RECONSTRUCTION USING LINE TOKENS IN P3(C)
Thomas Buchanan · 1992
We prove that in general the critical set for photogrammetric reconstruction using lines in P3(C) is a line congruence F of order 3 and class 6; F has 10 singular points and no singular planes. The general hyperplane sections of F (ruled surfaces formed by intersecting F with linear line complexes) have genus 5. F can be found in FanG's classification of congruences of order 3, and further properties of F can be found in the literature. Traditionally photogrammetry has used point tokens in 2-dimensional images to reconstruct a 3-dimensional scene. (See the references given in E11).) However, in recent years information scientists working in the field of computer vision have begun using line tokens from three images for reconstruction. (See (16); also (-3), (5), (81, (9).) In this paper we shah assume that the prerequisite correspondence between tokens in the images has been established. Moreover, we assume that a 3- dimensional reconstruction with respect to the given correspondence exists. For the sake of simplicity we consider only projective planes and spaces over the ground field C of complex numbers. We shall be concerned exclusively with reconstruction in P3 up to a collineation, i.e. up to an element of PGL(3, C). Images are isomorphic to P2. For many of the arguments in this paper, the images are identified with the stars at the respective centers of projection. The mathematical terms used in this paper can be found in the standard works dealing with line geometry, for example (15) or (19). I have announced the result of this paper in (2). The announcement- written for a broader audience-also contains a collection of definitions of concepts from line geometry. In addition, the announcement discusses the relationship between the set which defeats the algorithm described in (8) and the set described here.