On solving 2D and 3D puzzles using curve matching

We approach the problem of 2D and 3D puzzle solving by matching the geometric features of puzzle pieces three at a time. First, we define an affinity measure for a pair of pieces in two stages, one based on a coarse-scale representation of curves and one based on a fine-scale elastic curve matching method. This re-examination of the top coarse-scale matches at the fine scale results in an optimal relative pose as well as a matching cost which is used as the affinity measure for a pair of pieces. Pairings with overlapping boundaries are impossible and are removed from further consideration, resulting in a set of top valid candidate pairs. Second, triples arising from generic junctions are formed from this rank-ordered list of pairs. The puzzle is solved by a recursive grouping of triples using a best-first search strategy, with backtracking in the case of overlapping pieces. We also generalize aspects of this approach to matching of 3D pieces. Specifically, ridges of 3D fragments scanned using a laser range finder are detected using a dynamic programming method. A pair of ridges are matched using a generalization of the 2D curve matching approach to space curves by using an energy solution involving curvature and torsion, which are computed using a novel robust numerical method. The reconstruction of map fragments and broken tiles using this method is illustrated.