More Single View Geometry

Abstract

Chapter 6 introduced the projection matrix as the model for the action of a camera on points. This chapter describes the link between other 3D entities and their images under perspective projection. These entities include planes, lines, conics and quadrics; and we develop their forward and back-projection properties. The camera is dissected further, and reduced to its centre point and image plane. Two properties are established: images acquired by cameras with the same centre are related by a plane projective transformation; and images of entities on the plane at infinity, π ∞ , do not depend on camera position, only on camera rotation and internal parameters, K. The images of entities (points, lines, conics) on π ∞ are of particular importance. It will be seen that the image of a point on π ∞ is a vanishing point, and the image of a line on π ∞ a vanishing line; their images depend on both K and camera rotation. However, the image of the absolute conic, ω, depends only on K; it is unaffected by the camera's rotation. The conic ω is intimately connected with camera calibration, K, and the relation ω = (KK T ) −1 is established. It follows that ω defines the angle between rays back-projected from image points. These properties enable camera relative rotation to be computed from vanishing points independently of camera position. Further, since K enables the angle between rays to be computed from image points, in turn K may be computed from the known angle between rays.