From Reference Frames to Reference Planes: Multi-View Parallax Geometry and Applications

This paper presents a new framework for analyzing the geometry of multiple 3D scene points from multiple uncalibrated images, based on decomposing the projection of these points on the images into two stages: (i) the projection of the scene points onto a (real or virtual) physical reference planar surface in the scene; this creates a virtual “image” on the reference plane, and (ii) the re-projection of the virtual image onto the actual image plane of the camera. The positions of the virtual image points are directly related to the 3D locations of the scene points and the camera centers relative to the reference plane alone. All dependency on the internal camera calibration parameters and the orientation of the camera are folded into homographies relating each image plane to the reference plane. Bi-linear and tri-linear constraints involving multiple points and views are given a concrete physical interpretation in terms of geometric relations on the physical reference plane. In particular, the possible dualities in the relations between scene points and camera centers are shown to have simple and symmetric mathematical forms. In contrast to the plane+parallax (p+p) representation, which also uses a reference plane, the approach described here removes the dependency on a reference image plane and extends the analysis to multiple views. This leads to simpler geometric relations and complete symmetry in multi-point multiview duality. The simple and intuitive expressions derived in the reference-plane based formulation lead to useful applications in 3D scene analysis. In particular, simpler tri-focal constraints are derived that lead to simple methods for New View Synthesis. Moreover, the separation and compact packing of the unknown camera calibration and orientation into the 2D projection transformation (a homography) allows also partial reconstruction using partial calibration information.