This Article 
 Bibliographic References 
 Add to: 
A New Parameterization of Digital Straight Lines
August 1991 (vol. 13 no. 8)
pp. 847-852

A 1:1 correspondence is established between digital straight lines which start at a fixed point and a simple set of quadruples of integer parameters. Such a representation by parameters is useful for enumeration, First, the authors show a 1:1 correspondence between point pairs in a planar set of points and the linear dichotomies of this set. Then, from the equivalence between digital lines and linear dichotomies of points on the digitization grid, they prove a 1:1 and 'onto' correspondence between digital straight lines starting at a fixed point and a well-defined set of pairs of grid points. It follows that four parameters uniquely represent any given digital line with a fixed starting point. An O(N) algorithm is given for determining the parameters from the digital line, as well as O(log N) algorithms for transforming between these parameters and the parameters suggested by L. Dorst and A.W.M. Smeulders (1984).

[1] A. Rosenfeld, "Digital straight line segments,"IEEE Trans. Comput., vol. C-23, pp. 1264-1269, 1974.
[2] C. E. Kim and A. Rosenfeld, "Digital straight lines and convexity of digital regions,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-4, pp. 149-153, 1982.
[3] L. D. Wu, "On the chain code of a line,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-4, pp. 347-353, 1982.
[4] L. Dorst and A. W. M. Smeulders, "Discrete representation of straight lines,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-6, pp. 450-463, 1984.
[5] S. H. Y. Hung, "On the straightness of digital arcs,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-7, pp. 203-215, 1985.
[6] C. A. Berenstein and D. Lavine, "An enumeration problem in digital geometry,"IEEE Trans. Pattern Anal. Machine Intell., vol. 10, pp. 880-887, 1988.
[7] C. A. Berenstein, L. N. Kanal, D. Lavine, and E. Olson, "A geometric approach to subpixel registration accuracy,"Comput. Vision Graphics Image Processing, vol. 40, no. 3, pp. 334-360, 1987.
[8] M. D. McIlroy, "A note on discrete representation of lines,"AT&T Tech. J., vol. 64, no. 2, pp. 481-490, Feb. 1984.
[9] J. Koplowitz, M. Lindenbaum, and A. Bruckstein, "On the number of digital straight lines on a square grid,"IEEE Trans. Inform. Theory, vol. 36, pp. 192-197, 1990.
[10] M. Lindenbaum and J. Koplowitz, "Compression of chain codes using digital straight lines sequences,"Patt. Recognition Lett., vol. 7, pp. 167-171, 1988.
[11] F. P. Preparata and M. I. Shamos,Computational Geometry, an Introduction. New York: Springer-Verlag, 1985.
[12] T. Pavlidis,Algorithms for Graphics and Image Processing. Rockville, MD: Computer Science Press, 1982.
[13] G. H. Hardy and E. M. Wright,An Introduction to the Theory of Numbers. Oxford: Oxford University Press, 1979.
[14] D. E. Knuth,The Art of Computer Programming, Vol. 2, Seminumerical Algorithms. Reading, MA: Addison-Wesley, 1981.
[15] G. T. Toussaint and D. Avis, "On a convex hull algorithm for polygons and its application to triangulation algorithms,"Patt. Recognition, vol. 15, no. 1, pp. 23-29, 1982.
[16] G. T. Toussaint, "Solving geometric problems with a rotating calipers," inProc. MELECON'83(Athens, Greece), 1983.
[17] H. Rohnert, "Shortest paths in the planes with convex polygonal obstacles,"Inform. Processing Lett., vol. 23, pp. 71-76, Aug. 1986.
[18] J. Koplowitz and A. P. Sundar Raj, "A robust filtering algorithm for subpixel reconstruction of chain coded line drawings,"IEEE Trans. Patt. Anal. Machine Intell., vol. PAMI-9, pp. 451-457, 1987.

Index Terms:
computational geometry; computational complexity; parameterization; digital straight lines; 1:1 correspondence; point pairs; linear dichotomies; equivalence; computational complexity; computational geometry
M. Lindenbaum, J. Koplowitz, "A New Parameterization of Digital Straight Lines," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 13, no. 8, pp. 847-852, Aug. 1991, doi:10.1109/34.85678
Usage of this product signifies your acceptance of the Terms of Use.