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On Recursive, O(N) Partitioning of a Digitized Curve into Digital Straight Segments
September 1993 (vol. 15 no. 9)
pp. 949-953

A simple online algorithm for partitioning of a digital curve into digital straight-line segments of maximal length is given. The algorithm requires O(N) time and O(1) space and is therefore optimal. Efficient representations of the digital segments are obtained as byproducts. The algorithm also solves a number-theoretical problem concerning nonhomogeneous spectra of numbers.

[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. Patt. Anal. Machine Intell., vol. PAMI-4, pp. 149-153, 1982.
[3] L. D. Wu, "On the chain code of a line,"IEEE Trans. Patt. 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. Patt. Anal. Machine Intell., vol. PAMI-6, pp. 450-463, 1984.
[5] S. H. Y. Hung, "On the straightness of digital arcs,"IEEE Trans. Patt. Anal. Machine Intell., vol. PAMI-7, pp. 203-215, 1985.
[6] L. Dorst and A. W. M. Smeulders, "Best linear unbiased estimators for properties of digitized straight lines,"IEEE Trans. Pattern Anal. Machine Intell., vol. 8, pp. 276-282, Mar. 1986.
[7] 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.
[8] 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.
[9] M. Lindenbaum and J. Koplowitz, "Compression of chain codes using digital straight lines sequences,"Patt. Recognition Lett., vol. 7, pp. 167-171, 1988.
[10] M. Werman, A. Y. Wu, and R. A. Melter, "Recognition and characterization of digitized curves,"Patt. Recogn. Lett., vol. 5, pp. 207-213, 1987.
[11] M. D. McIlroy, "A note on discrete representation of lines,"AT&T Tech. J., vol. 64, no. 2, pp. 481-490, 1984.
[12] M. Lindenbaum and J. Koplowitz, "A new parametrization of digital straight lines,"IEEE Trans. Patt. Anal. Machine Intell., vol. 13, pp. 847-852, 1991.
[13] A. Bruckstein, "The selfsimilarity of digital straight lines," inVision Geometry(Melter, Rosenfeld, and Bhattacharya, Eds.). Contemporary Mathematics, AMS, 1991, pp. 1-20, vol. 119.
[14] 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.
[15] C. Ronse, "A bibliography on digital and computational convexity (1961-1988),"IEEE Trans. Patt. Anal Machine Intell., vol. 11, pp. 181-190, 1989.
[16] B. A. Venkov,Elementary Number Theory, (translated and edited by H. Anderson). Groningen: Wolters-Noordhoff, 1970.
[17] K. Stolarsky, "Beatty sequences, continued fractions, and certain shift operators,"Canadian Math. Bull., vol. 19, no. 4, pp. 473-482, 1976.
[18] A. S. Fraenkel, M. Mushkin, and U. Tassa, "Determination of [nθ] from its sequence of differences,"Canadian Math. Bull., vol. 21, no. 4, pp. 441-446, 1978.
[19] R. L. Graham, S. Lin, and C. S. Lin, "Spectra of numbers,"Math. Mag., vol. 51, pp. 174-176, 1978.
[20] M. Boshernitzan and A. S. Fraenkel, "A linear algorithm for nonhomogeneous spectra of numbers,"J. Algorithms, vol. 5, pp. 187-198, 1984.
[21] A. W. M. Smeulders and L. Dorst, "Decomposition of discrete curves into piecewise straight segments in linear time," inVision Geometry(Melter, Rosenfeld and Bhattacharya, Eds.). Contemporary Mathematics, AMS, 1991, pp. 169-195, vol. 119.
[22] G. H. Hardy and E. M. Wright ,An Introduction to the Theory of Numbers. Oxford: Oxford University Press, 1979.
[23] F. P. Preparata and M. I. Shamos,Computational Geometry, an Introduction. New York: Springer-Verlag, 1985.
[24] D. E. Knuth,The Art of Computer Programming, Vol. 2, Seminumerical Algorithms. Reading, MA: Addison-Wesley, 1981.
[25] L. Dorst and A. W. M. Smeulders, "Length estimators for digital contours,"Comput. Vision Graphics Image Processing, vol. 40, pp. 311-333, 1987.
[26] J. O'Rourke, "An on-line algorithm for fitting straight lines between data ranges,"Commun. ACM, vol. 24, no. 9, pp. 574-578, 1981.

Index Terms:
recursive partitioning; digitized curve segmentation; online algorithm; number-theoretical problem; nonhomogeneous spectra; computational complexity; image segmentation; number theory
Citation:
M. Lindenbaum, A. Bruckstein, "On Recursive, O(N) Partitioning of a Digitized Curve into Digital Straight Segments," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 15, no. 9, pp. 949-953, Sept. 1993, doi:10.1109/34.232082
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