionThisworkgivesthefoundationsofpanoramicstereovisionbypresentingtheanalysisofepipolargeometry[5]forpanoramiccamerasandexplainshowtodesignarealpanoramicvisionsensor.Ithasbeenprimarilymotivatedbylookingforanimprovementofthemotionestimationfromapairofimages[11,14,9,5]butitsresultsarealsoapplicableforstructurefrompanoramicstereoimages.1.1MotivationItiswellknownthatmotionestimationalgorithmscannot,insomecases,welldistinguishasmallpuretranslationofthecamerafromasmallrotation.Anexampleisatranslationparalleltotheimageplaneandarotationaroundanaxisperpendiculartothedirectionofthetranslation[4].Theconfusionbecomesdominantwhendepthvariationinthescenearesmalloriftheeldofviewisnarrow.Theconfusioncanberemovedifacamerawithlargeeldofviewisused[4].Ideallyonewouldliketouseapanoramiccamerawhichhascom-plete360eldofviewandseestoalldirections.Itcanbeimaginedasapinholecamerawithasphericalimagingsurface(insteadofplanaroneasitisusual)centeredatthefocalpointofthepinholecamera.Panoramiccam-eracan,inprinciple,obtaincorrespondencesfromeverywhereindependentlyofthedirectionofmotion.Theuncertaintyofthemotionestimationwillthereforealsobecomeindependentfromthedirectionofmotion.TheaboveintuitivereasoninghasbeencorroboratedbytheresultofBrodskyetal.[3]whoshowthatthemotionestimationisalmostneverambiguousifthespher-icalimagingsurfaceisassumed.2CHAPTER1.INTRODUCTION31.2TheproblemstobesolvedThealgorithmformotionestimationfromapairofperspectiveimageshasto(1)ndthecorrespondingpoints,(2)estimatetheFundamentalmatrixdescribingtheepipolargeometry,(3)calibratethecamera,and(4)decom-posetheFundamentalmatrixintoatranslationandrotation.Motionestimationfrompanoramicimagessolvessimilarproblems.Inpartitrequiresto1.designapracticalpanoramiccamerawithsimplemathematicalmodel,2.proposeamethodforitscalibration,3.developtheepipolargeometryforpanoramicimages,and4.workoutanalgorithmformotionestimation.Panoramicstereovisionalsoneedsecientsearchforthecorrespondencesinpanoramicimageswhichcallsfor:5.theanalysisoftheshapeoftheepipolarcurvesinordertoconstraintthelocationofcorrespondingpointsandfor6.thestudyofepipolaralignmentofthepanoramicimagesinordertospeedupthesearch.Ourcontributionstartswithanoverviewoftheworksmostrelatedtopanoramicstereo,giveninthenextsection.Inthesection2.1,wecommentondierentconstructionsofpanoramiccameras.Thesection2.2introducesthecentralpanoramiccamera,forwhichepipolargeometrycanbefoundandwhichisthereforesuitableformotionestimation.Section2.3describesthemathematicalmodelofthatcamera.Amethodofitsdesignisproposedinthesection2.4.Epipolargeometryofcentralpanoramiccamerasaswellastheanalysisoftheshapeofepipolarcurvesispresentedinsection3.2.Utilizationoftheproposedgeometryisexplainedinsection3.3.Section4providesanillustrativeexampleofepipolargeomet