Performance evolution of a fractal dimension estimated by an escape time algorithm
Résumé
The non-geometric and irregular objects are considered as complex patterns. The geometric complexity is measured as space lling capacity by a factor known as a fractal dimension. Dierent techniques are proposed to nd this complexity measure according to the properties of the pattern. This paper is aimed to introduce a method for counting the dimension of the lled Julia fractal set generated by the Escape Time Algorithm using the method of spreading the points inside the proposed window. The resulted dimension is called Escape Time dimension. A new method to compute a correlation dimension of the Filled Julia fractal set is also proposed based on the Grassberger-Procaccia algorithm by computing the correlation function. A log-log graph of the correlation function versus the distances between every pair of points in the lled Julia fractal set is an approximation of the correlation dimension. Finally, a comparison between these two fractal dimensions of the led Julia fractal set which is generated by the Escape Time Algorithm is presented to show the efficiency of the proposed method.
Téléchargements
Références
Barnsley, M. F. 2014. Fractals everywhere. Academic press.
Falconer, K. 2004. Fractal geometry: mathematical foundations and applications. John Wiley & Sons.
https://doi.org/10.1002/0470013850
Al-Saidi, N. M., Mohammed, A. J., & Ahmed, A. M. 2014. Fractal Fourier transforms based image authentication. In Proceedings of the World Congress on Engineering (Vol. 2)
Grassberger, P., & Procaccia, I. 1983. Measuring the strangeness of strange attractors.Physica D: Nonlinear Phenomena, 9(1-2), 189-208. https://doi.org/10.1016/0167-2789(83)90298-1
Mohammed, A. J. (2005)."On Dimensions of Some Fractals", Ph.D. thesis, Mustansiriyah University.
Al-Saidi, N. M., & Mohammed, A. J. 2012. A new approach for computing multi-fractal dimension based on Escape Time Method.Int. Journal of Math. Analysis,616, 761-773.
Al-Shameri, W. F. H. & Mohammed, A. J., 2012. Approximating the Correlation Dimension of the Fractal Attractor of Iterated Function System.Al-Mustansiriyah Journal of Science,23(3), 157-172.
Mohammed, A. J., & Mohammed, N.A, 2016. Estimation Correlation Dimension of the Filled Julia Set Generated by Escape Time Algorithm. Mathematics and statistic Journal, 2(3), 29-36.
Sisson, P. D. 2007. Fractal art using variations on escape time algorithms in the complex plane.Journal of Mathematics and the Arts,1(1), 41-45. https://doi.org/10.1080/17513470701210485
Liu, S., Che, X., & Wang, Z. 2011. Improvement of escape time algorithm by no-escape-point. Journal of Computers , 6(8), 1648-1653. https://doi.org/10.4304/jcp.6.8.1648-1653
Liu, S., Fu, W., Deng, H., Lan, C., & Zhou, J. 2013. Distributional fractal creating algorithm in parallel environment. International Journal of Distributed Sensor Networks,9(9), 281707. https://doi.org/10.1155/2013/281707
Liu, M., Liu, S., Fu, W., & Zhou, J. 2015. Distributional escape time algorithm based on generalized fractal sets in cloud environment.Chinese Journal of Electronics, 24(1), 124-127. https://doi.org/10.1049/cje.2015.01.020
Aslan, N., Saltan, M., & Demir, B.2018. A different construction of the classical fractals via the escape time algorithm. Journal of Abstract and Computational Mathemtics,3(4) , 1-15.
Al-Saidi, N. M., Mohammed, A. J., & Ahmed, A. M. 2014. Fuzzy fractal dimension based on escape time algorithm.Applied Mathematical Sciences, 8(3), 117-129. https://doi.org/10.12988/ams.2014.311647
Mohammed, A. J. (2019, April). A comparison of fractal dimension estimations for filled Julia fractal sets based on the escape time algorithm. In AIP Conference Proceedings(Vol. 2086, No. 1, p. 030026). AIP Publishing
https://doi.org/10.1063/1.5095111
Sun, Y., Zhao, X., & Hou, K. 2013. Calculation of Julia sets by equipotential point algorithm. International Journal of Bifurcation and Chaos, 23, (01), 1350015. https://doi.org/10.1142/S0218127413500156
Copyright (c) 2019 Boletim da Sociedade Paranaense de Matemática
Ce travail est disponible sous la licence Creative Commons Attribution 4.0 International .
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
