General methods for determining projective invariants in imagery

Abstract

This paper presents the results of a study of projective invariants and their applications in image analysis and object recognition. The familiar cross-ratio theorem, relating collinear points in the plane to the projections through a point onto a line, provides a starting point for their investigation. Methods are introduced in two dimensions for extending the cross-ratio theorem to relate noncollinear object points to their projections on multiple image lines. The development is further extended to three dimensions. It is well known that, for a set of points distributed in three dimensions, stereo pairs of images can be made and relative distances of the points from the film plane computed from measurements of the disparity of the image points in the stereo pair. These computations require knowledge of the effective focal length and baseline of the imaging system. It is less obvious, but true, that invariant metric relationships among the object points can be derived from measured relationships among the image points. These relationships are a generalization into three dimensions of the invariant cross-ratio of distances between points on a line. In three dimensions the invariants are cross-ratios of areas and volumes defined by the object points. These invariant relationships, which are independent of the parameters of the imaging system, are derived and demonstrated with examples.