1CS/ECE 181bMidterm ReviewMay 11, 2006• Image Formation• Projective Geometry• Camera Models•Stereo• Edge Detection2Image Formation• Pinhole camera geometry– Perspective projection• Vanishing point– What is it?– How do you geometrically construct thevanishing point of a line in 3-D• Orthographic and Parallel Projections2-D Projective Geometry• Homogeneous coordinates• Points from lines and lines from points– Intersection of parallel lines– Line at infinity• Duality principle• Conics and Conic Sections– Five points define a conic– Tangent lines and dual conics3Homography• Projective Transformations– Isometries– Similarities– Affine Transformation– General projective transformationCentral Projection• Pinhole camera geometry revisited– Central projection– Camera calibration matrix– Camera and World coordinate frames• Interior and Exterior Orientation– Finite Projective camera4Camera centerColumn pointsPrincipal planeAxis planePrincipal pointPrincipal rayCamera anatomy[][ ] =0010ppppp43212Column Vectors: image of the world origin.The columns of the projectivecamera are 3-vectors that have ageometric meaning as particularimage points.P1: vanishing point of the world coordinate x-axisP2: vanishing point of y-axisP3: vanishing point of z axis5Principal Planes• Row vectors of the camera projectionmatrix• Principal axis• Estimating the camera matrix PTopics• Image Formation• Projective Geometry• Camera Models•Stereo• Edge Detection6Stereo• Epipolar Geometry• Essential Matrix• Fundamental MatrixEpipolar geometry• Epipolar Plane• Epipoles• Epipolar Lines• BaselineC1C27geometric derivationxHx'=x'e'l' =[]FxxHe'==mapping from 2-D to 1-D family (rank 2)Fundamental Matrix FThe eight-point algorithm0=pFpTInvert and solve for F8The Correspondence Problem• Establishing correspondence between thetwo views• Disparity map and depth computations– Random dot
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