Geometric invariants for rational polynomial cameras

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. Relationships between object coordinates and image coordinates that are independent of the camera parameters are called geometric invariants. One example is the classical cross-ratios of volumes and areas. In practice, remote sensing systems are best modeled by rational functions of higher order polynomials with coefficients commonly referred to as RPCs. We derive some initial results on geometric invariants for RPC cameras, contrast these results with their central-projection analogues, and present examples of applications to remote sensing imagery.