Abstract
Given a fixed origin o in the d-dimensional grid, we give a novel definition of digital rays dig(op) from o to each grid point p. Each digital ray dig(op) approximates the Euclidean line segment \(\overline {op}\) between o and p. The set of all digital rays satisfies a set of axioms analogous to the Euclidean axioms. We measure the approximation quality by the maximum Hausdorff distance between a digital ray and its Euclidean counterpart and establish an asymptotically tight Θ(log n) bound in the n×n grid. The proof of the bound is based on discrepancy theory and a simple construction algorithm. Without a monotonicity property for digital rays the bound is improved to O(1). Digital rays enable us to define the family of digital star-shaped regions centered at o, which we use to design efficient algorithms for image processing problems.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Asano, T., Chen, D.Z., Katoh, N., Tokuyama, T.: Efficient algorithms for optimization-based image segmentation. Int. J. Comput. Geom. Appl. 11(2), 145–166 (2001)
Cgal: Computational Geometry Algorithms Library. http://www.cgal.org
Chen, D.Z., Chun, J., Katoh, N., Tokuyama, T.: Efficient algorithms for approximating a multi-dimensional voxel terrain by a unimodal terrain. In: Proc. 10th Ann. Internat. Conf. Computing and Combinatorics (COCOON’04). Lecture Notes in Comput. Sci., vol. 3106, pp. 238–248. Springer, Berlin (2004)
Chun, J., Sadakane, K., Tokuyama, T.: Efficient algorithms for constructing a pyramid from a terrain. In: Proc. Japanese Conf. Discrete and Comput. Geom. (JCDCG’02). Lecture Notes in Comput. Sci., vol. 2866, pp. 108–117. Springer, Berlin (2003)
Erdős, P.: Problems and results on Diophantine approximation. Compos. Math. 16, 52–65 (1964)
Goodman, J.E., Pollack, R., Sturmfels, B.: Coordinate representation of order types requires exponential storage. In: Proc. 21st Ann. ACM Symp. Theory Comput. (STOC’89), pp. 405–410. ACM, New York (1989)
Goodrich, M.T., Guibas, L.J., Hershberger, J., Tanenbaum, P.J.: Snap rounding line segments efficiently in two and three dimensions. In: Proc. 13th Ann. ACM Symp. Comput. Geom. (SoCG’97), pp. 284–293 (1997)
Greene, D.H., Yao, F.F.: Finite-resolution computational geometry. In: Proc. 27th Ann. IEEE Symp. Foundations Comput. Sci. (FOCS’86), pp. 143–152. IEEE, New York (1986)
Klette, R., Rosenfeld, A.: Digital straightness—a review. Discrete Appl. Math. 139(1–3), 197–230 (2004)
Matousěk, J.: Geometric Discrepancy: An Illustrated Guide. Springer, Berlin (1999)
Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63. SIAM, Philadelphia (1992)
Pach, J., Pollack, R., Spencer, J.: Graph distance and Euclidean distance on the grid. In: Bodendiek, R., Henn, R. (eds.) Topics in Graph Theory and Combinatorics, pp. 555–559. Physica-Verlag, Heidelburg (1990)
Schmidt, W.M.: Irregularities of distribution, VII. Acta Arithm. 21, 45–50 (1972)
Schmidt, W.M.: Lectures on Irregularities of Distribution. Tata Inst. Fund. Res., Bombay (1977)
Schwarzkopf, O.: The extensible drawing editor Ipe. In: Proc. 11th Ann. ACM Symp. Comput. Geom. (SoCG’95), pp. C10–C11 (1995)
Sugihara, K.: Robust geometric computation based on topological consistency. In: Alexandrov, V.N., Dongarra, J., Juliano, B.A., Renner, R.S., Tan, C.J.K. (eds.) Proc. Internat. Conf. Computational Science, Part 1 (ICCS’01). Lecture Notes in Comput. Sci., vol. 2073, pp. 12–26. Springer, Berlin (2001)
van der Corput, J.: Verteilungsfunktionen I & II. Nederl. Akad. Wetensch. Proc. 38, 813–820, 1058–1066 (1935)
Wu, X.: Efficient algorithms for the optimal-ratio region detection problems in discrete geometry with applications. In: Proc. 17th Internat. Symp. Algorithms and Computation (ISAAC’06). Lecture Notes in Comput. Sci., vol. 4288, pp. 289–299. Springer, Berlin (2006)
Wu, X., Chen, D.Z.: Optimal net surface problems with applications. In: Proc. 29th Internat. Coll. Automata, Languages and Programming (ICALP’02). Lecture Notes in Comput. Sci., vol. 2380, pp. 1029–1042. Springer, Berlin (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
Martin Nöllenburg is supported by the German Research Foundation (DFG) under grant WO 758/4-3.
Rights and permissions
About this article
Cite this article
Chun, J., Korman, M., Nöllenburg, M. et al. Consistent Digital Rays. Discrete Comput Geom 42, 359–378 (2009). https://doi.org/10.1007/s00454-009-9166-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-009-9166-2