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Line matching: solutions and unsolved problems

Matching sets of 3D lines is a basic tool in image registration, recognition, and other applications. In terms of line lengths, three basic cases arise in matching two sets of lines: (1) finite-finite; (2) finite-infinite; (3) infinite-infinite. Cases 1 and 2 have not been treated in the literature. We present a convergent, iterative algorithm that solves these two cases. Extensive tests show that the algorithm almost always converges to the best match, particularly when the hypothesized line correspondences are correct, or when noise does not substantially destroy the similarity of the two sets. Case 3 has been addressed by O.D. Faugeras and M. Hebert (see Int. J. Robotics Res., vol.5, no.3, p.27-52, 1986). They present an iterative technique that cannot be guaranteed to converge. To solve this case, we present a new method that converges. However, more importantly, we show that neither method is invariant to translation of the coordinate system. Therefore, a satisfactory solution for case 3 does not yet exist.