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 IssuesPowerPoint PresentationSlide 47Slide 48Camera Model SummaryNext3D Computer Visionand Video Computing3D Vision3D VisionTopic 5 of Part 2Camera ModelsCSC I6716Spring 2004Zhigang Zhu, NAC 8/203Ahttp://www-cs.engr.ccny.cuny.edu/~zhu/VisionCourse-2004.html3D Computer Visionand Video Computing3D Vision3D VisionClosely Related Disciplines Image processing – image to magePattern recognition – image to classesPhotogrammetry – obtaining accurate measurements from imagesWhat is 3-D ( three dimensional) Vision?Motivation: making computers see (the 3D world as humans do)Computer Vision: 2D images to 3D structureApplications : robotics / VR /Image-based rendering/ 3D videoLectures on 3-D Vision Fundamentals (Part 2)Camera Geometric Model (1 lecture – this class- topic 5)Camera Calibration (1 lecture – topic 6)Stereo (2 lectures – topic 7)Motion (2 lectures –topic 8)3D Computer Visionand Video ComputingLecture OutlineLecture OutlineGeometric Projection of a CameraPinhole camera modelPerspective projectionWeak-Perspective ProjectionCamera ParametersIntrinsic Parameters: define mapping from 3D to 2DExtrinsic parameters: define viewpoint and viewing directionBasic Vector and Matrix Operations, RotationCamera Models RevisitedLinear Version of the Projection Transformation EquationPerspective Camera ModelWeak-Perspective Camera ModelAffine Camera ModelCamera Model for PlanesSummary3D Computer Visionand Video ComputingLecture AssumptionsLecture AssumptionsCamera Geometric ModelsKnowledge about 2D and 3D geometric transformationsLinear algebra (vector, matrix)This lecture is only about geometryGoalBuild 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: Determing 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 ModelPin-hole is the basis for most graphics and visionDerived from physical construction of early camerasMathematics is very straightforward3D World projected to 2D ImageImage inverted, size reducedImage is a 2D plane: No direct depth informationPerspective projection f called the focal length of the lensgiven image size, change f will change FOV and figure sizesPinhole lensOptical AxisfImage Plane3D Computer Visionand Video ComputingFocal Length, FOVFocal Length, FOVConsider case with object on the optical axis:fzOptical axis: the direction of imagingImage plane: a plane perpendicular to the optical axisCenter of Projection (pinhole), focal point, viewpoint, nodal pointFocal length: distance from focal point to the image planeFOV : Field of View – viewing angles in horizontal and vertical directionsviewpointImage plane3D Computer Visionand Video ComputingFocal Length, FOVFocal Length, FOVConsider case with object on the optical axis:fzOut of viewImage planeOptical axis: the direction of imagingImage plane: a plane perpendicular to the optical axisCenter of Projection (pinhole), focal point, viewpoint, , nodal pointFocal length: distance from focal point to the image planeFOV : Field of View – viewing angles in horizontal and vertical directionsIncreasing f will enlarge figures, but decrease FOV3D Computer Visionand Video ComputingEquivalent GeometryEquivalent GeometryConsider case with object on the optical axis:fzMore convenient with upright image:fzProjection plane z = fEquivalent mathematically3D Computer Visionand Video ComputingPerspective ProjectionPerspective ProjectionCompute the image coordinates of p in terms of the world (camera) coordinates of P.Origin of camera at center of projectionZ axis along optical axisImage Plane at Z = f; x // X and y//YxyZP(X,Y,Z)p(x, y)Z = f0YXZYfyZXfx3D Computer Visionand Video ComputingReverse ProjectionReverse ProjectionGiven 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 linesizeparallelism/angleshapeshape of planes depth3D Computer Visionand Video ComputingPinhole camera imagePinhole camera imagePhoto by Robert Kosara, [email protected]://www.kosara.net/gallery/pinholeamsterdam/pic01.html Amsterdamstraight linesizeparallelism/angleshapeshape of planes depth3D Computer Visionand Video ComputingPinhole camera imagePinhole camera imagePhoto by Robert Kosara, [email protected]://www.kosara.net/gallery/pinholeamsterdam/pic01.html Amsterdamstraight linesizeparallelism/angleshapeshape of planes depth3D Computer Visionand Video ComputingPinhole camera imagePinhole camera imagePhoto by Robert Kosara, [email protected]://www.kosara.net/gallery/pinholeamsterdam/pic01.html Amsterdamstraight