Computer GraphicsComputer VisionCombinedPinhole camerasDistant objects are smallerConsequences: Parallel lines meetThe Effect of PerspectiveVanishing pointsSlide 9Slide 10Properties of ProjectionPinhole Camera TerminologyThe equation of projectionSlide 14The camera matrixWeak perspectiveWeak Perspective ProjectionThe Equation of Weak Perspective (scaled Orthographic)Slide 19Pictorial ComparisonSummary: Perspective LawsPros and Cons of These ModelsThe projection matrix for orthographic projectionSlide 24The Image Formation PipelineCamera parametersCamera calibrationOutlineNotes on NotationBlock Notation for Matrices2-D Transformations2-D ScalingSlide 33Slide 34Slide 35Slide 36Matrix form of 2-D ScalingSlide 382-D RotationSlide 40Slide 41Matrix form of 2-D RotationSlide 432-D Shear (Horizontal)Slide 45Slide 46Slide 472-D Shear (Vertical)2-D TranslationSlide 50Slide 51Slide 52Slide 53Slide 54Representing TransformationsExample: “Trick” of additional coordinate makes this possibleTranslation MatrixHomogeneous CoordinatesHomogeneous Coordinates: Projective SpaceLeaving Projective SpaceHomogeneous Coordinates: Rotations, etc.Example: Transformations Don’t CommuteSlide 63Slide 64Slide 65Slide 66Slide 67Slide 68Slide 69Slide 70Slide 71Computer Vision : CISC4/689Computer GraphicsImageOutputModelSyntheticCamera(slides courtesy of Michael Cohen)Computer Vision : CISC4/689Real SceneComputer VisionReal CamerasModelOutput(slides courtesy of Michael Cohen)Computer Vision : CISC4/689CombinedModelReal SceneReal CamerasImageOutputSyntheticCamera(slides courtesy of Michael Cohen)Computer Vision : CISC4/689Pinhole cameras•Abstract camera model - box with a small hole in it•Pinhole cameras work in practiceComputer Vision : CISC4/689Distant objects are smallerComputer Vision : CISC4/689Consequences: Parallel lines meet•There exist vanishing pointsMarc PollefeysComputer Vision : CISC4/689The Effect of PerspectiveComputer Vision : CISC4/689Vanishing pointsVPLVPRHVP1VP2VP3Different directions correspond to different vanishing pointsMarc PollefeysComputer Vision : CISC4/689Vanishing points•each set of parallel lines (=direction) meets at a different point–The vanishing point for this direction•Sets of parallel lines on the same plane lead to collinear vanishing points. –The line is called the horizon for that plane•Good ways to spot faked images–scale and perspective don’t work–vanishing points behave badly–supermarket tabloids are a great source.Computer Vision : CISC4/689Slide credit: David JacobsComputer Vision : CISC4/689Properties of Projection•Points project to points•Lines project to lines•Vanishing points for parallel lines•Parallel lines parallel to image plane donot converge•Closer objects appear bigger•Angles are not preserved•Degenerate cases–Line through focal point projects to a point.–Plane through focal point projects to lineComputer Vision : CISC4/689Pinhole Camera TerminologyCamera center/ pinholePrincipal point/image centerImage pointCamera pointFocal lengthOptical axisImage planeComputer Vision : CISC4/689The equation of projectionComputer Vision : CISC4/689The equation of projection•Cartesian coordinates:–We have, by similar triangles, that (x, y, z) -> (f x/z, f y/z, -f)–Ignore the third coordinate, and get(x, y, z)  ( fxz, fyz)Computer Vision : CISC4/689The camera matrix•Turn previous expression into HC’s–HC’s for 3D point are (X,Y,Z,T)–HC’s for point in image are (U,V,W)UVW1 0 0 00 1 0 00 01f0XYZTComputer Vision : CISC4/689Weak perspective•Issue–perspective effects, but not over the scale of individual objects–collect points into a group at about the same depth, then divide each point by the depth of its group–Adv: easy–Disadv: wrongComputer Vision : CISC4/689Weak Perspective ProjectionfZO-xZZXconstZfXxZComputer Vision : CISC4/689The Equation of Weak Perspective(scaled Orthographic)),(),,( yxszyx • s is constant for all points.• Parallel lines no longer converge, they remain parallel.Slide credit: David JacobsComputer Vision : CISC4/689Generalization of Orthographic ProjectionyYxXWhen the camera is at a(roughly constant) distancefrom the scene, take m=1.Marc PollefeysComputer Vision : CISC4/689Pictorial ComparisonWeak perspectivePerspectiveMarc PollefeysComputer Vision : CISC4/689camera theoflength focaldepthscoordinate world,,scoordinate image,fZZYXyxSummary: Perspective Laws1. Perspective2. Weak perspective3. OrthographicYconstyXco n stx ZfYyZfXx YyXx Computer Vision : CISC4/689Pros and Cons of These Models•Weak perspective has simpler math.–Accurate when object is small and distant.–Most useful for recognition.•Pinhole perspective much more accurate for scenes.–Used in structure from motion.•When accuracy really matters, we must model the real camera–Use perspective projection with other calibration parameters (e.g., radial lens distortion)Slide credit: David JacobsComputer Vision : CISC4/689The projection matrix for orthographic projectionUVW1 0 0 00 1 0 00 0 0 1XYZTComputer Vision : CISC4/689Affine camerasComputer Vision : CISC4/689The Image Formation PipelineComputer Vision : CISC4/689Camera parameters•Issue–camera may not be at the origin, looking down the z-axis•extrinsic parameters–one unit in camera coordinates may not be the same as one unit in world coordinates•intrinsic parameters - focal length, principal point, aspect ratio, angle between axes, etc.UVWTransformationrepresenting intrinsic parameters1 0 0 00 1 0 00 0 1 0Transformationrepresentingextrinsic parametersXYZTNote the matrix dimensionsComputer Vision : CISC4/689Camera