Geometric Invariants for Rational Polynomial Cameras

• Authors: Eamon B. Barrett;Paul M. Payton
• Venue: AIPR '00 Proceedings of the 29th Applied Imagery Pattern Recognition Workshop
• Year: 2000

Remote sensing imaging systems map object points, located at 3D coordinates (x,y,z) in object space, to image points located at 2D (line,sample) coordinates in image space. For "central projection" imaging systems such as conventional cameras the object-to-image mapping may be modeled as ratios of linear polynomials: line coordinate is P(x,y,z)/R(x,y,z), and sample coordinate is Q(x,y,z)/R(x,y,z), where P(x,y,z), Q(x,y,z), and R(x,y,z) are linear in x, y, and z. The polynomial coefficients are functions of the camera parameters, i.e. the focal length, position and orientation of the camera at the moment the image is captured. The characteristics of these "linear fractional" transformations are studied extensively in classical projective geometry [1,2]. Relationships between object coordinates and image coordinates that are independent of the camera parameters are called geometric invariants; the classical "cross-ratios" of volumes and areas being one example. During the last two decades a rich theory of geometric invariants has been developed for central projection cameras [3,4]. Practical applications to cartography and 3D object reconstruction from imagery are also described in the literature [5]. In practice, remote sensing systems are best modeled by rational functions of higher order polynomials with coefficients commonly referred to as "RPCs". The invariants theory for such rational polynomial cameras is substantially more challenging than for the central projection model. In this paper we will derive some initial results on geometric invariants for RPC cameras, contrast these results with their central-projection analogues, and we will present examples of applications to remote sensing imagery.