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A Generalization of Magic Squares with Applications to Digital Halftoning

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Algorithms and Computation (ISAAC 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3341))

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Abstract

A semimagic square of order n is an n × n matrix containing the integers 0, ..., n 2–1 arranged in such a way that each row and column add up to the same value. We generalize this notion to that of a zero k × k-discrepancy matrix by replacing the requirement that the sum of each row and each column be the same by that of requiring that the sum of the entries in each k × k square contiguous submatrix be the same. We show that such matrices exist if k and n are both even, and do not if k and n are are relatively prime. Further, the existence is also guaranteed whenever n = k m, for some integers k,m ≥ 2. We present a space-efficient algorithm for constructing such a matrix.

Another class that we call constant-gap matrices arises in this construction. We give a characterization of such matrices.

An application to digital halftoning is also mentioned.

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References

  1. Aggarwal, A., Klawe, M., Moran, S., Shor, P., Wilber, R.: Geometric Applications of a Matrix Searching Algorithm. In: Proc. 2nd ACM Symposium on Computational Geometry, pp. 285–292 (1986)

    Google Scholar 

  2. Asano, T., Katoh, N., Obokata, K., Tokuyama, T.: Combinatorial and geometric problems related to digital halftoning. In: Asano, T., Klette, R., Ronse, C. (eds.) Geometry, Morphology, and Computational Imaging. LNCS, vol. 2616, pp. 58–71. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  3. Asano, T.: Digital Halftoning: Algorithm Engineering Challenges. IEICE Trans. on Inf. and Syst., E86-D 2, 159–178 (2003)

    Google Scholar 

  4. Asano, T., Obokata, K., Katoh, N., Tokuyama, T.: Matrix rounding under the L p -discrepancy measure and its application to digital halftoning. In: Proc. ACM-SIAM Symposium on Discrete Algorithms, San Francisco, pp. 896–904 (2002)

    Google Scholar 

  5. Bayer, B.E.: An optimum method for two-level rendition of continuous-tone pictures. In: Conference Record, IEEE International Conference on Communications, vol. 1, pp. (26-11)–(26-15) (1973)

    Google Scholar 

  6. Chazelle, B.: The Discrepancy Method: Randomness and Complexity. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  7. Floyd, R.W., Steinberg, L.: An adaptive algorithm for spatial gray scale. In: SID 75 Digest, Society for Information Display, pp. 36–37 (1975)

    Google Scholar 

  8. Heinz, H.D.: Magic Squares, Magic Stars & Other Patterns. web site, http://www.geocities.com/CapeCanaveral/Launchpad/4057/

  9. Graham, R.L., Lubachevsky, B.D., Nurmela, K.J., Östergård, P.R.J.: Dense packings of congruent circles in a circle. Discrete Math 181, 139–154 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  10. Knuth, D.E.: The Art of Computer Programming. In: Fundamental Algorithms, 3rd edn., vol. 1, Addison-Wesley, Reading (1997)

    Google Scholar 

  11. Kodera Ed.: Practical Design and Evaluation of Halftoned Images (in Japanese) Trikepps (2000)

    Google Scholar 

  12. Matoušek, J.: Geometric Discrepancy. Springer, Heidelberg (1991)

    Google Scholar 

  13. Mitsa, T., Parker, K.J.: Digital halftoning technique using a blue-noise mask. J. Opt. Soc. Am. 9(11), 1920–1929 (1992)

    Article  Google Scholar 

  14. Nurmela, K.J., Östergård, P.R.J.: Packing up to 50 Equal Circles in a Square. Discrete Comput. Geom. 18, 111–120 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ulichney, R.A.: Dithering with blue noise. Proc. IEEE 76(1), 56–79 (1988)

    Article  Google Scholar 

  16. Ulichney, R.: The void-and-cluster method for dither array generation. In: Allebach, J.W. (ed.) IS&T/SPIE Symposium on Electronic Imaging Science and Technology, Proceedings of Conf. Human Vision, Visual Processing and Digital Display IV. SPIE, pp. 332–343 (1993)

    Google Scholar 

  17. Yao, M., Parker, K.J.: Modified approach to the construction of a blue noise mask. J. Electronic Imaging 3, 92–97 (1994)

    Article  Google Scholar 

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© 2004 Springer-Verlag Berlin Heidelberg

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Aronov, B., Asano, T., Kikuchi, Y., Nandy, S.C., Sasahara, S., Uno, T. (2004). A Generalization of Magic Squares with Applications to Digital Halftoning. In: Fleischer, R., Trippen, G. (eds) Algorithms and Computation. ISAAC 2004. Lecture Notes in Computer Science, vol 3341. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30551-4_10

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  • DOI: https://doi.org/10.1007/978-3-540-30551-4_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-24131-7

  • Online ISBN: 978-3-540-30551-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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