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Journal articles on the topic 'Box-counting dimension'

Author: Grafiati

Published: 4 June 2021

Last updated: 21 February 2023

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1

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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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

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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4

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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5

OLSON, ERIC J., JAMES C. ROBINSON, and NICHOLAS SHARPLES. "Generalised Cantor sets and the dimension of products." Mathematical Proceedings of the Cambridge Philosophical Society 160, no. 1 (October 30, 2015): 51–75. http://dx.doi.org/10.1017/s0305004115000584.

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AbstractIn this paper we consider the relationship between the Assouad and box-counting dimension and how both behave under the operation of taking products. We introduce the notion of ‘equi-homogeneity’ of a set, which requires a uniformity in the cardinality of local covers at all length-scales and at all points, and we show that a large class of homogeneous Moran sets have this property. We prove that the Assouad and box-counting dimensions coincide for sets that have equal upper and lower box-counting dimensions provided that the set ‘attains’ these dimensions (analogous to ‘s-sets’ when considering the Hausdorff dimension), and the set is equi-homogeneous. Using this fact we show that for any α ∈ (0, 1) and any β, γ ∈ (0, 1) such that β + γ ⩾ 1 we can construct two generalised Cantor sets C and D such that dimBC = αβ, dimBD = α γ, and dimAC = dimAD = dimA (C × D) = dimB (C × D) = α.

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6

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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7

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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8

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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9

Falconer, K. J. "The dimension of self-affine fractals II." Mathematical Proceedings of the Cambridge Philosophical Society 111, no. 1 (January 1992): 169–79. http://dx.doi.org/10.1017/s0305004100075253.

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AbstractA family {S1, ,Sk} of contracting affine transformations on Rn defines a unique non-empty compact set F satisfying . We obtain estimates for the Hausdorff and box-counting dimensions of such sets, and in particular derive an exact expression for the box-counting dimension in certain cases. These estimates are given in terms of the singular value functions of affine transformations associated with the Si. This paper is a sequel to 4, which presented a formula for the dimensions that was valid in almost all cases.

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10

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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11

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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12

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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13

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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14

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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15

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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16

CHEN, WEN-SHIUNG, and SHANG-YUAN YUAN. "SOME FRACTAL DIMENSION ESTIMATE ALGORITHMS AND THEIR APPLICATIONS TO ONE-DIMENSIONAL BIOMEDICAL SIGNALS." Biomedical Engineering: Applications, Basis and Communications 14, no. 03 (June 25, 2002): 100–108. http://dx.doi.org/10.4015/s1016237202000152.

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Fractals can model many classes of time-series data. The fractal dimension is an important characteristic of fractals that contains information about their geometrical structure at multiple scales. The covering methods are a class of efficient approaches, e.g., box-counting (BC) method, to estimate the fractal dimension. In this paper, the differential box-counting (DBC) approach, originally for 2-D applications, is modified and applied to 1-D case. In addition, two algorithms, called 1-D shifting-DBC (SDBC-1D) and 1-D scanning-BC (SBC-1D), are also proposed for 1-D signal analysis. The fractal dimensions for 1-D biomedical pulse and ECG signals are calculated.

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17

FATHALLAH-SHAYKH, HASSAN M. "FRACTAL DIMENSION OF THE DROSOPHILA CIRCADIAN CLOCK." Fractals 19, no. 04 (December 2011): 423–30. http://dx.doi.org/10.1142/s0218348x11005476.

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Fractal geometry can adequately represent many complex and irregular objects in nature. The fractal dimension is typically computed by the box-counting procedure. Here I compute the box-counting and the Kaplan-Yorke dimensions of the 14-dimensional models of the Drosophila circadian clock. Clockwork Orange (CWO) is transcriptional repressor of direct target genes that appears to play a key role in controlling the dynamics of the clock. The findings identify these models as strange attractors and highlight the complexity of the time-keeping actions of CWO in light-day cycles. These fractals are high-dimensional counterexamples of the Kaplan-Yorke conjecture that uses the spectrum of the Lyapunov exponents.

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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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WANG, JUN, and KUI YAO. "DIMENSION ANALYSIS OF CONTINUOUS FUNCTIONS WITH UNBOUNDED VARIATION." Fractals 25, no. 01 (February 2017): 1730001. http://dx.doi.org/10.1142/s0218348x1730001x.

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In this paper, we mainly discuss fractal dimensions of continuous functions with unbounded variation. First, we prove that Hausdorff dimension, Packing dimension and Modified Box-counting dimension of continuous functions containing one UV point are [Formula: see text]. The above conclusion still holds for continuous functions containing finite UV points. More generally, we show the result that Hausdorff dimension of continuous functions containing countable UV points is [Formula: see text] also. Finally, Box dimension of continuous functions containing countable UV points has been proved to be [Formula: see text] when [Formula: see text] is self-similar.

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20

Deng, Yuqian, Xiuxiong Liu, and Yongping Zhang. "Fractal Dimension Analysis of the Julia Sets of Controlled Brusselator Model." Discrete Dynamics in Nature and Society 2016 (2016): 1–13. http://dx.doi.org/10.1155/2016/8234108.

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Fractal theory is a branch of nonlinear scientific research, and its research object is the irregular geometric form in nature. On account of the complexity of the fractal set, the traditional Euclidean dimension is no longer applicable and the measurement method of fractal dimension is required. In the numerous fractal dimension definitions, box-counting dimension is taken to characterize the complexity of Julia set since the calculation of box-counting dimension is relatively achievable. In this paper, the Julia set of Brusselator model which is a class of reaction diffusion equations from the viewpoint of fractal dynamics is discussed, and the control of the Julia set is researched by feedback control method, optimal control method, and gradient control method, respectively. Meanwhile, we calculate the box-counting dimension of the Julia set of controlled Brusselator model in each control method, which is used to describe the complexity of the controlled Julia set and the system. Ultimately we demonstrate the effectiveness of each control method.

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21

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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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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23

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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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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PARKINSON, IAN H., and NIC L. FAZZALARI. "GOODNESS OF FIT ON A MODIFIED RICHARDSON PLOT BY RESIDUAL ANALYSIS." Fractals 08, no. 03 (September 2000): 261–65. http://dx.doi.org/10.1142/s0218348x00000275.

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Modified Richardson plots obtained by a box counting method on outlines of trabecular bone were tested for linearity. The degree of deviation from true linearity was quantified. The results showed that although there was evidence of nonlinearity or serial correlation in the Richardson plots, the magnitude of deviation from true linearity was less than 0.3% for the residuals and less than 4% for the standard deviation of the residuals. This study shows that the modified box counting method for estimating overall fractal dimension or sectional fractal dimensions of trabecular bone is efficacious. The low magnitude of deviation from linearity confirms that over a defined range of scale the Richardson plot provides an accurate estimation of the fractal dimension of trabecular bone.

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Górski, A. Z., M. Stróż, P. Oświȩcimka, and J. Skrzat. "Accuracy of the box-counting algorithm for noisy fractals." International Journal of Modern Physics C 27, no. 10 (August 29, 2016): 1650112. http://dx.doi.org/10.1142/s0129183116501126.

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The box-counting (BC) algorithm is applied to calculate fractal dimensions of four fractal sets. The sets are contaminated with an additive noise with amplitude [Formula: see text]. The accuracy of calculated numerical values of the fractal dimensions is analyzed as a function of [Formula: see text] for different sizes of the data sample. In particular, it has been found that even in case of pure fractals ([Formula: see text]) as well as for tiny noise ([Formula: see text]) one has considerable error for the calculated exponents of order 0.01. For larger noise the error is growing up to 0.1 and more, with natural saturation limited by the embedding dimension. This prohibits the power-like scaling of the error. Moreover, the noise effect cannot be cured by taking larger data samples.

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Lim, Michael, Alit Kartiwa, and Herlina Napitupulu. "Estimation of Citarum Watershed Boundary’s Length Based on Fractal’s Power Law by the Modified Box-Counting Dimension Algorithm." Mathematics 11, no. 2 (January 11, 2023): 384. http://dx.doi.org/10.3390/math11020384.

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This research aimed to estimate the length of the Citarum watershed boundary because the data are still unknown. We used the concept of fractal’s power law and its relation to the length of an object, which is still not described in other research. The method that we used in this research is the Box-Counting dimension. The data were obtained from the geographic information system. We found an equation that described the relationship between the length and fractal dimension of an object by substituting equations. Following that, we modified the algorithm of Box-Counting dimension by consideration of requiring a high-resolution image, using the Canny edge detection so that the edges look sharper and the dimension values are more accurate. A Box-Counting program was created with Python based on the modified algorithm and used to execute the Citarum watershed boundary’s image. The values of ε and N were used to calculate the fractal dimension and the length for each scale by using the value of C=1, assuming the ε as the ratio between the length of box and the length of plane. Finally, we found that the dimension of Citarum watershed boundary is approximately 1.1109 and its length is 770.49 km.

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FALCONER, KENNETH, and TOM KEMPTON. "Planar self-affine sets with equal Hausdorff, box and affinity dimensions." Ergodic Theory and Dynamical Systems 38, no. 4 (October 20, 2016): 1369–88. http://dx.doi.org/10.1017/etds.2016.74.

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Using methods from ergodic theory along with properties of the Furstenberg measure we obtain conditions under which certain classes of plane self-affine sets have Hausdorff or box-counting dimensions equal to their affinity dimension. We exhibit some new specific classes of self-affine sets for which these dimensions are equal.

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De Bartolo, S. G., S. Gabriele, and R. Gaudio. "Multifractal behaviour of river networks." Hydrology and Earth System Sciences 4, no. 1 (March 31, 2000): 105–12. http://dx.doi.org/10.5194/hess-4-105-2000.

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Abstract. A numerical multifractal analysis was performed for five river networks extracted from Calabrian natural basins represented on 1:25000 topographic sheets. The spectrum of generalised fractal dimensions, D(q), and the sequence of mass exponents, τ(q), were obtained using an efficient generalised box-counting algorithm. The multi-fractal spectrum, f(α), was deduced with a Legendre transform. Results show that the nature of the river networks analysed is multifractal, with support dimensions, D(0), ranging between 1.76 and 1.89. The importance of the specific number of digitised points is underlined, in order to accurately define, the geometry of river networks through a direct generalised box-counting measure that is not influenced by their topology. The algorithm was also applied to a square portion of the Trionto river network to investigate border effects. Results confirm the multifractal behaviour, but with D(0) = 2. Finally, some open mathematical problems related to the assessment of the box-counting dimension are discussed. Keywords: River networks; measures; multifractal spectrum

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30

Falconer, K. J., and J. D. Howroyd. "Projection theorems for box and packing dimensions." Mathematical Proceedings of the Cambridge Philosophical Society 119, no. 2 (February 1996): 287–95. http://dx.doi.org/10.1017/s0305004100074168.

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AbstractWe show that if E is an analytic subset of ℝn thenfor almost all m–dimensional subspaces V of ℝn, where projvE is the orthogonal projection of E onto V and dimp denotes packing dimension. The same inequality holds for lower and upper box counting dimensions, and these inequalities are the best possible ones.

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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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Jiang, Hongxiang, Changlong Du, Songyong Liu, and Kuidong Gao. "Fractal Characteristic of Rock Cutting Load Time Series." Discrete Dynamics in Nature and Society 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/915136.

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A test-bed was developed to perform the rock cutting experiments under different cutting conditions. The fractal theory was adopted to investigate the fractal characteristic of cutting load time series and fragment size distribution in rock cutting. The box-counting dimension for the cutting load time series was consistent with the fractal dimension of the corresponding fragment size distribution, which indicated that there were inherent relations between the rock fragmentation and the cutting load. Furthermore, the box-counting dimension was used to describe the fractal characteristic of cutting load time series under different conditions. The results show that the rock compressive strength, cutting depth, cutting angle, and assisted water-jet types all have no significant effect on the fractal characteristic of cutting load. The box-counting dimension can be an evaluation index to assess the extent of rock crushing or cutting. Rock fracture mechanism would not be changed due to water-jet in front of or behind the cutter, but it would be changed when the water-jet was in cutter.

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Sun, Hong Quan, and Jun Ding. "The Comparison of Fractal Dimensions of Cracks on Reinforced Concrete Beam." Advanced Materials Research 291-294 (July 2011): 1126–30. http://dx.doi.org/10.4028/www.scientific.net/amr.291-294.1126.

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In the paper, fractal geometry is used to study the crack evolving process of reinforce concrete beams. The fractal dimensions on surface of the reinforced concrete beam and the mechanical properties of the beam have the linear relationships perfectly. In order to compare the accuracy of the fractal dimensions, box counting method and the digital image box method in practical engineering are used to calculate the fractal dimension separately. The advantages and the disadvantages of these methods are analyzed. And the calculating conditions of these two methods are obtained. The research result gives a better way for determining the fractal dimension of the cracks on the reinforced concrete beam.

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WEI, CHUN, SHENGYOU WEN, and ZHIXIONG WEN. "REMARKS ON DIMENSIONS OF CARTESIAN PRODUCT SETS." Fractals 24, no. 03 (August 30, 2016): 1650031. http://dx.doi.org/10.1142/s0218348x16500316.

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Given metric spaces [Formula: see text] and [Formula: see text], it is well known that [Formula: see text] [Formula: see text] and [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] denote the Hausdorff, packing, lower box-counting, and upper box-counting dimension of [Formula: see text], respectively. In this paper, we shall provide examples of compact sets showing that the dimension of the product [Formula: see text] may attain any of the values permitted by the above inequalities. The proof will be based on a study on dimension of products of sets defined by digit restrictions.

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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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YUN, CHOL-HUI, HUI-CHOL CHOI, and HYONG-CHOL O. "CONSTRUCTION OF RECURRENT FRACTAL INTERPOLATION SURFACES WITH FUNCTION SCALING FACTORS AND ESTIMATION OF BOX-COUNTING DIMENSION ON RECTANGULAR GRIDS." Fractals 23, no. 04 (December 2015): 1550030. http://dx.doi.org/10.1142/s0218348x15500309.

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We consider a construction of recurrent fractal interpolation surfaces (RFISs) with function vertical scaling factors and estimation of their box-counting dimension. A RFIS is an attractor of a recurrent iterated function system (RIFS) which is a graph of bivariate interpolation function. For any given dataset on rectangular grids, we construct general RIFSs with function vertical scaling factors and prove the existence of bivariate functions whose graph are attractors of the above-constructed RIFSs. Finally, we estimate lower and upper bounds for the box-counting dimension of the constructed RFISs.

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Tiwari, Ram Krishna, and Harihar Paudyal. "Fractal Structure of Seismic Signals of 2015 Gorkha-Kodari Earthquakes: A Box Counting Method." BMC Journal of Scientific Research 5, no. 1 (December 31, 2022): 18–26. http://dx.doi.org/10.3126/bmcjsr.v5i1.50667.

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Fractal dimension analysis is a computational image processing technique that allows assessing the degree of complexity in patterns. In seismology, fractal dimensions can be used to describe fractured surfaces quantitatively. The larger the fractal dimension the more rugged is the surface, the more irregular is the line, and the more complex is the pore space. For the present investigation the seismicwave signal of 40 earthquakes including one foreshock, main shock and 38 aftershocks (mb ≥ 5.0) of 2015 Gorkha-Kodari earthquakes from 2015/4/21 to 2016/11/27 were considered. The seismograms were retrieved from the archived waveform data of Incorporated Research Institutions of Seismology (IRIS). The fractal dimension (D) was evaluated by the Python program for box counting. It is found that the fractal dimensions of the seismic wave signal during the active seismic period do not show sudden variation and they are almost identical. The maximum value was noticed to be 1.99±0.006 and the minimum value to be 1.95±0.007. The estimated fractal dimension is greater or equal to 1.95 with an average value of 1.98 which signify the presence of high grade of fractality in seismic wave time series. This suggests that the fractal characteristics of the seismic wave signal of 2015 central Himalayan earthquakes occurrence behavior is nonlinear and coplanar.

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Stratmann, B., and Mariusz Urbański. "The box-counting dimension for geometrically finite Kleinian groups." Fundamenta Mathematicae 149, no. 1 (1996): 83–93. http://dx.doi.org/10.4064/fm-149-1-83-93.

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AHAMMER, H., and M. MAYRHOFER-REINHARTSHUBER. "IMAGE PYRAMIDS FOR CALCULATION OF THE BOX COUNTING DIMENSION." Fractals 20, no. 03n04 (September 2012): 281–93. http://dx.doi.org/10.1142/s0218348x12500260.

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The fractal dimensions of real world objects are commonly investigated using digital images. Unfortunately, these images are unable to represent an infinitesimal range of scales. In addition, a proper evaluation of the applied methods that encompass the image processing techniques is often missing. Several mathematical well-defined fractals with theoretically known fractal dimensions, represented by digital images, were investigated in this work. The very popular Box counting method was compared to a new image pyramid approach as well as to the Minkowski dilation method. Effects from noise and altered aspect ratios were also considered. The new Pyramid method is quite identical to the Box counting method, but it is easier to implement. Additionally, the calculation times are much shorter and memory requirements are almost comparable.

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So, Paul, Ernest Barreto, and Brian R. Hunt. "Box-counting dimension without boxes: ComputingD0from average expansion rates." Physical Review E 60, no. 1 (July 1, 1999): 378–85. http://dx.doi.org/10.1103/physreve.60.378.

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Margaris, Alexandros, and James C. Robinson. "Embedding properties of sets with finite box-counting dimension." Nonlinearity 32, no. 10 (August 20, 2019): 3523–47. http://dx.doi.org/10.1088/1361-6544/ab1b7f.

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James C. Robinson and Nicholas Sharples. "Strict Inequality in the Box-Counting Dimension Product Formulas." Real Analysis Exchange 38, no. 1 (2013): 95. http://dx.doi.org/10.14321/realanalexch.38.1.0095.

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Takeo, Fukiko. "Box-counting dimension of graphs of generalized Takagi series." Japan Journal of Industrial and Applied Mathematics 13, no. 2 (June 1996): 187–94. http://dx.doi.org/10.1007/bf03167241.

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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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TANAKA, MANABU, ATSUSHI KAYAMA, RYUICHI KATO, and YOSHIAKI ITO. "ESTIMATION OF THE FRACTAL DIMENSION OF FRACTURE SURFACE PATTERNS BY BOX-COUNTING METHOD." Fractals 07, no. 03 (September 1999): 335–40. http://dx.doi.org/10.1142/s0218348x99000335.

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In the box-counting method, positioning of images do not significantly affect the estimation of the fractal dimension of river pattern on the brittle fracture surface, and that of dimple pattern on the ductile fracture surface of materials. A reasonable estimation of the fractal dimension can be made using the box-counting method by a single measurement on the fracture surface pattern. The fractal dimension of dimple pattern in pure Zn polycrystals (about 1.50) is larger than that of river pattern in soda-lime glass (about 1.30). Personal difference in image processing does not have a large influence on the estimation of the fractal dimension of grain-boundary fracture surface profile, compared with the effects of local variation in fracture pattern concerning image size.

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Ma, Li, Li Shang, Long Zhang, and Wei Shi Shao. "An Effective Image Edge Detection Algorithm – Fuzzy Box-Counting." Applied Mechanics and Materials 543-547 (March 2014): 2711–15. http://dx.doi.org/10.4028/www.scientific.net/amm.543-547.2711.

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Edge detection plays an important role in computer vision and image processing. Fractal and Fuzzy theory show significant effect in the edge detection and have attracted much attention. Compared with traditional edge detection methods, this paper proposes a Fuzzy Box-counting Dimension Method (FBDM). This algorithm introduces the pre-judging mechanism to improve the speed of image segmentation, and the self-adaptive dimension threshold and the voting mechanism under multi-windows to improve the accuracy of the determination of edge points. Finally, closest principle is used to clear edge and reduce noise. Experimental results show FBDM can improve the precision of image edge detection effectively without pretreatment, and it has a very superior de-noising performance.

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Fan, Youping, Dai Zhang, and Jingjiao Li. "Study on the Fractal Dimension and Growth Time of the Electrical Treeing Degradation at Different Temperature and Moisture." Advances in Materials Science and Engineering 2018 (November 1, 2018): 1–10. http://dx.doi.org/10.1155/2018/6019269.

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The paper aims to understand how the fractal dimension and growth time of electrical trees change with temperature and moisture. The fractal dimension of final electrical trees was estimated using 2-D box-counting method. Four groups of electrical trees were grown at variable moisture and temperature. The relation between growth time and fractal dimension of electrical trees were summarized. The results indicate the final electrical trees can have similar fractal dimensions via similar tree growth time at different combinations of moisture level and temperature conditions.

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Alsaidi, Nadia M. G., Arkan J. Mohammed, and Wael J. Abdulaal. "Fingerprints Authentication Using Grayscale Fractal Dimension." Al-Mustansiriyah Journal of Science 29, no. 3 (March 10, 2019): 106. http://dx.doi.org/10.23851/mjs.v29i3.627.

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Characterizing of visual objects is an important role in pattern recognition that can be performed through shape analysis. Several approaches have been introduced to extract relevant information of a shape. The complexity of the shape is the most widely used approach for this purpose where fractal dimension and generalized fractal dimension are methodologies used to estimate the complexity of the shapes. The box counting dimension is one of the methods that used to estimate fractal dimension. It is estimated basically to describe the self-similarity in objects. A lot of objects have the self-similarity; fingerprint is one of those objects where the generalized box counting dimension is used for recognizing of the fingerprints to be utilized for authentication process. A new fractal dimension method is proposed in this paper. It is verified by the experiment on a set of natural texture images to show its efficiency and accuracy, and a satisfactory result is found. It also offers promising performance when it is applied for fingerprint recognition.

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Braverman, Boris, and Mauro Tambasco. "Scale-Specific Multifractal Medical Image Analysis." Computational and Mathematical Methods in Medicine 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/262931.

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Fractal geometry has been applied widely in the analysis of medical images to characterize the irregular complex tissue structures that do not lend themselves to straightforward analysis with traditional Euclidean geometry. In this study, we treat the nonfractal behaviour of medical images over large-scale ranges by considering their box-counting fractal dimension as a scale-dependent parameter rather than a single number. We describe this approach in the context of the more generalized Rényi entropy, in which we can also compute the information and correlation dimensions of images. In addition, we describe and validate a computational improvement to box-counting fractal analysis. This improvement is based on integral images, which allows the speedup of any box-counting or similar fractal analysis algorithm, including estimation of scale-dependent dimensions. Finally, we applied our technique to images of invasive breast cancer tissue from 157 patients to show a relationship between the fractal analysis of these images over certain scale ranges and pathologic tumour grade (a standard prognosticator for breast cancer). Our approach is general and can be applied to any medical imaging application in which the complexity of pathological image structures may have clinical value.

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Tanaka, M., Y. Kimura, A. Kayama, L. Chouanine, Reiko Kato, and J. Taguchi. "Image Reconstruction and Analysis of Three-Dimensional Fracture Surfaces Based on the Stereo Matching Method." Key Engineering Materials 261-263 (April 2004): 1593–98. http://dx.doi.org/10.4028/www.scientific.net/kem.261-263.1593.

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A computer program of the fractal analysis by the box-counting method was developed for the estimation of the fractal dimension of the three-dimensional fracture surface reconstructed by the stereo matching method. The image reconstruction and fractal analysis were then made on the fracture surfaces of materials created by different mechanisms. There was a correlation between the fractal dimension of the three-dimensional fracture surface and the fractal dimensions evaluated by other methods on ceramics and metals. The effects of microstructures on the fractal dimension were also experimentally discussed.

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