We prove that euclidean minimal spanning trees correctly reconstruct differentiable arcs from sufficiently dense samples. The proof is based on a combinatorial characterization of minimal spanning paths and on a description of the local geometry of ares inside tubular neighborhoods. We also present simple heuristics for reconstruting more general curves.
Key wordsComputational morphology Curve reconstruction Minimal spanning trees
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- 1.Arnon DS (1983) Topologically reliable display of algebraic curves. Comput Graph 17:219–227Google Scholar
- 2.Boissonnat J-D (1984) Geometric structures for three-dimensional shape representation. ACM Trans Graph 3:266–286Google Scholar
- 3.Duda RO, Hart PE (1973) Pattern Classification and Scene Analysis. WileyGoogle Scholar
- 4.Figueiredo LH (1992) Computational Morphology of Implicit Curves. Doctoral thesis, IMPA, Rio de JaneiroGoogle Scholar
- 5.Hoppe H, DeRose T, Duchamp T, McDonald J, Stuetzle W (1992) Surface reconstruction from unorganized points. Comput Graph 26:71–78Google Scholar
- 6.Hirsh MW (1976) Differential Topology, Graduate Texts in Mathematics 33, Springer, Berlin, Heidelberg, New YorkGoogle Scholar
- 7.Kirkpatrick DG, Radke JD (1985) A framework for computational morphology. In: Toussaint G (ed) Computational Geometry. North-Holland, pp 217–248Google Scholar
- 8.Prim RC (1957) Shortest connection matrix network and some generalizations. Bell Syst Tech J 36:1389–1401Google Scholar
- 9.Toussaint G (1980) Pattern recognition and computational complexity. Proceedings of the Fifth International Conference on Pattern Recognition, pp 1324–1347Google Scholar
- 10.Veltkamp RC (1992) The ψ-neighborhood graph. Comput Geom Theory Appl 1:227–246Google Scholar
- 11.Zahn CT (1971) Graph-theoretical methods for detecting and describing gestalt clusters. IEEE Trans Comput C 20:68–86Google Scholar