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Journal articles on the topic 'Box-counting'

Author: Grafiati
Published: 4 June 2021
Last updated: 31 January 2023

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1

Meisel, L. V., Mark Johnson, and P. J. Cote. "Box-counting multifractal analysis." Physical Review A 45, no. 10 (May 1, 1992): 6989–96. http://dx.doi.org/10.1103/physreva.45.6989.

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2

Chen, Xiang, Jingchao Li, Hui Han, and Yulong Ying. "Improving the signal subtle feature extraction performance based on dual improved fractal box dimension eigenvectors." Royal Society Open Science 5, no. 5 (May 2018): 180087. http://dx.doi.org/10.1098/rsos.180087.

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Because of the limitations of the traditional fractal box-counting dimension algorithm in subtle feature extraction of radiation source signals, a dual improved generalized fractal box-counting dimension eigenvector algorithm is proposed. First, the radiation source signal was preprocessed, and a Hilbert transform was performed to obtain the instantaneous amplitude of the signal. Then, the improved fractal box-counting dimension of the signal instantaneous amplitude was extracted as the first eigenvector. At the same time, the improved fractal box-counting dimension of the signal without the Hilbert transform was extracted as the second eigenvector. Finally, the dual improved fractal box-counting dimension eigenvectors formed the multi-dimensional eigenvectors as signal subtle features, which were used for radiation source signal recognition by the grey relation algorithm. The experimental results show that, compared with the traditional fractal box-counting dimension algorithm and the single improved fractal box-counting dimension algorithm, the proposed dual improved fractal box-counting dimension algorithm can better extract the signal subtle distribution characteristics under different reconstruction phase space, and has a better recognition effect with good real-time performance.
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3

GRASSBERGER, PETER. "ON EFFICIENT BOX COUNTING ALGORITHMS." International Journal of Modern Physics C 04, no. 03 (June 1993): 515–23. http://dx.doi.org/10.1142/s0129183193000525.

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We present two variants of a fast and storage efficient algorithm for box counting of fractals and fractal measures. In contrast to recently proposed algorithms, no sorting of the data is done. With comparable storage demands, CPU times are between 1 and 2 orders of magnitude lower than those needed with the latter algorithms.
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4

KYRIACOS, S., S. BUCZKOWSKI, F. NEKKA, and L. CARTILIER. "A MODIFIED BOX-COUNTING METHOD." Fractals 02, no. 02 (June 1994): 321–24. http://dx.doi.org/10.1142/s0218348x94000417.

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Fractal geometry has been widely used to characterize irregular structures. Our interest in applying this concept in biomedical research leads us to the conclusion that there are no standard methods. In order to objectively set parameters involved in the estimation of fractal dimension, a significantly more accurate and efficient box-counting method based on a new algorithm was developed. Measurements of mathematical objects with known fractal dimension was performed using the traditional method and the proposed modification. The latter always yields results with less than 1% difference from the theoretical value, which represents a significant improvement.
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5

ALEVIZOS, PANAGIOTIS D., and MICHAEL N. VRAHATIS. "OPTIMAL DYNAMIC BOX-COUNTING ALGORITHM." International Journal of Bifurcation and Chaos 20, no. 12 (December 2010): 4067–77. http://dx.doi.org/10.1142/s0218127410028197.

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An optimal box-counting algorithm for estimating the fractal dimension of a nonempty set which changes over time is given. This nonstationary environment is characterized by the insertion of new points into the set and in many cases the deletion of some existing points from the set. In this setting, the issue at hand is to update the box-counting result at appropriate time intervals with low computational cost. The proposed algorithm tackles the dynamic box-counting problem by using computational geometry methods. In particular, we use a sequence of compressed Box Quadtrees to store the data points. This storage permits the fast and efficient application of our box-counting approach to compute what we call the "dynamic fractal dimension". For a nonempty set of points in the d-dimensional space ℝd (for constant d ≥ 1), the time complexity of the proposed algorithm is shown to be O(n log n) while the space complexity is O(n), where n is the number of considered points. In addition, we show that the time complexity of an insertion, or a deletion is O( log n), and that the above time and space complexity is optimal. Experimental results of the proposed approach illustrated on the well-known and widely studied Hénon map are presented.
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6

von Hardenberg, J., R. Thieberger, and A. Provenzale. "A box-counting red herring." Physics Letters A 269, no. 5-6 (May 2000): 303–8. http://dx.doi.org/10.1016/s0375-9601(00)00265-6.

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7

Nikolaidis, Nikolaos S., and Ioannis N. Nikolaidis. "The box-merging implementation of the box-counting algorithm." Journal of the Mechanical Behavior of Materials 25, no. 1-2 (April 1, 2016): 61–67. http://dx.doi.org/10.1515/jmbm-2016-0006.

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AbstractFractal analysis is a powerful tool for the classification of materials. However, until now, there were no efficient tools that could process large or color images due to the processing time required. In this article we present a fast, easy to implement and very easily expandable to any number of dimensions variation, the box merging method. It is applied here to RGB test images which are considered as sets in 5-D space.
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8

Ndiaye, M. "Combining Fractals and Box-Counting Dimension." Applied Mathematics 12, no. 09 (2021): 818–34. http://dx.doi.org/10.4236/am.2021.129055.

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9

Górski, Andrzej Z. "Pseudofractals and the box counting algorithm." Journal of Physics A: Mathematical and General 34, no. 39 (September 21, 2001): 7933–40. http://dx.doi.org/10.1088/0305-4470/34/39/302.

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10

Jiménez, J., and J. Ruiz de Miras. "Fast box-counting algorithm on GPU." Computer Methods and Programs in Biomedicine 108, no. 3 (December 2012): 1229–42. http://dx.doi.org/10.1016/j.cmpb.2012.07.005.

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11

Miyata, Takahisa, and Tadashi Watanabe. "Approximate resolutions and box-counting dimension." Topology and its Applications 132, no. 1 (July 2003): 49–69. http://dx.doi.org/10.1016/s0166-8641(02)00362-0.

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12

Bingham, Stuart. "Box-counting with base bn numerals." Physics Letters A 163, no. 5-6 (March 1992): 419–24. http://dx.doi.org/10.1016/0375-9601(92)90849-h.

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13

Balogh, Zoltán M., and Roger Züst. "Box-counting by Hölder’s traveling salesman." Archiv der Mathematik 114, no. 5 (December 12, 2019): 561–72. http://dx.doi.org/10.1007/s00013-019-01415-5.

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14

Liu, HongYan, Addie Bahi, and Frank K. Ko. "A one dimensional heat transfer model for wolverine (gulo-gulo) hair." International Journal of Clothing Science and Technology 30, no. 4 (August 6, 2018): 548–58. http://dx.doi.org/10.1108/ijcst-08-2017-0108.

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Purpose Wolverine hairs with superior heat transfer properties have been used as fur ruffs for extreme cold-weather clothing. In order to understand the exclusive mechanism of wolverine surviving in the cold areas of circumpolar, the purpose of this paper is to establish a one-dimensional fractional heat transfer equation to reveal the hidden mechanism for the hairs, and also calculate the fractal dimension of the wolverine hair using the box counting method to verify the proposed theory. The observed results (from the proposed model) found to be in good agreement with the box counting method. This model can explain the phenomenon which offers the theoretical foundation for the design of extreme cold weather clothing. Design/methodology/approach The authors calculated the fractal dimension of the wolverine hair using the box counting method to verify the proposed theory. The observed results (from the proposed model) found to be in good agreement with the box counting method. Findings The box counting method proves that the theoretical model is applicable. Originality/value The authors propose the first heat transfer model for the wolverine hair.
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15

BARBÉ, ANDRÉ, FRITZ VON HAESELER, and GENCHO SKORDEV. "LIMIT SETS OF RESTRICTED RANDOM SUBSTITUTIONS." Fractals 14, no. 01 (March 2006): 37–47. http://dx.doi.org/10.1142/s0218348x06003076.

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We consider a certain type of random substitution and show that the sets generated by it have almost surely the same box-counting and Hausdorff dimension, and that box-counting and Hausdorff dimension coincide.
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16

PERFECT, E., and B. DONNELLY. "BI-PHASE BOX COUNTING: AN IMPROVED METHOD FOR FRACTAL ANALYSIS OF BINARY IMAGES." Fractals 23, no. 01 (March 2015): 1540010. http://dx.doi.org/10.1142/s0218348x15400101.

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Many natural systems are irregular and/or fragmented, and have been interpreted to be fractal. An important parameter needed for modeling such systems is the fractal dimension, D. This parameter is often estimated from binary images using the box-counting method. However, it is not always apparent which fractal model is the most appropriate. This has led some researchers to report different D values for different phases of an analyzed image, which is mathematically untenable. This paper introduces a new method for discriminating between mass fractal, pore fractal, and Euclidean scaling in images that display apparent two-phase fractal behavior when analyzed using the traditional method. The new method, coined "bi-phase box counting", involves box-counting the selected phase and its complement, fitting both datasets conjointly to fractal and/or Euclidean scaling relations, and examining the errors from the resulting regression analyses. Use of the proposed technique was demonstrated on binary images of deterministic and stochastic fractals with known D values. Traditional box counting was unable to differentiate between the fractal and Euclidean phases in these images. In contrast, bi-phase box counting unmistakably identified the fractal phase and correctly estimated its D value. The new method was also applied to three binary images of soil thin sections. The results indicated that two of the soils were pore-fractals, while the other was a mass fractal. This outcome contrasted with the traditional box counting method which suggested that all three soils were mass fractals. Reclassification has important implications for modeling soil structure since different fractal models have different scaling relations. Overall, bi-phase box counting represents an improvement over the traditional method. It can identify the fractal phase and it provides statistical justification for this choice.
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17

Troscheit, Sascha. "The box-counting dimension of random box-like self-affine sets." Indiana University Mathematics Journal 67, no. 2 (2018): 495–535. http://dx.doi.org/10.1512/iumj.2018.67.7295.

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18

Jiang, Shiguo, and Desheng Liu. "Box-Counting Dimension of Fractal Urban Form." International Journal of Artificial Life Research 3, no. 3 (July 2012): 41–63. http://dx.doi.org/10.4018/jalr.2012070104.

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The difficulty to obtain a stable estimate of fractal dimension for stochastic fractal (e.g., urban form) is an unsolved issue in fractal analysis. The widely used box-counting method has three main issues: 1) ambiguities in setting up a proper box cover of the object of interest; 2) problems of limited data points for box sizes; 3) difficulty in determining the scaling range. These issues lead to unreliable estimates of fractal dimensions for urban forms, and thus cast doubt on further analysis. This paper presents a detailed discussion of these issues in the case of Beijing City. The authors propose corresponding improved techniques with modified measurement design to address these issues: 1) rectangular grids and boxes setting up a proper box cover of the object; 2) pseudo-geometric sequence of box sizes providing adequate data points to study the properties of the dimension profile; 3) generalized sliding window method helping to determine the scaling range. The authors’ method is tested on a fractal image (the Vicsek prefractal) with known fractal dimension and then applied to real city data. The results show that a reliable estimate of box dimension for urban form can be obtained using their method.
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19

Yang, Zon-Yee, and Jian-Liang Juo. "Interpretation of sieve analysis data using the box-counting method for gravelly cobbles." Canadian Geotechnical Journal 38, no. 6 (December 1, 2001): 1201–12. http://dx.doi.org/10.1139/t01-052.

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In fractal theory, the fractal dimension (D) is a measure of the complexity of particle distribution in nature. It can provide a description of how much space a particle set fills. The box-counting method uses squared grids of various sizes to cover the particles to obtain a box dimension. This sequential counting concept is analogous to the sieve analysis test using stacked sieves. In this paper the box-counting method is applied to describe the particle-size distribution of gravelly cobbles. Three approaches to obtain the fractal dimension are presented. In the first approach the data obtained from a classic laboratory sieve analysis are rearranged into a double-logarithmic plot, according to a fractal model, to obtain the fractal dimension of the particle collection. In addition, an equivalent number of covered grids on each sieve during the sieve analysis are counted to produce the box dimension. According to the box-counting method concept, a photo-sieving technique used in scanning electron microscope microstructure analysis is adopted for use on gravelly cobbles in the field. The box-counting method concept is capable of explaining the sieve analysis data to clarify the information on the particle-size distribution. Using photo-sieving to produce the fractal dimension from field photographs can provide another approach for understanding the particle-size distribution. However, the representative cross-profile should be chosen carefully. The composition of the particle-size distribution for gravelly cobbles with higher D values is more complicated than those at sites with smaller D values.Key words: sieve analysis, box-counting method, fractal dimension, particle-size distribution, gravelly cobbles.
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20

Tan, Piqiang, Yifan Yin, Deyuan Wang, Diming Lou, and Zhiyuan Hu. "The microscopic characteristics of particle matter and image algorithm based on fractal theory." E3S Web of Conferences 360 (2022): 01003. http://dx.doi.org/10.1051/e3sconf/202236001003.

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The effects of ash and sulfur content on the morphology of particulate matter (PM) in diesel particle filter (DPF) were investigated with five different components of lubricants. The aggregate morphology of primary particles in diesel were analyzed using transmission electron microscopy (TEM). The fractal dimensions of carbon particles were calculated by box-counting method (BCM), differential box-counting method (DBC), relative differential box-counting method (RDBC) and MAD-based box counting method (MAD-DBC), and the results were compared. The results showed that the microstructure of PM developed from chain-like structure to agglomerate structure with the increase of sulfur and ash content in lubricating oil. The fractal dimension of carbon particles increased with the increase of sulfur and ash content. The SSE of RDBC fitting results was smaller, and the R-square is larger. MAD-DBC fitting results had stronger anti-noise interference performance.
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21

Górski, A. Z., S. Drożdż, A. Mokrzycka, and J. Pawlik. "Accuracy Analysis of the Box-Counting Algorithm." Acta Physica Polonica A 121, no. 2B (February 2012): B—28—B—30. http://dx.doi.org/10.12693/aphyspola.121.b-28.

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22

Yamaguti, Marcos, and Carmen P. C. Prado. "Smart covering for a box-counting algorithm." Physical Review E 55, no. 6 (June 1, 1997): 7726–32. http://dx.doi.org/10.1103/physreve.55.7726.

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23

Laucelli Meana, Marco, M. A. R. Osorio, and Jesús Puente Peñalba. "Counting closed string states in a box." Physics Letters B 408, no. 1-4 (September 1997): 183–91. http://dx.doi.org/10.1016/s0370-2693(97)00789-2.

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24

Tominaga, Yukio. "Data Structure Comparison Using Box Counting Analysis." Journal of Chemical Information and Computer Sciences 38, no. 5 (September 1998): 867–75. http://dx.doi.org/10.1021/ci9802070.

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25

Ruiz de Miras, Juan. "Fast differential box-counting algorithm on GPU." Journal of Supercomputing 76, no. 1 (October 17, 2019): 204–25. http://dx.doi.org/10.1007/s11227-019-03030-1.

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26

Donnelly, Harold. "Resonance counting function in black box scattering." Journal of Mathematical Physics 47, no. 10 (October 2006): 102105. http://dx.doi.org/10.1063/1.2356910.

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27

Kruger, A. "Implementation of a fast box-counting algorithm." Computer Physics Communications 98, no. 1-2 (October 1996): 224–34. http://dx.doi.org/10.1016/0010-4655(96)00080-x.

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28

Miličić, Siniša. "Box-counting dimensions of generalised fractal nests." Chaos, Solitons & Fractals 113 (August 2018): 125–34. http://dx.doi.org/10.1016/j.chaos.2018.05.025.

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29

Hou, Xin-Jun, Robert Gilmore, Gabriel B. Mindlin, and Hernán G. Solari. "An efficient algorithm for fast box counting." Physics Letters A 151, no. 1-2 (November 1990): 43–46. http://dx.doi.org/10.1016/0375-9601(90)90844-e.

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30

Maryenko, N. І., and O. Yu Stepanenko. "Fractal analysis of anatomical structures linear contours: modified Caliper method vs Box counting method." Reports of Morphology 28, no. 1 (February 23, 2022): 17–26. http://dx.doi.org/10.31393/morphology-journal-2022-28(1)-03.

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Fractal analysis estimates the metric dimension and complexity of the spatial configuration of different anatomical structures. This allows the use of this mathematical method for morphometry in morphology and clinical medicine. Two methods of fractal analysis are most often used for fractal analysis of linear fractal objects: the Box counting method (Grid method) and the Caliper method (Richardson’s method, Perimeter stepping method, Ruler method, Divider dimension, Compass dimension, Yard stick method). The aim of the research is a comparative analysis of two methods of fractal analysis – Box counting method and author's modification of Caliper method for fractal analysis of linear contours of anatomical structures. A fractal analysis of three linear fractals was performed: an artificial fractal – a Koch snowflake and two natural fractals – the outer contours of the pial surface of the human cerebellar vermis cortex and the cortex of the cerebral hemispheres. Fractal analysis was performed using the Box counting method and the author's modification of the Caliper method. The values of the fractal dimension of the artificial linear fractal (Koch snowflakes) obtained by the Caliper method coincide with the true value of the fractal dimension of this fractal, but the values of the fractal dimension obtained by the Box counting method do not match the true value of the fractal dimension. Therefore, fractal analysis of linear fractals using the Caliper method allows you to get more accurate results than the Box counting method. The values of the fractal dimension of artificial and natural fractals, calculated using the Box counting method, decrease with increasing image size and resolution; when using the Caliper method, fractal dimension values do not depend on these image parameters. The values of the fractal dimension of linear fractals, calculated using the Box counting method, increase with increasing width of the linear contour; the values calculated using the Caliper method do not depend on the contour line width. Thus, for the fractal analysis of linear fractals, preference should be given to the Caliper method and its modifications.
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31

MASTERS, BARRY R. "FRACTAL ANALYSIS OF NORMAL HUMAN RETINAL BLOOD VESSELS." Fractals 02, no. 01 (March 1994): 103–10. http://dx.doi.org/10.1142/s0218348x94000090.

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The fractal dimension of the pattern of retinal blood vessels in the normal human eye was calculated. Photomontages were constructed from 10 red-free retinal photographs. Manual tracings of the vessels were made. Digital images of the tracings were analyzed on a computer using the box-counting method to determine the fractal dimension. The mean value and standard deviation of the fractal dimension (box-counting dimension), computed as described in the Methods section, is 1.70 ± 0.02 (N = 10). The use of standard methods for both data acquisition and computer assisted box-counting to determine the fractal dimension, resulted in reduced variance in the calculated data.
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32

Saa, A., G. Gascó, J. B. Grau, J. M. Antón, and A. M. Tarquis. "Comparison of gliding box and box-counting methods in river network analysis." Nonlinear Processes in Geophysics 14, no. 5 (September 12, 2007): 603–13. http://dx.doi.org/10.5194/npg-14-603-2007.

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Abstract. We use multifractal analysis to estimate the Rényi dimensions of river basins by two different partition methods. These methods differ in the way that the Euclidian plane support of the measure is covered, partitioning it by using mutually exclusive boxes or by gliding a box over the plane. Images of two different drainage basins, for the Ebro and Tajo rivers, located in Spain, were digitalized with a resolution of 0.5 km, giving image sizes of 617×1059 pixels and 515×1059, respectively. Box sizes were chosen as powers of 2, ranging from 2×4 pixels to 512×1024 pixels located within the image, with the purpose of covering the entire network. The resulting measures were plotted versus the logarithmic value of the box area instead of the box size length. Multifractal Analysis (MFA) using a box counting algorithm was carried out according to the method of moments ranging from −5<q<5, and the Rényi dimensions were calculated from the log/log slope of the probability distribution for the respective moments over the box area. An optimal interval of box sizes was determined by estimating the characteristic length of the river networks and by taking the next higher power of 2 as the smallest box size. The optimized box size for both river networks ranges from 64×128 to 512×1024 pixels and illustrates the multiscaling behaviour of the Ebro and Tajo. By restricting the multifractal analysis to the box size range, good generalized dimension (Dq) spectra were obtained but with very few points and with a low number of boxes for each size due to image size restrictions. The gliding box method was applied to the same box size range, providing more consistent and representative Dq values. The numerical differences between the results, as well as the standard error values, are discussed.
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33

Grau, J., V. Méndez, A. M. Tarquis, M. C. Díaz, and A. Saa. "Comparison of gliding box and box-counting methods in soil image analysis." Geoderma 134, no. 3-4 (October 2006): 349–59. http://dx.doi.org/10.1016/j.geoderma.2006.03.009.

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34

OSAKA, MOTOHISA, and NOBUYASU ITO. "LOCAL BOX-COUNTING TO DETERMINE FRACTAL DIMENSION OF HIGH-ORDER CHAOS." International Journal of Modern Physics C 11, no. 08 (December 2000): 1519–26. http://dx.doi.org/10.1142/s0129183100001474.

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To determine the attractor dimension of chaotic dynamics, the box-counting method has the difficulty in getting accurate estimates because the boxes are not weighted by their relative probabilities. We present a new method to minimize this difficulty. The local box-counting method can be quite effective in determining the attractor dimension of high-order chaos as well as low-order chaos.
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35

O'Brien, Frank. "The Poisson 2-Space Box-Counting Method Revisited." Perceptual and Motor Skills 80, no. 3_suppl (June 1995): 1318. http://dx.doi.org/10.2466/pms.1995.80.3c.1318.

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The author's box-counting method for assessing stochastic randomness in two-dimensional space can be enhanced when the area of residence is a square figure. A large improvement in the detection probability can be realized under the modification proposed even for very small populations.
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36

Juwitarty, Novita Anggraini, Kosala Dwidja Purnomo, and Kiswara Agung Santoso. "PENDETEKSIAN CITRA DAUN TANAMAN MENGGUNAKAN METODE BOX COUNTING." Majalah Ilmiah Matematika dan Statistika 20, no. 1 (March 16, 2020): 35. http://dx.doi.org/10.19184/mims.v20i1.17221.

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Different types of plants make identification difficult. Therefore, we need a system that can identify the similarity of the leaves based on a reference leaf. Extraction can be done by taking one part of the plant and the most easily obtained part is the leaf part. Natural objects such as leaves have irregular shapes and are difficult to measure, but this can be overcome by using fractal dimensions. In this research, image detection of plant leaves will be carried out using the box counting method. The box counting method is a method of calculating fractal dimensions by dividing images into small boxes in various sizes. Image detection using fractal dimension values, we know which leaves the match with the reference. In this study,10 species of leave were tested, with each species 10 samples of plant leaves. Image testing of plant leaves uses a variety of r box size, namely 1/2 ,1/4 , 1/8 , 1/16 ,1/32 , 1/64 , 128which obtain an average match accuracy of 44%. Keywords: Box Counting, Fractal dimension
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37

Fujiwara, Masahiko. "Counting points in a small box on varieties." Proceedings of the Japan Academy, Series A, Mathematical Sciences 64, no. 8 (1988): 267–70. http://dx.doi.org/10.3792/pjaa.64.267.

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38

GARCÍA, G., G. MORA, and D. A. REDTWITZ. "BOX-COUNTING DIMENSION COMPUTED BY α-DENSE CURVES." Fractals 25, no. 05 (September 4, 2017): 1750039. http://dx.doi.org/10.1142/s0218348x17500396.

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We introduce a method to reduce to the real case the calculus of the box-counting dimension of subsets of the unit cube [Formula: see text], [Formula: see text]. The procedure is based on the existence of special types of [Formula: see text]-dense curves (a generalization of the space-filling curves) in [Formula: see text] called [Formula: see text]-uniform curves.
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39

Molteno, T. C. A. "FastO(N) box-counting algorithm for estimating dimensions." Physical Review E 48, no. 5 (November 1, 1993): R3263—R3266. http://dx.doi.org/10.1103/physreve.48.r3263.

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40

Vinodchandran, N. V. "Counting Complexity of Solvable Black-Box Group Problems." SIAM Journal on Computing 33, no. 4 (January 2004): 852–69. http://dx.doi.org/10.1137/s0097539703420651.

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41

Gonzato, Guido. "A practical implementation of the box counting algorithm." Computers & Geosciences 24, no. 1 (January 1998): 95–100. http://dx.doi.org/10.1016/s0098-3004(97)00137-4.

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42

Block, A., W. von Bloh, and H. J. Schellnhuber. "Efficient box-counting determination of generalized fractal dimensions." Physical Review A 42, no. 4 (August 1, 1990): 1869–74. http://dx.doi.org/10.1103/physreva.42.1869.

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43

ROSENBERG, ERIC. "Lower Bounds on Box Counting for Complex Networks." Journal of Interconnection Networks 14, no. 04 (December 2013): 1350019. http://dx.doi.org/10.1142/s0219265913500199.

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Determining the fractal dimension dB of a complex network requires computing N(s), the minimal number of boxes of size s needed to cover the network. While effective approximation methods for this problem are known, the computation of a lower bound on N(s) has not been studied. We show that a lower bound can be obtained by formulating the covering problem as an uncapacitated facility location problem, and applying dual ascent to the dual of its linear programming relaxation. We illustrate the method on a small example, and provide numerical results on some larger problems. The upper and lower bounds on N(s) can be used to define a linear program which yields upper and lower bounds on dB.
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44

PÖTZELBERGER, KLAUS. "The quantization dimension of distributions." Mathematical Proceedings of the Cambridge Philosophical Society 131, no. 3 (November 2001): 507–19. http://dx.doi.org/10.1017/s0305004101005357.

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We show that the asymptotic behaviour of the quantization error allows the definition of dimensions for probability distributions, the upper and the lower quantization dimension. These concepts fit into standard geometric measure theory, as the upper quantization dimension is always between the packing and the upper box-counting dimension, whereas the lower quantization dimension is between the Hausdorff and the lower box-counting dimension.
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45

Yudantoro, Tri Raharjo, Liliek Triyono, Dwi Desy Nur Hani'ah, and Moudina Risma Slodia. "Arduino-based Charity Box Safety, Tracking, and Counter System." JAICT 4, no. 2 (June 26, 2020): 17. http://dx.doi.org/10.32497/jaict.v4i2.1891.

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<p>The charity box is a supporting facility that can be used by pilgrims to distribute their infaq. In this modern era, it turns out that the mosque still counts the contents of the charity box manually and uses the usual padlock key for the safety of the charity box. The purpose of this research is to build a system and tool that can simplify the performance of mosque administrators in counting money and maintaining the security of the charity box. ArduinoBased Charity Box Safety, Tracking, and Counter System is a charity box equipped with automatic counting and security features using RFID and GPS. The method used in making this system is the waterfall method. The features in this system are automatic counting and security features using RFID keys, GPS, buzzers, and infrared sensors. This system is also equipped with a notification to the mosque management regarding the amount of money in the charity box and the location of the charity box using GSM / GPRS. From the testing of the system, it was produced that the Arduino-Based Charity Box Safety, Tracing, and Counter System was able to detect banknotes and coins well. The SMS feature also works well where there will be an SMS message regarding the amount of money and location of the charity box. The safety of the charity box is enhanced by using RFID, GPS, buzzers, and infrared sensors. From the user satisfaction test results obtained by the percentage of user satisfaction by 85%, which means the Arduino-Based Charity Box Safety, Tracing, and Counter System is quite attractive to users.</p>
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46

Meng, Xianmeng, Pengju Zhang, Jing Li, Chuanming Ma, and Dengfeng Liu. "The linkage between box-counting and geomorphic fractal dimensions in the fractal structure of river networks: the junction angle." Hydrology Research 51, no. 6 (October 15, 2020): 1397–408. http://dx.doi.org/10.2166/nh.2020.082.

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Abstract In the past, a great deal of research has been conducted to determine the fractal properties of river networks, and there are many kinds of methods calculating their fractal dimensions. In this paper, we compare two most common methods: one is geomorphic fractal dimension obtained from the bifurcation ratio and the stream length ratio, and the other is box-counting method. Firstly, synthetic fractal trees are used to explain the role of the junction angle on the relation between two kinds of fractal dimensions. The obtained relationship curves indicate that box-counting dimension is decreasing with the increase of the junction angle when geomorphic fractal dimension keeps constant. This relationship presents continuous and smooth convex curves with junction angle from 60° to 120° and concave curves from 30° to 45°. Then 70 river networks in China are investigated in terms of their two kinds of fractal dimensions. The results confirm the fractal structure of river networks. Geomorphic fractal dimensions of river networks are larger than box-counting dimensions and there is no obvious relationship between these two kinds of fractal dimensions. Relatively good non-linear relationships between geomorphic fractal dimensions and box-counting dimensions are obtained by considering the role of the junction angle.
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47

Jamaludin, Iqbal, Mohd Zulfaezal Che Azemin, Mohd Izzuddin Mohd Tamrin, and Abdul Halim Sapuan. "Volume of Interest-Based Fractal Analysis of Huffaz’s Brain." Fractal and Fractional 6, no. 7 (July 19, 2022): 396. http://dx.doi.org/10.3390/fractalfract6070396.

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The robust process in memorising the Quran is expected to cause neuroplasticity changes in the brain. To date, the analysis of neuroplasticity is limited in binary images because greyscale analysis requires the usage of more robust processing techniques. This research work aims to explore and characterise the complexity of textual memorisation brain structures using fractal analysis between huffaz and non-huffaz applying global box-counting, global Fourier fractal dimension (FFD), and volume of interest (VOI)-based analysis. The study recruited 47 participants from IIUM Kuantan Campus. The huffaz group had their 18 months of systematic memorisation training. The brain images were acquired by using MRI. Global box-counting and FFD analysis were conducted on the brain. Magnetic resonance imaging (MRI) found no significant statistical difference between brains of huffaz and non-huffaz. VOI-based analysis found nine significant areas: two for box-counting analysis (angular gyrus and medial temporal gyrus), six for FFD analysis (BA20, BA30, anterior cingulate, fusiform gyrus, inferior temporal gyrus, and frontal lobe), and only a single area (BA33) showed significant volume differences between huffaz and non-huffaz. The results have highlighted the sensitivity of VOI-based analysis because of its local nature, as compared to the global analysis by box-counting and FFD.
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JIMÉNEZ-HORNERO, FRANCISCO J., ANA B. ARIZA-VILLAVERDE, and EDUARDO GUTIÉRREZ DE RAVÉ. "MULTIFRACTAL DESCRIPTION OF SIMULATED FLOW VELOCITY IN IDEALISED POROUS MEDIA BY USING THE SANDBOX METHOD." Fractals 21, no. 01 (March 2013): 1350006. http://dx.doi.org/10.1142/s0218348x13500060.

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The spatial description of flows in porous media is a main issue due to their influence on processes that take place inside. In addition to descriptive statistics, the multifractal analysis based on the Box-Counting fixed-size method has been used during last decade to study some porous media features. However, this method gives emphasis to domain regions containing few data points that spark the biased assessment of generalized fractal dimensions for negative moment orders. This circumstance is relevant when describing the flow velocity field in idealised three-dimensional porous media. The application of the Sandbox method is explored in this work as an alternative to the Box-Counting procedure for analyzing flow velocity magnitude simulated with the lattice model approach for six media with different porosities. According to the results, simulated flows have multiscaling behaviour. The multifractal spectra obtained with the Sandbox method reveal more heterogeneity as well as the presence of some extreme values in the distribution of high flow velocity magnitudes as porosity decreases. This situation is not so evident for the multifractal spectra estimated with the Box-Counting method. As a consequence, the description of the influence of porous media structure on flow velocity distribution provided by the Sandbox method improves the results obtained with the Box-Counting procedure.
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49

He, Tao, Long Fei Cheng, Qing Hua Wu, Zheng Jia Wang, Lian Gen Yang, and Lang Yu Xie. "An Image Segmentation Calculation Based on Differential Box-Counting of Fractal Geometry." Applied Mechanics and Materials 719-720 (January 2015): 964–68. http://dx.doi.org/10.4028/www.scientific.net/amm.719-720.964.

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Differential box-counting of fractal geometry has been widely used in image processing.A method which uses the differential box-counting to segment the gathered images is discussed in this paper . It is to construct a three-dimensional gray space and use the same size boxes to contain the three dimensional space.The number of boxes needed to cover the entire image are calculated .Different sizes of boxes can receive different number of boxes, so least squares method is used to calculate the fractal dimension. According to the fractal dimension parameters, appropriate threshold is chose to segment the image by using binarization .From the handle case of bearing pictures can be seen that image segmentation based on differential box-counting method can get clear image segmentation .This method is easy to understand, to operate, and has important significance on computer image segmentation .
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Xiang, Ming, Zhen Dong Cui, and Yuan Hong Wu. "A Fingerprint Image Segmentation Method Based on Fractal Dimension." Advanced Materials Research 461 (February 2012): 299–301. http://dx.doi.org/10.4028/www.scientific.net/amr.461.299.

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Fractal analysis is becoming more and more popular in image segmentation community, in which the box-counting based fractal dimension estimations are most commonly used. In this paper, a novel fractal estimation algorithm is proposed. Both the proposed algorithm and the box-counting based methods have been applied to the segmentation of texture images. The comparison results demonstrate that the fractal estimation can differentiate texture images more effectively and provide more robust segmentations
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