Normalized Shape

In this note, we would like to explore some of the issues suggested by the discussion paper of Pizlo et. al. The context is the study of image of planar objects taken with calibrated cameras. However, in this note, we shall consider the question using notation that has become somewhat standard in the literature of computer vision, particularly in the study of uncalibrated cameras. The projection from three-dimensional Euclidean space (or projective space) to the two dimensional image is conveniently expressed using formalism of projective geometry. Representing points in Euclidean space and in the image by homogeneous coordinates allows the mapping to be expressed in a very simple form as : u = M x (1) where u = (u, v, w) are the homogeneous coordinates in the image, x = (x, y, z, t) is the point in space, expressed also in homogeneous coordinates, and M is a 3 × 4 matrix. Since we are using homogeneous coordinates, the matrix M is defined up to a constant factor only, and hence has 11 degrees of freedom. The matrix M may be split up as a product M = K 3×3 (I 3×3 | 0) R 3×3 t 0 1 = K(R | t). (2) In this equation R 3×3 is a rotation matrix, and the right-hand term in the equation is simply a Euclidean change of coordinates in object space. The central term, (I | 0) represents the projection from 3 to 2 dimensions, and the left hand term K is a 3 × 3 upper triangular matrix representing an affine transformation of the image. The matrix K encodes the information about the calibration of the camera. In talking of calibrated cameras, we imply that K is known. Once again we may count 11 degrees of freedom : 6 for the Euclidean change of coordinates (three for rotation and three for translation) and 5 for the entries of the matrix K, which we defined only up to an irrelevant scale factor. All this is fairly standard. One may see [2] for a few more details.