  This article presents for the first time a global method for registering 3D
curves with 3D surfaces without requiring an initialization. The algorithm
works with 2-tuples point+vector that consist in pairs of points augmented with
the information of their tangents or normals. A closed-form solution for
determining the alignment transformation from a pair of matching 2-tuples is
proposed. In addition, the set of necessary conditions for two 2-tuples to
match is derived. This allows fast search of correspondences that are used in
an hypothesise-and-test framework for accomplishing global registration.
Comparative experiments demonstrate that the proposed algorithm is the first
effective solution for curve vs surface registration, with the method achieving
accurate alignment in situations of small overlap and large percentage of
outliers in a fraction of a second. The proposed framework is extended to the
cases of curve vs curve and surface vs surface registration, with the former
being particularly relevant since it is also a largely unsolved problem.
