Geometric interpretations of algebraic invariants in images of 3D scenes

We will demonstrate for central-projection imaging systems a natural progression of cross-ratio invariant theorems extending from one through three dimensions. In each dimension there is an invariant quantitative relationship between combinations of geometric entities in image space, and combinations of corresponding geometric entities in object space. In one dimension, when the object points and image points are colinear, these entities are line segments formed by corresponding pairs of object and image points. The ''mother of all invariants'' is the invariant relationship between cross-ratios of products of the lengths of these corresponding line segments in object and image. In two dimensions these geometric entities are triangles formed by corresponding triplets of points in the object and in the image. There exists an invariant relationship between cross-ratios of products of areas of these corresponding triangles in object and image. The one- and two-dimensional results are well known. Not so well-known is the fact that for the case of multiple images of three-dimensional scenes and objects the geometric entities are triangles (in the images) and tetrahedra (in the objects), and that there exist invariant linear relationships between cross-ratios of products of the areas of image-triangles and volumes of object-tetrahedra(1,2). One objective of our paper is to demonstrate that these linear relationships are established by a uniform pattern of algebraic arguments that extends the cross-ratio invariants in a natural progression from lower to higher dimensions. A second objective is to demonstrate that the resulting cross-ratio invariants can be interpreted as metric properties of geometric entities (e.g. areas of triangles formed by triplets of image points, and volumes of tetrahedra formed by four-tuples of object points). A third objective is to demonstrate that these cross-ratios of points in the images, which we can observe directly, are equal to the corresponding cross-ratios of points in the objects, which may not be directly accessible. We will use computer simulations to validate the algebraic results we derive in this paper, and 3D graphics to visualize them.