Projective Geometry and Camera ModelsNote about HW1Last class: introNext two classes: Single-view GeometryToday’s classImage formationPinhole cameraPinhole cameraCamera obscura: the pre-cameraFirst PhotographDimensionality Reduction Machine (3D to 2D)Projection can be tricky…Projection can be tricky…Projective GeometryLength is not preservedProjective GeometryProjective GeometryVanishing points and linesVanishing points and linesVanishing points and linesVanishing points and linesNote on estimating vanishing points(more vanishing points on board)Vanishing objectsProjection: world coordinatesimage coordinatesHomogeneous coordinatesHomogeneous coordinatesBasic geometry in homogeneous coordinatesAnother problem solved by homogeneous coordinatesSlide Number 33Interlude: when have I used this stuff?When have I used this stuff?When have I used this stuff?When have I used this stuff?When have I used this stuff?When have I used this stuff?Slide Number 40Remove assumption: known optical centerRemove assumption: square pixelsRemove assumption: non-skewed pixelsOriented and Translated CameraAllow camera translation3D Rotation of PointsAllow camera rotationDegrees of freedomScaled Orthographic ProjectionOther things to be aware ofRadial DistortionFocal length, aperture, depth of fieldDepth of fieldVarying the aperture Shrinking the apertureShrinking the apertureRelation between field of view and focal lengthThings to rememberNext classQuestionsProjective Geometry and Camera ModelsComputer VisionCS 543 / ECE 549 University of IllinoisDerek Hoiem01/21/10Note about HW1• Out before next Tues• Prob1: covered today, Tues• Prob2: covered next Thurs• Prob3: covered following weekLast class: intro• Overview of vision, examples of state of art• LogisticsNext two classes: Single‐view GeometryHow tall is this woman?Which ball is closer?How high is the camera?What is the camera rotation?What is the focal length of the camera?Today ’s classMapping between image and world coordinates– Pinhole camera model– Projective geometry• homogeneous coordinates and vanishing lines– Camera matrix– Other camera parametersImage formationLet’s design a camera– Idea 1: put a piece of film in front of an object– Do we get a reasonable image?Slide source: SeitzPinhole camer aIdea 2: add a barrier to block off most of the rays– This reduces blurring– The opening known as the apertureSlide source: SeitzPinhole camer aFigure from Forsythff = focal lengthc = center of the cameracCamera obscura: the pre‐camer a• First idea: Mo‐Ti, China (470BC to 390BC)• First built: Alhacen, Iraq/Egypt (965 to 1039AD)Illustration of Camera Obscura Freestanding camera obscura at UNC Chapel HillPhoto by Seth IlysFirst PhotographFirst photograph– Took 8 hours on pewter plateJoseph Niepce, 1826Photograph of the first photographStored at UT AustinPoint of observationFigures © Stephen E. Palmer, 2002Dimensionality Reduction Machine (3D to 2D)3D world 2D imageProjection can be tricky…Slide source: SeitzProjection can be tricky…Slide source: SeitzProjective GeometryWhat is lost?• LengthWhich is closer?Who is taller?Length is not preservedFigure by David ForsythB’C’A’Projective GeometryWhat is lost?• Length• AnglesPerpendicular?Parallel?Projective GeometryWhat is preserved?• Straight lines are still straightVanishing points and linesParallel lines in the world intersect in the image at a “vanishing point”Vanishing points and linesoVanishing PointoVanishing PointVanishing LineVanishing points and linesVanishingpointVanishinglineVanishingpointVertical vanishingpoint(at infinity)Slide from Efros, Photo from CriminisiVanishing points and linesPhoto from online Tate collectionNote on estimating vanishing pointsUse multiple lines for better accuracy… but lines will not intersect at exactly the same point in practiceOne solution: take mean of intersecting pairs… bad idea!Instead, minimize angular differences(more vanishing points on board)Vanishing objectsProjection: world coordinatesÆimage coordinates(work on board)Homogeneous coordinatesConversionConverting to homogeneous coordinateshomogeneous image coordinateshomogeneous scene coordinatesConverting from homogeneous coordinatesHomogeneous coordinatesInvariant to scalingPoint in Euclidean is ray in Homogeneous⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡⇒⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡wywxkwkykwkxkwkykxwyxkHomogeneous CoordinatesEuclidean CoordinatesBasic geometry in homogeneous coordinates• Line equation: ax + by + c = 0• Append 1 to pixel coordinate to get homogeneous coordinate• Line given by cross product of two points• Intersection of two lines given by cross product of the lines⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=1iiivupjiijppline×=jiijlinelineq×=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=iiiicbalineAnother problem solved by homogeneous coordinatesEuclidean: (Inf, Inf)Homogeneous: (1, 1, 0)Intersection of parallel linesEuclidean: (Inf, Inf) Homogeneous: (1, 2, 0)Slide Credit: SavereseProjection matrix[]XtRKx =x: Image Coordinates: (u,v,1)K: Intrinsic Matrix (3x3)R: Rotation (3x3) t: Translation (3x1)X: World Coordinates: (X,Y,Z,1)OwiwkwjwR,TInterlude: when have I used this stuff?When have I used this stuff?Object Recognition (CVPR 2006)When have I used this stuff?Single‐view reconstruction (SIGGRAPH 2005)When have I used this stuff?Getting spatial layout in indoor scenes (ICCV 2009)When have I used this stuff?Inserting photographed objects into images (SIGGRAPH 2007)Original CreatedWhen have I used this stuff?Inserting synthetic objects into images (submitted)[]X0IKx =⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡101000000001zyxffvukKSlide Credit: SavereseProjection matrixIntrinsic Assumptions• Unit aspect ratio•