# Application of fractal algorithms of coastline echo’s generation on marine radar simulator

- Shuguang Ji
^{1}Email author, - Zhang Zhang
^{2}, - Hongbiao Yang
^{1}, - Dan Liu
^{3}and - Rapinder Sawhney
^{1}

**4**:8

**DOI: **10.1186/s40327-016-0037-7

© Ji et al. 2016

**Received: **22 October 2015

**Accepted: **10 March 2016

**Published: **12 May 2016

## Abstract

### Background

Marine radar simulator is a useful approach endorsed by International Maritime Organization (IMO) to train the seafarers on how to operate marine radar equipment and use marine radar equipment for positioning and collision avoidance in laboratory. To fulfill all of the marine radar simulator training requirements, a high performance simulator is necessary. However, imperfections with currently available marine radar simulators require simulator developers to make improvements.

### Case description

In this study, improved fractal algorithms (random Koch curve, fractional Brownian motion, and Weierstrass-Mandelbrot function) are applied to generate natural-looking radar echoes on a marine radar simulator.

### Discussion and evaluation

From the results of the simulations, we can observe that the structures of the coastline echoes generated by improved fractal algorithms, especially by fractional Brownian motion algorithm, outperform the echoes generated by conventional method in representing a natural coastline feature.

### Conclusions

Based on evaluations from a panel of experienced mariners, we conclude that the coastline echoes simulated by fractal algorithms better represent a natural coastline feature than those generated by conventional methods.

### Keywords

Marine radar simulator Coastline echo Fractal algorithm## Background

The International Convention on Standards of Training, Certification and Watchkeeping for Seafarers 78/95 (STCW Convention 78/95) of International Maritime Organization (IMO) requires using a marine radar simulator to train seafarers. A marine radar simulator is the only acceptable approach in the laboratory for seafarers to learn how to operate radar equipment and use radar for positioning, navigation, and anti-collision. Upon completing all requirements set forth in the training, the trainee will receive certification for qualifications to work on board. Currently, marine radar simulators are widely used by the members of IMO as one of many useful tools for seafarer education and training (Ali 2006; Organization 2006; Teel et al. 2009; Xiuwen et al. 2010). However, due to the limitation of the simulation technology, the marine radar simulators on the market are unable to replicate the performance of real radar equipment. Taking the simulation of coastline echoes as an example, the coastline echo is generated by raw data, which are acquired from the digitalized chart and consist of a collection of coordinate points. By connecting two adjacent coordinate points, a straight line is generated to approximate a real coastline echo. This method works well for generating a coastline echo under a large radar range (say 6 nautical miles (NM)). However, when the radar range is adjusted to a smaller radar range (such as 0.25 NM), the shape of coastline echo will lose its natural structure and look quite artificial (Ji et al. 2015; Zhang 2007). In addition, it should take around three seconds for the scan line of the marine radar to rotate a round. By adopting a traditional generation method for the coastline echo under small radar range, the time for the scan line to rotate around is much more likely to exceed three seconds, since extra sampling coordinate points (if available) have to be inserted to generate a high quality coastline echo. Imperfections like the artificial coastline shape and slow rotation of the radar scan line may have negative impacts on the training effectiveness for users (Ji et al. 2015; Zhang 2007). In order to overcome the problems associated with the conventional coastline echo simulation approach, we apply fractal theory to the coastline echo simulation process, since fractal theory is widely used as a graphics tool for generating natural-looking shapes like coastlines, rivers, mountains, and other natural features (Pentland 1984). The simulation results are evaluated and scored by 30 experienced mariners to validate that the coastline echoes generated by fractal algorithms look more natural than those generated by conventional method. Furthermore, an improved fractal algorithm is designed to guarantee the scan line can finish a round of rotation within three seconds, which is difficult to be achieved using the conventional method, especially under a larger radar range.

## Literature review

The STCW Convention provides required components for seafarer training, which use the radar simulator as a tool of training and assessment. These highlights include (Ali 2006; Organization 2006; Teel et al. 2009): factors affecting performance and accuracy; detection of misrepresentation of information, including false echoes and sea returns; setting up and maintaining displays; range and bearing; plotting techniques and relative motion concepts; identification of critical echoes; course and speed of other ships; time and distance of the closest approach to crossing, meeting or overtaking ships; detecting course and speed changes of other ships; effects on the changes of the own ship’s course or speed or both; and application of the International Regulations for Preventing Collisions at Sea. To fulfill all of these training requirements, a high performance marine radar simulator is needed. In the current marine radar simulator market, the major developing teams include Nautical Software (2016), Bridge Command (2016), Kongsberg Maritime (2016), Landfall (2016), and Dalian Maritime University Institute of Navigational Technology (2016). In addition, some previous research investigates methods to improve the marine radar simulator. For example, Arnold-Bos et al. developed a versatile bistatic and polarimetric marine radar simulator. In their simulator, realistic sea surfaces are generated using the two-scale model on a semi-deterministic basis, so as to incorporate the presence of ship wakes in the simulation (Arnold-Bos et al. 2006). Yin et al. designed a radar simulator using a PC to generate radar echoes and a radar interface board to generate radar signals. Their simulator has a more flexible and realistic operation interface than other simulators (Yin et al. 2007). Zhang et al. put forward a coastline echo intensity algorithm based on RGB and HIS color models and applied this algorithm on the marine radar simulator. The simulation results from this model are consistent with the electronic chart (Zhang et al. 2010).

In this study, we incorporate fractal theory, a branch of non-linear mathematics, to improve coastline echo simulation. The research targets of fractal theory are irregular objects and non-linear systems in the nature. The term “fractal”? was first used by mathematician Benoit Mandelbrot in 1975 to extend the concept of theoretical fractional dimensions to geometric patterns in nature (Mandelbrot 1983). In the 1980s, fractal theory was applied into the signal processing for radar because the echoes reflected into radar system have many fractal patterns (Ji et al. 2015; Zhang 2007). Even though fractal theory has been widely applied in fields such as virtual reality, image processing, and time series analysis, etc. (Ji et al. 2005; Zhang et al. 2005), there are few studies to apply it into the simulation of coastline echo for marine radar simulator. This research aims to close this gap. Partial of findings reported in this article were originally presented at the 94th Transportation Research Board Annual Meeting. We improved the research methodology in Ji et al. (2015) in this article. Especially, a full control of the physical parameters involved fractal function, Weierstrass-Mandelbrot function (WMF), is used to simulate coastline echoes. In addition, a quantitative validation of the simulation results is designed to assess the fidelity of the simulation outcomes and comprehend possible values of the introduced parameters among the simulation algorithms.

## Methods

The echo reflection on radar simulator can be classified into three types (Ji et al. 2015; Zhang 2007). A Type I Echo is the echo reflected by artificial architectures such as berth and breakwater. Type I Echoes have regular shape and can be used for positioning because of its clear boundaries and fixed position. A Type II Echo is the echo reflected by rocky coast. Type II Echoes have a realistic pattern as well as fixed position. A Type III Echo is an echo reflected by flat coast such as sand coast. Type III Echoes have a large echo reflection zone and relatively weak reflection. Additionally, the shape and position of a Type III Echo will change with the motion of the waves. This study focuses only on simulation of Type II Coastline Echoes because of its natural fractal features. Three different fractal algorithms are adopted to simulate coastline echoes. A comparison among these three simulation algorithms is included.

### Random Koch curve algorithm

### Fractional Brownian Motion algorithm

*∞*<

*t*<

*∞*and

*w*belongs to a set of values of a random function and if the intervals (

*t*

_{1},

*t*

_{2}) and (

*t*

_{3},

*t*

_{4}) do not overlap, the ordinary Brownian motion

*B*(

*t,w*) is a real random function with independent Gaussian increments. Therefore,

*B*(

*t*

_{2},

*w*)−

*B*(

*t*

_{1},

*w*) has a mean of zero and a variance |

*t*

_{2}−

*t*

_{1}|. Also,

*B*(

*t*

_{2},

*w*)−

*B*(

*t*

_{1},

*w*) is independent of

*B*(

*t*

_{4},

*w*)−

*B*(

*t*

_{3},

*w*). Let 0<

*H*<1 and

*b*

_{0}be an arbitrary real number. For

*t*>0, the random function

*B*

_{ H }(

*t,w*) below is called a reduced FBM with Hurst coefficient

*H*and starting value

*b*

_{0}at time 0 (Dieker 2004; Mandelbrot and Van Ness 1968).

where *Γ* represents the Gamma function: \(\Gamma (\alpha) = \int _{0}^{\infty }x^{\alpha -1}e^{-x}dx\) and *B*
_{
H
}(0,*w*)=*b*
_{0} (Mandelbrot and Van Ness 1968).

*h*

_{ random }is a random offset. In our simulation, the coordinates of the midpoints are calculated by the formula below (Boyle et al. 2007):

*σ*is the standard deviation of the heights of sample points.

*H*is Hurst index and

*G*

*a*

*u*

*s*

*s*(·) is a random number generated by standard normal distribution which has a mean of zero and a standard deviation of one. The curves generated by the RMD method under different Hurst coefficients are shown in Fig. 5. The dimension of a fractal curve generated by RMD method equals to 2−

*H*(Huang et al. 1992). As we mentioned above, the dimensions of coastlines range from 1 to 1.5. Therefore, we can select Hurt coefficient in the range of 0.5 and 1 to simulate coastline by RMD method. In this study, Hurst coefficient is set to 0.6.

*H*=0.6) under a different number of iterations. It can be seen that the FBM method works better than the conventional method in terms of coastline echo simulation since more details are provided and the shape is approximate to a real coastline. In addition, we notice that there are no obvious differences between the coastline echoes with more than three iterations via visual inspection. We suggest that three iterations are sufficient for coastline echo simulation by the FBM method as well.

### Weierstrass-Mandelbrot function algorithm

*a*<1,

*b*is a positive odd integer, and

*a*

*b*>1+1.5

*π*(Weierstrass 1967; Zhang et al. 2015). In 1977, Mandelbrot extended the Weierstrass function to the following form,which is called Weierstrass-Mandelbrot function. He also pointed out that the WMF is a fractal and has no smallest scale (Berry and Lewis 1980; Mandelbrot 1979). WMF has been widely adopted to simulate various phenomena in real world (Harrouni 2008; Ma et al. 2015; Shanhua et al. 2015; Wang et al. 2015; Zhang et al. 2015).

*D*is the fractal dimension of the graph of

*W*(

*t*) and 1<

*D*<2.

*γ*is a parameter larger than 1.

*ϕ*

_{ n }is an arbitrary phase. When

*ϕ*

_{ n }=0, the form of WMF is:

*ϕ*

_{ n }=

*n*

*π*, the form of WMF becomes:

*ϕ*

_{ n }=0 to generate cosine series to simulate the coastline echo (Zhang et al. 2015). By inspecting the simulation results under various combinations of

*D*and

*γ*, we notice that the fractal dimension

*D*has much larger impact on the fluctuation frequency of the cosine series than parameter

*γ*. Additionally, in order to keep consistent with the fractal dimension of actual signals, parameter

*D*should be selected between 1 and 1.5, since the fractal dimensions of actual signals range from 1 to 1.5. Therefore, in this study,

*γ*and

*D*are set to 1.5 for both. Figure 7 presents the cosine series simulated in Matlab using aforementioned parameters. It can be seen that the cosine series generated by WMF have infinitely complex patterns across different scales. Based on the characteristics and previous applications of fractal theory (Majumdar and Tien 1990; Voss 1988; Yang et al.), the cosine series generated by WMF can be used to simulate the natural pattern of a coastline.

### Algorithm improvement

*L*=10), the fractal algorithms will be applied. Otherwise, the conventional simulation method will be used to generate the coastline echo. This is because the conventional simulation method works well enough under large radar ranges. Again, the value L can be adjusted in accordance with computer’s performance. We can set L to a smaller value or even zero in the improved algorithm on a high performance computer. For those who are using similar computers as the authors do,

*L*=10 is a good starting point. Figure 8 illustrates the flow chart of improved fractal algorithms for the generation of coastline echo.

## Result and discussion

### Simulation outcomes on marine radar simulator

*v*.

*s*. random Koch curve, conventional method

*v*.

*s*. FBM, and conventional method

*v*.

*s*. WMF). The mariners need to select the best simulation result from each panel based on their own judgements. In the second part of the survey, the mariners are required to score the coastline echoes simulated by conventional methods and three fractal algorithms. Score scale is using a likert scale from 1 to 5, where 1 is “bad” and 5 is “excellent”. The evaluation results are summarized in Table 1. For example, comparing the coastline echo simulated by conventional method with the one by FBM, 30 % of the mariners consider the conventional method is better and 70 % of the mariners think otherwise. Regarding the evaluation scores, the median evaluation scores on the coastline echoes generated by conventional method and FBM are 3 points and 4 points respectively. In general, the results of the evaluation from experienced mariners validate the advantages of fractal algorithms in terms of simulation of coastline echoes.

Summary of evaluations from experience mariners on the simulated coastline echoes by various methods

Survey | Simulation method | Evaluation result |
---|---|---|

Part I | Conventional Method | 30 % |

Conventional Method | 30 % | |

Conventional Method | 33 % | |

Part II | Conventional Method | [2.00, 3.00, 4.00] |

Random Koch Curve | [3.00, 3.00, 4.00] | |

FBM | [3.00, 4.00, 4.25] | |

WMF | [2.00, 3.50, 4.00] |

### Performance analysis of improved fractal algorithms

Speed of scan line to rotate a round by different algorithms, iterations, and radar ranges (unit in second/round)

Algorithm | Iteration | Radar range (in NM) | |||||||
---|---|---|---|---|---|---|---|---|---|

6 | 5 | 4 | 3 | 2 | 1 | 0.5 | 0.25 | ||

Random Koch Curve | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

2 | 12 | 11 | 8 | 6 | 5 | 3 | 3 | 3 | |

3 | − | − | − | − | 8 | 7 | 6 | 5 | |

FBM | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |

3 | 5 | 4 | 3 | 3 | 3 | 3 | 3 | 3 | |

4 | 12 | 11 | 8 | 6 | 5 | 5 | 5 | 3 | |

WMF | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

2 | 31 | 25 | 11 | 8 | 7 | 5 | 4 | 3 | |

3 | − | − | − | − | 14 | 11 | 8 | 6 | |

Improved | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

Random Koch Curve/ | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

FBM/WMF | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

4 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

### Comparison of three fractal algorithms

Based on calculations under the same number of iterations (i) and amount of original sample points of a coastline, the total number of sample points used by a random Koch curve algorithm in the simulation process is 2^{
i
} times of those used by the FBM and WMF algorithm. Since the number of the sample points can directly affect the rotation speed of the scan line, the FBM and WMF algorithm performs better than the random Koch curve algorithm from this aspect.

In addition, the pattern of the coastline echoes generated by FBM algorithm can be adjusted by Hurst index *H* conveniently. The pattern of the coastline echoes simulated by WMF algorithm can be adjusted by the different combinations of *D*, *γ*, and *ϕ*
_{
n
} as well. By contrast, the pattern of the coastline echoes generated by a random Koch curve algorithm can be changed very little. According to the simulation outcomes in this study, the coastline echoes generated by the FBM algorithm more closely resemble real coastline echoes than the echoes generated by a random Koch curve and WMF algorithm.

## Conclusion

In this paper, fractal algorithms are applied into the simulation of coastline echoes on marine radar simulator. The simulation outcomes from different methods are compared as well. In order to guarantee the rotating speed of radar scan line, threshold value L is introduced into the simulation process. Based on our evaluation, the improved FBM algorithm is the best choice for the simulation of coastline echoes on marine radar simulators. Natural-looking coastline echoes generated by the algorithms introduced in this study can improve the quality of the training significantly. The fractal algorithms developed in this paper are packaged into a dynamic link library (dll) with well documented application programmable interface (API), which means that the algorithms are decoupled from the simulator program. This brings great benefits and convenience when transplanting the algorithm to other simulators, as long as the simulator is able to load a dll library. Since a dll library is supported by most of Windows based programs, we believe that the adoption of our algorithms on other simulators will be effortless. One limitation of this study is that we only simulated Type II coastline echoes. Another limitation of this study is that we aim at applying the fractal algorithms into the simulation of the coastline echoes in radar simulator rather than finding the best parameters of the fractal algorithms for the simulation. Last but not least, since the authors are unable to collect real data from marine radar on board, a survey method is employed to evaluate the simulation results by fractal algorithm. In the future, simulation of other radar echo types will be considered; how to choose the parameters for the fractal algorithms should be investigated; and a more objective evaluation approach should be designed to evaluate the simulation results.

## Declarations

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- Ali, A (2006). Simulator instructor-stcw requirements and reality.
*Pomorstvo: Scientific Journal of Maritime Research*,*20*(2), 23–32.Google Scholar - Arnold-Bos, A, Martin, A, Khenchaf, A (2006). A versatile bistatic polarimetric marine radar simulator. In Radar, 2006 IEEE Conference On, (pp. 605–612).Google Scholar
- Aviles, C, & Scholz, C (1987). Fractal analysis applied to characteristic segments.
*Journal of Geophysical Research*,*92*(B1), 331–344.View ArticleGoogle Scholar - Baliarda, C.P, Romeu, J, Cardama, A (2000). The koch monopole: A small fractal antenna.
*Antennas and Propagation, IEEE Transactions on*,*48*(11), 1773–1781.View ArticleGoogle Scholar - Berry, M, & Lewis, Z (1980). On the weierstrass-mandelbrot fractal function. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, (Vol. 370. The Royal Society, pp. 459–484).Google Scholar
- Boyle, R, Parvin, B, Koracin, D, Paragios, N, Tanveer, S.-M (2007). Advances in visual computing.Google Scholar
- Bridge Command (2016). Interactive 3D Ship Simulator. https://www.bahookie.org/.
- Cross, S.S (1994). The application of fractal geometric analysis to microscopic images.
*Micron*,*25*(1), 101–113.MathSciNetView ArticleGoogle Scholar - Dalian Maritime University Institute of Navigational Technology (2016). Marine Radar Simulator. http://nvc.dlmu.edu.cn/list.php?fid=3.
- Dieker, T. (2004).
*Simulation of fractional brownian motion*. The Netherlands: MSc theses, University of Twente, Amsterdam.Google Scholar - Falconer, K (2013). Fractals: A very short introduction.Google Scholar
- Filoche, M, & Sapoval, B (2000). Transfer across random versus deterministic fractal interfaces.
*Physical review letters*,*84*(25), 5776.View ArticleGoogle Scholar - Harrouni, S (2008). Fractal classification of typical meteorological days from global solar irradiance: application to five sites of different climates, 29–54.Google Scholar
- Huang, S, Oelfke, S, Speck, R (1992). Applicability of fractal characterization and modelling to rock joint profiles. In International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, (Vol. 29. Elsevier, pp. 89–98).Google Scholar
- Ji, S, Liu, D, Zhang, Z (2005). A review on fractal image compression coding and some improvement measures. In PDPTA’05, (pp. 151–156).Google Scholar
- Ji, S, Zhang, Z, Yang, H, Liu, D, Sawhney, R (2015). Simulation of coastline’s echo on marine radar simulator based on fractal theory. In Transportation Research Board 94th Annual Meeting.Google Scholar
- Keddam, M, & Takenouti, H (1988). Impedance of fractal interfaces: new data on the von koch model.
*Electrochimica acta*,*33*(3), 445–448.View ArticleGoogle Scholar - Kongsberg Maritime (2016). Polaris Ships Bridge Simulator. http://www.km.kongsberg.com/ks/web/nokbg0240.nsf/AllWeb/B2F29B3742D75297C1257315003C3F6F?OpenDocument.
- Landfall (2016). Marine Radar Simulator. http://www.landfallnavigation.com/radarsimulator.html.
- Mandelbrot, B.B, & Van Ness, J.W (1968). Fractional brownian motions, fractional noises and applications.
*SIAM review*,*10*(4), 422–437.MathSciNetView ArticleMATHGoogle Scholar - Mandelbrot, B (1979). Fractals: form, chance and dimension. Fractals: form, chance and dimension., by Mandelbrot, BB, 1, 16+ 365.Google Scholar
- Mandelbrot, B.B (1983). The fractal geometry of nature, 173.Google Scholar
- Majumdar, A, & Tien, C (1990). Fractal characterization and simulation of rough surfaces.
*Wear*,*136*(2), 313–327.View ArticleGoogle Scholar - Ma, C, Yang, J, Zhao, L, Mei, X, Shi, H (2015). Simulation and experimental study on the thermally induced deformations of high-speed spindle system.
*Applied Thermal Engineering*,*86*, 251–268.View ArticleGoogle Scholar - Nautical Software (2016). Marine Radar Simulator. http://www.nauticalsoftware.com/training-software/marine-radar-simulator.html.
- Organization, I.M (2006). Internatinoal Convention on Standards of Training, Certification and Watchkeeping for Seafarers, 1978, as amended in 2006 London.Google Scholar
- Pentland, A.P (1984). Fractal-based description of natural scenes. Pattern Analysis and Machine Intelligence, IEEE Transactions on (6), 661–674.Google Scholar
- Rostek, S, & Schöbel, R (2013). A note on the use of fractional brownian motion for financial modeling.
*Economic Modelling*,*30*, 30–35.View ArticleGoogle Scholar - Shanhua, X, Songbo, R, Youde, W (2015). Three-dimensional surface parameters and multi-fractal spectrum of corroded steel.
*PloS one*,*10*(6), 0131361.View ArticleGoogle Scholar - Teel, S, Sanders, J, Parrott, D.S, Wade, L, Gervais, T, Rovinski, K, Stone, L.C, Murai, K, Hayashi, Y (2009). Evaluation of marine simulator training based on heart rate variability. In Systems, Man and Cybernetics, 2009. SMC 2009. IEEE International Conference On. IEEE, (pp. 233–238).Google Scholar
- Voss, R.F. (1988). Fractals in Nature: from Characterization to Simulation: Springer.Google Scholar
- Wang, J, Wu, C, Liu, C, Wei, J (2015). Fractal simulation on random rough surface. In 2015 International Conference on Automation, Mechanical Control and Computational Engineering. Atlantis Press.Google Scholar
- Weierstrass, K (1967). Über continuirliche funktionen eines reellen arguments, die für keinen werth des letzteren einen bestimmten differentialquotienten besitzen, 1872. Karl Weiertrass Mathematische Werke.Google Scholar
- Xiuwen, L, Yong, Y, Yicheng, J, Xinyu, Z (2010). Design radar signal interface for navigation radar/arpa simulator using radar display. In Circuits, Communications and System (PACCS), 2010 Second Pacific-Asia Conference On, (Vol. 1. IEEE, pp. 442–445).Google Scholar
- Yin, Y, Liu, X, Li, Z (2007). Key technologies of navigation radar simulator using real radar monitor.
*Journal of System Simulation*,*19*(5), 1014–1017.Google Scholar - Yang, X, Qin, K, Wu, C, Chen, L. Simulation of coastlines based on cloud fractal.Google Scholar
- Zhang, Z, Liu, D, Han, Y, Ji, S (2005). Characteristics, Applications and the Prospects of DSP. In Parallel and Distributed Processing Techniques and Applications, (pp. 161–165).Google Scholar
- Zhang, Z. (2007).
*Research On The Algorithm Of Generating Coastline Echo In Radar Based On Fractal Theory*. China: Master’s thesis, Dalian Maritime University.Google Scholar - Zhang, C, Zhang, D, Quan, D (2010). Echo image generation method for marine radar based on opengl modeling and simulation technology.
*Journal of Dalian Maritime University*,*3*, 017.Google Scholar - Zhang, L, Yu, C, Sun, J (2015). Generalized weierstrass–mandelbrot function model for actual stocks markets indexes with nonlinear characteristics. Fractals, 1550006.Google Scholar