Generating irregular fractals based on iterated function systems

Baoxing Zhang; Yunkun Zhang; Yuanyuan Xie; Baoxing Zhang; Yunkun Zhang; Yuanyuan Xie

doi:10.3934/math.2024651

AIMS Mathematics

2024, Volume 9, Issue 5: 13346-13357. doi: 10.3934/math.2024651

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Generating irregular fractals based on iterated function systems

1.
College of Statistics and Mathematics, Hebei University of Economics and Business, Shijiazhuang 050061, China
2.
Institute of Science and Technology, Tianjin University of Finance and Economics, Tianjin 300222, China

Received: 13 February 2024 Revised: 01 April 2024 Accepted: 07 April 2024 Published: 11 April 2024
MSC : 28A80, 47J26

The iterated function system (IFS) is important in different fields like image compression. An important feature of such systems is that they can be used to generate fractals. Yet, for the obtained fractals, it is difficult to locally control them to generate new ones with desired structures at specific places. In this paper, we gave an attempt to solve this problem based on a nonuniform multiple function system. For this, we first analyzed the multiple function systems needed in the generation of the final desired fractals. Based on such analysis, the final fractals with desired structures at specific places can be generated using the nonuniform multiple function system. Moreover, these two procedures were summarized into two algorithms for convenience. Examples were also given to illustrate the performance of the nonuniform multiple function system and the two algorithms in this paper.
- iterated function system,
- local control,
- fractal,
- subdivision,
- multiple function system
Citation: Baoxing Zhang, Yunkun Zhang, Yuanyuan Xie. Generating irregular fractals based on iterated function systems[J]. AIMS Mathematics, 2024, 9(5): 13346-13357. doi: 10.3934/math.2024651

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Abstract

The iterated function system (IFS) is important in different fields like image compression. An important feature of such systems is that they can be used to generate fractals. Yet, for the obtained fractals, it is difficult to locally control them to generate new ones with desired structures at specific places. In this paper, we gave an attempt to solve this problem based on a nonuniform multiple function system. For this, we first analyzed the multiple function systems needed in the generation of the final desired fractals. Based on such analysis, the final fractals with desired structures at specific places can be generated using the nonuniform multiple function system. Moreover, these two procedures were summarized into two algorithms for convenience. Examples were also given to illustrate the performance of the nonuniform multiple function system and the two algorithms in this paper.

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