1Camera ModelsAcknowledgementsUsed slides/content with permission fromMarc Pollefeys for the slidesHartley and Zisserman: book figures from the webMatthew Turk: for the slidesApril 2004 Camera Models 2Camera modelCamera calibrationSingle view geom.Single view geometry2April 2004 Camera Models 3Pinhole camera geometry• A general projective camera P maps world points X toimage points x according to x = PX.April 2004 Camera Models 4TTZfYZfXZYX )/,/(),,( a=101001ZYXffZfYfXZYXaCentral projection in homogeneous coordinates3April 2004 Camera Models 5=10100ZYXffZfYfX=10101011ZYXffZfYfXPXx =[ ]0|I)1,,(diagP ff=Camera projection matrix PP: principal pointPrincipal planeApril 2004 Camera Models 6TyxTpZfYpZfXZYX )/,/(),,( ++aprincipal pointTyxpp ),(=++101001ZYXpfpfZZpfYZpfXZYXyxxxaPinhole point offsetImage (x,y) and camera(x_cam, y_cam) coordinate systems.4April 2004 Camera Models 7=++10100ZYXpfpfZZpfYZpfXyxxx€ x = K I | 0[ ]Xcam=1yxpfpfKcalibration matrixcamera is assumed to be located at the center of aEuclidean coordinate system with the principal axisof the camera point in the direction of z-axis.Camera calibration matrix KApril 2004 Camera Models 8( )C~-X~RX~cam=€ Xcam=R −R˜ C 0 1      XYZ1            =R −R˜ C 0 1      X€ x = K I | 0[ ]Xcam= KR I | -˜ C [ ]X[ ]t|RKP =C~Rt −=PXx =Camera rotation and translationEuclidean transformation between worldand camera coordinate framesInhomogeneous 3-vectorof coordinates of a point inthe world coordinate frame.Same point in the cameracoordinate frameCoordinates of cameracenter in worldcoordinates5April 2004 Camera Models 9Internal and exterior orientation• has 9 dof– 3 for K (f, px, py)– 3 for R– 3 for• Parameters contained in K are called the internal cameraparameters, or the internal orientation of the camera.• The parameters of R and which relate the cameraorientation and position to a world coordinate system arecalled the external parameters or exterior orientation.• Often convenient not to make the camera center explicit,and instead to represent the world->image transformationas , whereApril 2004 Camera Models 10€ K =αxx0αyy01          € K =mxmy1          f pxf py1          CCD camera: 10 dofCCD CamerasCCD Cameras: may havenon-square pixels!6April 2004 Camera Models 11Finite projective cameraS: skew parameter;0 for most normal camerasA camerawith K as above is called a a finite projective camera.A finite projective camera has 11 degrees of freedom. This isthe same number of degrees of freedom as a 3 x 4 matrix,defined up to an arbitrary scale.Note that the left hand 3 x 3 submatrix of P, equal to KR, is non-singular.any 3 x 4 matrix P for which the left hand 3 x 3 submatrix isnon-singular is the camera matrix for some finite projectivecamera.April 2004 Camera Models 12Camera centerColumn pointsPrincipal planeAxis planePrincipal pointPrincipal rayCamera anatomy7April 2004 Camera Models 130PC =null-space camera projection matrixConsider:For all A all points on ray AC project on imageof A, therefore C is camera centerImage of camera center is (0,0,0)T, i.e. undefinedCamera CenterConsider the line containing C and any other point A in 3-space. April 2004 Camera Models 14[ ] [ ]=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 axis8April 2004 Camera Models 15Row Vectors and the Principal PlaneThe principal plane is the plane through the camera centerparallel to the image plane. It consists of the set of points Xwhich are imaged on the line at infinity of the image.i.e.,⇒A point X lies on the image plane iff⇒In particular, the camera center C lies on the principal plane.P3 is the vector representing the principal plane of the camera,April 2004 Camera Models 16Principal Plane9April 2004 Camera Models 17Axis planesnote: p1,p2 dependent on image x and y axis(choice of image coordinage system).Consider the set of points X on plane P1. This set satisfies:These are imaged at PX = (0,y,w)^Tthese are points on the image y-axis.Plane P1 is defined by the camera center and theline x=0 in the image.Similarly, P2 is given by P2.X =0,April 2004 Camera Models 18principal point( )0,,,pˆ3332313ppp=∞The principal pointPrincipal axis: is the line passing through the camera center C,with direction perpendicular to the principal plane P3.The axis intersects the image plane at the principal point.10April 2004 Camera Models 19iixX ↔? PResectioningEstimating the camera projectionmatrix from corresponding 3-spaceand image measurements ->resectioning.⇒Similar to the 2D projective transformation H.⇒H was 3x3 whereas P is 3x4.April 2004 Camera Models 20iiPXx =0Ap =Basic equations: is a 4-vector, the i-th row of P.Each point correspondence gives 2independent equations.A = 2n x 12 matrixp: 12 x 1 column vector.11April 2004 Camera Models 210Ap =minimal solutionOver-determined solution⇒ 5.5 correspondences needed (say 6) P has 11 dof, 2 independent eq./pointsn ≥ 6 pointsApminimize subject to constraint 1p =1pˆ3=3pˆ=PCamera matrix PApril 2004 Camera Models 22HW #3: Computing P• Will be posted soon.• Will be due next