Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T15:18:22.005Z Has data issue: false hasContentIssue false
  • English
  • Français

CHARACTERIZING DIGITAL STRAIGHTNESS AND DIGITAL CONVEXITY BY MEANS OF DIFFERENCE OPERATORS

Part of: Difference equations Discrete geometry Special classes of linear operators Difference and functional equations

Published online by Cambridge University Press:  31 May 2011

Christer O. Kiselman
Christer O. Kiselman*
Affiliation:
Department of Mathematics, Uppsala University, P.O. Box 480, SE-751 06 Uppsala, Sweden (email: kiselman@math.uu.se, christer@kiselman.eu)
Get access

Abstract

We characterize straightness of digital curves in the integer plane by means of difference operators. Earlier definitions of digital rectilinear segments have used, respectively, Rosenfeld’s chord property, word combinatorics, Reveillès’ double Diophantine inequalities, and the author’s refined hyperplanes. We prove that all these definitions are equivalent. We also characterize convexity of integer-valued functions on the integers with the help of difference operators.

MSC classification

Secondary: 52C99: None of the above, but in this section 39A05: General theory 39A70: Difference operators 47B39: Difference operators
Type
Research Article
Information
Mathematika , Volume 57 , Issue 2 , July 2011 , pp. 355 - 380
Copyright
Copyright © University College London 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bédaride, N., Domenjoud, E., Jamet, D. and Rémy, J.-L., On the number of balanced words of given length and height over a two-letter alphabet. Discrete Math. Theor. Comput. Sci. 12(3) (2010), 41–62.Google Scholar
[2]Berthé, V., Discrete geometry and symbolic dynamics. In Complex Analysis and Digital Geometry. Proceedings from the Kiselmanfest, 2006 (Acta Universitatis Upsaliensis 86) (ed. Passare, M.), Uppsala University (Uppsala, 2009), 81–110, 364 pp; ISSN: 0502-7454, ISBN: 978-91-554-7672-4.Google Scholar
[3]Bruckstein, A. M., Self-similarity properties of digitized straight lines. In Vision Geometry (Hoboken, NJ, 1989) (Contemporary Mathematics 119), American Mathematical Society (Providence, RI, 1991), 1–20.Google Scholar
[4]Debled, I., Étude et reconnaissance des droites et plans discrets. Doctoral Thesis, Université Louis Pasteur, Strasbourg, 1995, 209 pp.Google Scholar
[5]Eckhardt, U., Digital lines and digital convexity. In Digital and Image Geometry (Lecture Notes in Computer Science 2243), Springer (Berlin, 2001), 209–228.CrossRefGoogle Scholar
[6]Heath, T. L., The Thirteen Books of Euclid’s Elements, Dover (New York, 1956), Volume I: Introduction and Books I, II, Translated from the text of Heiberg, with Introduction and Commentary by Sir Thomas L. Heath.Google Scholar
[7]Hübler, A., Klette, R. and Voss, K., Determination of the convex hull of a finite set of planar points within linear time. Elektron. Informationsverarb. Kybernet 17(2–3) (1981), 121–139.Google Scholar
[8]Hung, S. H. Y. and Kasvand, T., On the chord property and its equivalences, Proceedings of the 7th International Conference on Pattern Recognition (1984), 116–119.Google Scholar
[9]Kim, C. E. and Rosenfeld, A., Digital straight lines and convexity of digital regions. IEEE Trans. Pattern Anal. Mach. Intell. PAMI-4(2) (1982), 149–153.CrossRefGoogle Scholar
[10]Kiselman, C. O., Convex functions on discrete sets. In Combinatorial Image Analysis, 10th International Workshop, IWCIA 2004, Auckland, New Zealand, 1–3 December 2004 (Lecture Notes in Computer Science 3322) (eds Klette, R. and Žunić, J.), Springer (Berlin, 2004), 443–457.Google Scholar
[11]Kiselman, C. O. and Samieinia, S., Convexity of marginal functions in the discrete case. In [23, Paper VI], 2010.Google Scholar
[12]Klein, F., Ueber eine geometrische Auffassung der gewöhnliche Kettenbruchentwicklung. Nachrichten von der Königl. Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1895), 357–359. Available at: www.digizeitschriften.de.Google Scholar
[13]Klette, R. and Rosenfeld, A., Digital Geometry: Geometric Methods for Digital Picture Analysis, Elsevier (Amsterdam, 2004).Google Scholar
[14]Lothaire, M., Algebraic Combinatorics on Words, Cambridge University Press (Cambridge, 2002).CrossRefGoogle Scholar
[15]Morse, M. and Hedlund, G. A., Symbolic dynamics II. Sturmian trajectories. Amer. J. Math. 62(1) (1940), 1–42.CrossRefGoogle Scholar
[16]Murota, K., Discrete Convex Analysis, Society for Industrial and Applied Mathematics (SIAM) (Philadelphia, PA, 2003).CrossRefGoogle Scholar
[17]Pytheas Fogg, N., Substitutions in Dynamics, Arithmetics and Combinatorics (Lecture Notes in Mathematics 1794), Springer (Berlin, 2002).CrossRefGoogle Scholar
[18]Reveillès, J.-P., Géométrie discrète, calcul en nombres entiers et algorithmique. Thèse d’État, Université Louis Pasteur, Strasbourg, 1991.Google Scholar
[19]Rosenfeld, A., Digital straight line segments. IEEE Trans. Comput. c-32(12) (1974), 1264–1269.CrossRefGoogle Scholar
[20]Rosenfeld, A. and Klette, R., Digital straightness. Electron. Notes Theor. Comput. Sci. 46 (2001), 32.CrossRefGoogle Scholar
[21]Samieinia, S., Digital straight line segments and curves. Licentiate Thesis presented at Stockholm University 2007-09-25 (2007), www2.math.su.se/reports/2007/6/.Google Scholar
[22]Samieinia, S., Chord properties of digital straight line segments. Math. Scand. 107 (2010), 1–27.Google Scholar
[23]Samieinia, S., Digital geometry, combinatorics, and discrete optimization. Doctoral Thesis defended 2011-01-21. Department of Mathematics, Stockholm University, Stockholm, 2010.Google Scholar
[24]Uscka-Wehlou, H., Two equivalence relations on digital lines with irrational slopes. A continued fraction approach to upper mechanical words. Theoret. Comput. Sci. 410(38–40) (2009), 3655–3669.CrossRefGoogle Scholar
[25]Uscka-Wehlou, H., Digital lines, Sturmian words, and continued fractions. Doctoral Thesis defended 2009-09-25, Uppsala University, Uppsala, 2009.Google Scholar
[26]Voss, K., Coding of digital straight lines by continued fractions. Comput. Artif. Intell. 10(1) (1991), 75–80.Google Scholar
[27]Vuillon, L., Balanced words. Bull. Belg. Math. Soc. Simon Stevin 10(suppl.) (2003), 787–805.CrossRefGoogle Scholar