An open problem in matching sets of 3D lines

Methods for matching sets of 3D lines depend on whether line lengths are finite or infinite. In terms of line lengths, three basic cases arise in matching sets of lines: (1) finite to finite, (2) finite to infinite, and (3) infinite to infinite. For cases 1 and 2, which have not been treated in the literature, we present convergent iterative algorithms that (almost) always find the best match. For case 3, O.D. Faugeras and M. Hebert (FH) (1986) have proposed a popular iterative method that cannot be guaranteed to converge. We present an alternative approach that does converge. However, we also show that neither the FH solution, nor our solution is invariant with respect to coordinate transforms, which renders any best match meaningless. Thus, a satisfactory solution to case 3 does not yet exist. We discuss the underlying problem, which is the representation of infinite lines, and suggest alternatives that may lead to an invariant solution.