MEASURING IN IMAGES WITH PROJECTIVE GEOMETRY
Keywords: projective geometry, affine geometry, barycentric coordinates, barycentric ratio, cross-ratio, camera pose estimation, 3D point reconstruction
Abstract. There is a fundamental relationship between projective geometry and the perspective imaging geometry of a pinhole camera. Projective scales have been used to measure within images from the beginnings of photogrammetry, mostly the cross-ratio on a straight line. However, there are also projective frames in the plane with interesting connections to affine and projective geometry in three dimensional space that can be utilized for photogrammetry. This article introduces an invariant on the projective plane, describes its relation to affine geometry, and how to use it to reduce the complexity of projective transformations. It describes how the invariant can be use to measure on projectively distorted planes in images and shows applications to this in 3D reconstruction. The article follows two central ideas. One is to measure coordinates in an image relatively to each other to gain as much invariance of the viewport as possible. The other is to use the remaining variance to determine the 3D structure of the scene and to locate the camera centers. For this, the images are projected onto a common plane in the scene. 3D structure not on the plane occludes different parts of the plane in the images. From this, the position of the cameras and the 3D structure are obtained.