Model-based recognition and motion tracking depends upon the ability to solve for projection and model parameters that will best fit a 3-D model to matching 2-D image features. This paper extends current methods of parameter solving to handle objects with

Figure2:Anexampleofamodelwithcurvedsurfacesandaninternalparameterspecifyingrotationofthehandle.Theunderlyingapproximatingpatchesareshownontheleft,andthegeneratedcontoursformatchingareshownontheright.

4Solvingforviewpointandmodelparameters

Projectionfrom3-Dto2-Disanon-linearoperation.Fortunately,however,itisasmoothandwell-behavedtransformation.Rotationindepthpriortoprojectiontransformstheprojectedpointsasafunctionofthecosineoftherotationangle.Translationtowardsorawayfromthecameraintroducesperspectivedistortionasafunctionoftheinverseofthedistance.Translationparalleltotheimageplaneisalmostentirelylinear.Translationsandrotationsassociatedwithinternalmodelparametershaveeffectsthatareidenticaltotheviewpointparameters,butappliedtoonlyasubsetofthemodelpoints.Allofthesetransformationsaresmoothandwellbehaved.Therefore,thisproblemisapromisingcandidatefortheapplicationofNewton’smethod,whichisbasedonassumingthatthefunctionislocallylinear.Whilethisdoesrequirestartingwithanappropriateinitialchoicefortheunknownparametersandfacestheriskofconvergingtoafalselocalminimum,wewillseebelowthatstabilizationmethodscanbeusedtomakethismethodhighlyeffectiveinpractice.

7