Algorithms for the recognition of 2D images of m points and n lines in 3D
Let alpha be an ordered collection consisting of m points and n lines, m + n greater than or equal to 6, in three-dimensional space. Let beta be an ordered collection consisting of m points and n lines, m + n greater than or equal to 6, in two-dimensional space. Using alpha and beta, we construct a matrix M(alpha, beta) and, then, give two algorithms for determining whether beta is an image of a taken with a pinhole camera. These algorithms apply to any configuration, degenerate or not, consisting of points and lines. The first algorithm depends on the calculation of the nullspace of M(alpha, beta) and gives necessary and sufficient conditions for beta to be an image of alpha, if the data can be read exactly. The second algorithm depends only on the rank of M(alpha, beta) and can be used if errors arise in reading the data; this algorithm also gives equations which any image of a must satisfy. In the special case m + n = 6, there is a single equation J(alpha,beta) = 0 which must hold if beta is an image of alpha. We show that J(alpha, beta) is simply the determinant of M(alpha,beta). We conclude with a discussion of some computational aspects of the theory. (C) 2003 Elsevier Science B.V. All rights reserved.
recognition image nullspace rank image equations