3D VisionSlide 2Lecture OutlineLecture AssumptionsImage FormationSlide 6Pinhole Camera ModelFocal Length, FOVSlide 9Equivalent GeometryPerspective ProjectionReverse ProjectionPinhole camera imageSlide 14Slide 15Slide 16Slide 17Slide 18Slide 19Yet other pinhole camera imagesSlide 21It’s real!Weak Perspective ProjectionCamera ParametersIntrinsic Parameters (I)Intrinsic Parameters (II)Extrinsic ParametersLinear Algebra: Vector and MatrixSlide 29Rotation: from Angles to MatrixSlide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Rotation- Axis and AngleLinear Version of Perspective ProjectionLinear Matrix Equation of perspective projectionThree Camera ModelsCamera Models for a PlaneSlide 43Slide 44Applications and IssuesCamera Model SummaryNext3D Computer Visionand Video Computing3D Vision3D VisionTopic 1 of Part IICamera ModelsCSC I6716Spring 2008Zhigang Zhu, City College of New York [email protected] Computer Visionand Video Computing3D Vision3D VisionClosely Related Disciplines Image Processing – images to magesComputer Graphics – models to imagesComputer Vision – images to modelsPhotogrammetry – obtaining accurate measurements from imagesWhat is 3-D ( three dimensional) Vision?Motivation: making computers see (the 3D world as humans do)Computer Vision: 2D images to 3D structureApplications : robotics / VR /Image-based rendering/ 3D videoLectures on 3-D Vision FundamentalsCamera Geometric Models (1.5 lectures)Camera Calibration (1.5 lectures)Stereo (2 lectures)Motion (2 lectures)3D Computer Visionand Video ComputingLecture OutlineLecture OutlineGeometric Projection of a CameraPinhole camera modelPerspective projectionWeak-Perspective ProjectionCamera ParametersIntrinsic Parameters: define mapping from 3D to 2DExtrinsic parameters: define viewpoint and viewing directionBasic Vector and Matrix Operations, RotationCamera Models RevisitedLinear Version of the Projection Transformation EquationPerspective Camera ModelWeak-Perspective Camera ModelAffine Camera ModelCamera Model for PlanesSummary3D Computer Visionand Video ComputingLecture AssumptionsLecture AssumptionsCamera Geometric ModelsKnowledge about 2D and 3D geometric transformationsLinear algebra (vector, matrix)This lecture is only about geometryGoalBuild up relation between 2D images and 3D scenes-3D Graphics (rendering): from 3D to 2D-3D Vision (stereo and motion): from 2D to 3D-Calibration: Determning the parameters for mapping3D Computer Visionand Video ComputingImage FormationImage FormationLight (Energy) SourceSurfacePinhole LensImaging PlaneWorld Optics Sensor SignalB&W FilmColor FilmTV CameraSilver DensitySilver densityin three colorlayersElectrical3D Computer Visionand Video ComputingImage FormationImage FormationLight (Energy) SourceSurfacePinhole LensImaging PlaneWorld Optics Sensor SignalCamera: Spec & Pose3D Scene2D Image3D Computer Visionand Video ComputingPinhole Camera ModelPinhole Camera ModelPin-hole is the basis for most graphics and visionDerived from physical construction of early camerasMathematics is very straightforward3D World projected to 2D ImageImage inverted, size reducedImage is a 2D plane: No direct depth informationPerspective projection f called the focal length of the lensgiven image size, change f will change FOV and figure sizesPinhole lensOptical AxisfImage Plane3D Computer Visionand Video ComputingFocal Length, FOVFocal Length, FOVConsider case with object on the optical axis:fzOptical axis: the direction of imagingImage plane: a plane perpendicular to the optical axisCenter of Projection (pinhole), focal point, viewpoint, nodal pointFocal length: distance from focal point to the image planeFOV : Field of View – viewing angles in horizontal and vertical directionsviewpointImage plane3D Computer Visionand Video ComputingFocal Length, FOVFocal Length, FOVConsider case with object on the optical axis:fzOut of viewImage planeOptical axis: the direction of imagingImage plane: a plane perpendicular to the optical axisCenter of Projection (pinhole), focal point, viewpoint, , nodal pointFocal length: distance from focal point to the image planeFOV : Field of View – viewing angles in horizontal and vertical directionsIncreasing f will enlarge figures, but decrease FOV3D Computer Visionand Video ComputingEquivalent GeometryEquivalent GeometryConsider case with object on the optical axis:fzMore convenient with upright image:fzProjection plane z = fEquivalent mathematically3D Computer Visionand Video ComputingPerspective ProjectionPerspective ProjectionCompute the image coordinates of p in terms of the world (camera) coordinates of P.Origin of camera at center of projectionZ axis along optical axisImage Plane at Z = f; x // X and y//YxyZP(X,Y,Z)p(x, y)Z = f0YXZYfyZXfx3D Computer Visionand Video ComputingReverse ProjectionReverse ProjectionGiven a center of projection and image coordinates of a point, it is not possible to recover the 3D depth of the point from a single image.In general, at least two images of the same point taken from two different locations are required to recover depth.All points on this linehave image coordi-nates (x,y).p(x,y)P(X,Y,Z) can be any-where along this line3D Computer Visionand Video ComputingPinhole camera imagePinhole camera imagePhoto by Robert Kosara, [email protected]://www.kosara.net/gallery/pinholeamsterdam/pic01.html Amsterdam : what do you see in this picture?straight linesizeparallelism/angleshapeshape of planes depth3D Computer Visionand Video ComputingPinhole camera imagePinhole camera imagePhoto by Robert Kosara, [email protected]://www.kosara.net/gallery/pinholeamsterdam/pic01.html Amsterdamstraight linesizeparallelism/angleshapeshape of planes depth3D Computer Visionand Video ComputingPinhole camera imagePinhole camera imagePhoto by Robert Kosara, [email protected]://www.kosara.net/gallery/pinholeamsterdam/pic01.html Amsterdamstraight linesizeparallelism/angleshapeshape of planes depth3D Computer Visionand Video ComputingPinhole camera imagePinhole camera imagePhoto by Robert Kosara, [email protected]://www.kosara.net/gallery/pinholeamsterdam/pic01.html Amsterdamstraight linesizeparallelism/angleshapeshape of planes depth3D Computer Visionand Video
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