Translation of a Digital Line into another according to various Digitization Processes

Leroy, J. ;  Rigo, M. ;  Charlier, E. (éd.)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Special issue dedicated to the 16th "Journées Montoises d’Informatique Théorique", Tome 51 (2017) no. 4, pp. 169-180.

We introduce unusual methods for the digitization process of a line. A square pixel of the computer screen is blackened when the line crosses a special part of this pixel, called the active pixel. The shape of this active pixel is discussed, in the following sense: can we obtain the new Freeman Code of the line, using of a mechanical transformation of the initial Freeman Code, which is the classical Cutting Sequence? Our choice is to limit mechanical transformations to the existence of a given transducer, so that everytime we put in (a power of) the classical Freeman Code of a line, the output recovers the new Freeman Code. Then we prove that such a transducer exists if and only if the active pixel is a polygon with rational vertices and big enough. The same result can be proved if we introduce several grey levels in the representation of the line. Then we get some antialising effects.

Reçu le :
Accepté le :
DOI : 10.1051/ita/2017016
Classification : 68R15, 68Q68
Mots-clés : Digital lines, digitization processes, Freeman codes, cutting sequences
Borel, Jean-Pierre 1

1 
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     editor = {Leroy, J. and Rigo, M. and Charlier, E.},
     title = {Translation of a {Digital} {Line} into another according to various {Digitization} {Processes}},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
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     publisher = {EDP-Sciences},
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Borel, Jean-Pierre. Translation of a Digital Line into another according to various Digitization Processes. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Special issue dedicated to the 16th "Journées Montoises d’Informatique Théorique", Tome 51 (2017) no. 4, pp. 169-180. doi : 10.1051/ita/2017016. http://www.numdam.org/articles/10.1051/ita/2017016/

[1] J.-P. Allouche, J. Shallit, Automatic Sequences: Theory and Applications. Cambridge University Press, Cambridge (2003). | DOI | MR | Zbl

[2] J. Berstel, Tracé de droites, fractions continues et morphismes réels, in M. Lothaire, “Mots”, Mélanges offerts á M.P. Schutzenberger, Hermès (1990). | MR

[3] J. Berstel and P. Seébold, Sturmian words, in Algebraic Combinatorics on Words, edited by M. Lothaire. Cambridge University Press (2002). | MR

[4] J.-P. Borel and C. Reutenauer, Palindromic factors of billiard words. Theor. Comput. Sci. 340–342 (2005) 334–348. | DOI | MR | Zbl

[5] J.-P. Borel, How to build Billiard Words using Decimations. RAIRO: ITA 44 (2010) 59–77. | Numdam | MR | Zbl

[6] J.E. Bresenham, Algorithm for computer control of a digital plotter. IBM Syst. J. 4 (1965) 25–30. | DOI

[7] D. Crisp, W. Moran, A. Pollington and P. Shive, Substitution invariant cutting sequences. J. Théorie des Nombres Bordeaux 5 (1993) 123–137. | DOI | Numdam | MR | Zbl

[8] J.D. Foley and A. Van Dam, Fundamentals of Interactive Computer Graphics. Addison-Wesley, Cambridge (1982).

[9] H. Freeman, On the encoding of arbitrary geometric configuration. IRE Trans. Electron. Comput. 10 (1961) 260–268. | DOI | MR

[10] G. Hegron, Synthèse d’images : algorithmes élémentaires. Dunod, Paris (1985).

[11] J. Justin and G. Pirillo, Decimations and Sturmian words. Theor. Inform. Appl. 31 (1997) 271–290. | DOI | Numdam | MR | Zbl

[12] J. Koplowitz, On the performance of chain codes for quantization of line drawings. IEEE Trans Pattern Anal. Machine Intell. PAMI-3 (1981) 357–393. | DOI | Zbl

[13] W.M. Newman and R.F. Sproull, Principles of Interactive Computer Graphics. McGraw-Hill (1985).

[14] M.L.V. Pitteway and R.A. Earnshax, in Euclid’s Algorithm and Line Drawing, Fundamental Algorithms in Computer Graphics. Springer Verlag (1985) 101–105.

[15] J.-P. Reveilles, Géométrie discréte, calcul en nombres entiers et algorithmique. Thesis, Univ. Louis Pasteur – Strasbourg, France (1991). | Zbl

[16] A. Rosenfeld, Digital straight line segments. IEEE Trans. Comput. 32 (1974) 1264–1269. | DOI | MR | Zbl

[17] C. Series,The geometry of Markoff numbers. Math. Intell. 7 (1985) 20–29. | DOI | MR | Zbl

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