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Computer Vision Lecture 4. Information From Two or More Cameras Lecture Outline Geometrical tools and methods (2.1) Vectors, inner and outer products Homogeneous coordinates Projective transformations Cameras, coordinates, and projection matrices (2.2) Two projections Stereo vision Cartesian Vectors Vectors in two dimensions are pairs of numbers Vectors in three dimensions are triplets of numbers Vectors correspond to pairs of points We define vector addition, subtraction, and scalar multiplication Equations Geometrical Properties Coordinate Changes Homogeneous Coordinates Camera Equations Camera Calibration Chapter 3 Given: set of (Pi, pi), i = 1, ... , k Object and projection point coordinates Find: M, the 3x4 projection matrix Method: See chapter 3. Inverse Problem Given: Coordinates u, v and calibration matrix M Find: Coordinates x, y, z No solution Two Camera Inverse Problem Given: p, p’, projection coordinates of point P from two cameras. Given M1, M2. Find: P Problem: four equations, three unknowns. Epipolar Geometry O, O’ are poles. l is the epipolar line of p’, l’ is the epipolar line of p. OPxO’P is normal to plane, OO’ lies in plane (OPxO’P)?OO’ = 0 Constraint on coordinates of projections Epipolar Equation Write p = (u, v, 1)T, p’ = (u’, v’, 1)T. Then the epipolar equation is pTEp = 0. E can be found from the projection matrices, if known. If not, E can be found from measurements (weak calibration). If the two projection planes are parallel, the epipolar lines are parallel to each other and to the line between the poles. Stereo Vision Two views of a point allow us to find the third dimension (left). With many points in the field of view we must establish correspondence. Image Pair Rectification By transforming the images we can make the projection planes coincide. The epipolar lines now coincide and are parallel. Coincident Projection Planes We see that v’=v. Let u’(u) be the location of a point on each epipolar line in the second image correspon