- Volume 11 Issue 2
We propose a new distance measure between 2-dimensional points to provide a total order for an entire point set and to reflect the correct geometric meaning of the naturalness of the point ordering. In general, there is no total order for 2-dimensional point sets, so curve reconstruction algorithms do not solve the self-intersection problem because the distance used in the previous methods is the Euclidean distance. A natural distance based on Brownian motion was previously proposed to solve the self-intersection problem. However, the distance reflects the wrong geometric meaning of the naturalness. In this paper, we correct the disadvantage of the natural distance by introducing a polar-natural distance, and we also propose a new curve reconstruction algorithm that is based on the polar-natural distance. Our experiments show that the new distance adequately reflects the correct geometric meaning, so non-simple curve reconstruction can be solved.
Point Ordering;Curve Reconstruction;Polar-Natural Distance
- T. K. Dey and T. K. Kumar, "A simple provable algorithm for curve reconstruction," Proc. 10th. ACM-SIAM Symp. Discr. Algorithms, 1999, pp. 893-894.
- L. Fang and D. C. Gossard, "Fitting 3D curves to unorganized data points using deformable curves," Proc. of CG International'92, 1992, pp. 535-543.
- J. P. Dedieu and C. H. Favardin, “Algorithms for ordering unorganized points along parametrized curves,” Numerical Algorithms, vol. 6, 1994, pp. 160-200. https://doi.org/10.1007/BF02149768
- G. Taubin and R. Ronfard, “Implicit simplical models for adaptive curve reconstruction,” IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 18, 1996, pp. 321-325. https://doi.org/10.1109/34.485559
- Thanh An Nguyen and Yong Zeng, “VICUR: A human-vision-based algorithm for curve reconstruction,” Robotics and Computer Integrated manufacturing, vol. 24, 2008, pp. 824-834. https://doi.org/10.1016/j.rcim.2008.03.007
- H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle, "Surface Reconstruction from unorganized points," Proc. SIGGRAPH' 92, 1992, pp. 71-78.
- J. D. Boissonnat and B. Geiger, "Three cimentional reconstruction of complex shapes based on the Delaunay triangulation," Proc. of Biomedical Image Process, Biomed. Visualizaiton, 1993, pp. 964-975.
- Philsu Kim and Hyoung-Seok Kim, “Point ordering with the natural distance based on Brownian motion,” Mathematical problems in engineering, vol. 2010, article ID 450460, 2010, p. 17.
- J. D. Boissonnat and F. Cazals, “Smooth surface reconstruction via natural neighbor interpolation of distance functions,” Comp. Geom. Theo. Appl., vol. 22, 2002, pp. 185-203. https://doi.org/10.1016/S0925-7721(01)00048-7
- H. Edelsbrunner, D. G. Kirkpatrick, and R. Seidel, “On the shape of a set of points in the plane,” IEEE Trans. Information Theory, 1983, pp. 551-559. https://doi.org/10.1109/TIT.1983.1056714
- D. Attali, "r-regular shape reconstruction from unorganized points," Proc. of 13th Ann. Sympos. Comput. Geom., 1997, pp. 248-253.
- N. Amenta, M. Bern, and D. Eppstein, “The crust and the beta-skeleton: combinatorial curve reconstruction,” Graphical Models, vol. 60, 1998, pp. 125-135. https://doi.org/10.1006/gmip.1998.0465