Binary States Cellular Automata with Reflexive and Periodic Boundaries and Image problem

Year 2018, Volume: 9, 14 - 24, 28.12.2018

Shovkat Redjepov Selman Uguz

Abstract

The present paper focuses on the theory of two-dimensional (2D) linear cellular automata (CA) with
respect to uniform reflexive and periodic boundary conditions. It is investigated the theoretical aspects of 2D linear
CA over binary states field with image problem. We consider geometrical and visual aspects of images generated
by these CA transition rules. Multiple copies of any arbitrary images corresponding to CA can be studied further
by considering theses transition rules of von Neumann and Moore CAs. An important note that these special types
of CAs can be applied many different special problems e.g. computability theory, applied mathematics, theoretical
chemistry and biology, DNA and genetics research, image science, textile design.  

Keywords

Cellular automata, Binary states, Reflexive and periodic boundary

References

• Choudhury, P.P., Nayak, K., Sahoo, S., Rath, P., Theory and applications of two-dimensional, null-boundary, nine-neighborhood, cellular automata linear rules, Journal of Current Computer Science and Technology, 2(2012), 50–54. arXiv:0804.2346.
• Choudhury, P.P., Sahoo, S., Chakraborty, M., Bhandari, S., Pal, A., Investigation of the global dynamics of cellular automata using Boolean derivatives, Computers and Mathematics with Applications, 57(2009), 1337-1351. arXiv:0804.2346.
• Choudhury, P.P., Sahoo, S., Hassan, S. S., Basu, S., Ghosh, D., Kar, D., Ghosh, Ab., Ghosh, Av., Ghosh A.K., Classification of cellular automata rules based on their properties, Int. J. of Comp. Cogn., 8(2010), 50–54.
• Chou H.H., Reggia J.A., Emergence of self-replicating structures in a cellular automata space, Phys. D, 110(1997), 252–276.
• Dihidar K., Choudhury P.P., Matrix algebraic formulae concerning some exceptional rules of two dimensional cellular automata, Inf. Sci., 165(2004), 91–101.
• Mohammed, J., Mohanty, B., Sahoo, S., Two dimensional cellular automata and its reduced rule matrix, Int. Journal of The Orissa Mathematical Society, 1(2012), 97-104.
• Mohammed, J., Sahoo, S., Design and analysis of matrices for two dimensional cellular automata linear rules in hexagonal neighborhood, Mathematica Aeterna, 1(2011), 537–545.
• Mohammed, J., Nayak, D.R., An ecient edge detection technique by two dimensional rectangular cellular automata, Information Communication and Embedded Systems (ICICES) (2014).
• Nayak, D.R., Sahu, S.K., Mohammed, J., A cellular automata based optimal edge detection technique using twenty-five neighborhood model, arXiv:1402.1348.
• Nayak, D.R., Patra,P.K., Mahapatra, A., A survey on two dimensional cellular automata and its application in image processing, arXiv:1407.7626. Padhan, U., Tripathy, S. K., Sahoo, S., Nayak, B. K. Jena, O.P., Sweeper’s algorithm and its application on image clustering, International Journal of Computer Applications, 119 (2015), 30–37.
• Pattanaik, S.R., Nayak, B.K., Sahoo, S., Application of two-dimensional periodic cellular automata in image processing, Int. Journal of Computer, Mathematical Sciences and Applications, 5(2011), 49–55.
• Sahin, U., Sahin, F., Uguz, S., Hybridized fuzzy cellular automata thresholding algorithm for edge detection optimized by PSO, High Capacity Optical Networks and Enabling Technologies (HONET-CNS), 10th International Conference IEEE, (2013), 228–232.
• Sahin U., Uguz S., Akin H., The transition rules of 2D linear cellular automata over ternary field and self- replicating patterns, Int. J. Bif. and Chaos, 25(2015), 1550011.
• Sahin U., Uguz S., Akin H., Siap, I., Three-state von Neumann cellular automata and pattern generation, App. Math. Mod., 39(2015), 2003–2024.
• Sahin, U., Uguz, S., Sahin F., Salt and pepper noise filtering with fuzzy-cellular automata, Comp. and Elect. Eng., 40(2014), 59–69.
• Sahoo, S., Choudhury, P.P., Pal, A., Nayak, B.K., Solutions on 1-D and 2-D density classification problem using programmable cellular automata, J. Cellular Automata, 9(2014), 59–88.
• Sahoo, S., Sahoo, S.K., Nayak, B.K., Choudhury, P.P., Encompression using two-dimensional cellular automata rules, arXiv: 0808.1470.
• Sahoo, S., Choudhury, P.P., Chakraborty, M., Nayak, B.K., Characterization of any non-linear Boolean function using a set of linear operators, Journal of Orissa Mathematical Society, 29 (2010), 111–133.
• Siap, I., Akin, H., Uguz, S., Structure and reversibility of 2D hexagonal cellular automata, Comput. Math. App., 62(2011), 4161–4169.
• Uguz, S., Akin H., Siap I., Reversibility algorithms for 3-state hexagonal cellular automata with periodic boundaries, Int. J. Bif. and Chaos, 23(2013), 1350101-1-15.
• Uguz, S., Sahin , U., Akin, H., Siap I., Self-replicating patterns in 2D linear cellular automata, Int. J. Bif. and Chaos, 24(2014), 1430002.
• Uguz, S., Sahin , U., Akin, H., Siap I., 2D cellular automata with an image processing application, Acta Physica Polonica A, 125(2014), 435–438.
• Uguz, S., Sahin, U., Sahin, F., Edge detection with fuzzy cellular automata transition function optimized by PSO, Comp. and Elect. Eng., 43(2015), 180-192.
• Uguz, S., Akin H., Siap, I., Sahin, U., On the irreversibility of Moore cellular automata over the ternary field and image application, App. Math. Mod., 40(2016), 8017–8032.
• Uguz, S., Redjepov, S., Acar E., Akin H., Structure and reversibility of 2D von Neumann cellular automata over triangular lattice, Int. J. Bif. and Chaos, 27(2017), 1750083.
• von Neumann, J., The Theory of Self-Reproducing Automata, (Edited by A. W. Burks), Univ. of Illinois Press, Urbana, 1966.
• Wolfram, S., Cellular automata , Rev. Mod. Phys., 55(1983), 601–644.