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1
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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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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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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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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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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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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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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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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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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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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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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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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Wang, Ji Zhe, and Qing Jie Guan. "Numerical Simulation of Cell Growth Pattern and Determination of Fractal Dimension of Cell Cluster." Advanced Materials Research 690-693 (May 2013): 1229–33. http://dx.doi.org/10.4028/www.scientific.net/amr.690-693.1229.
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Life system behaves self-similar properties from microcosms to macrostructure. Based on the cell growth roles, the cell cluster growth process is simulated. The sandbox method and box counting are used for determining the fractal dimension of cell associated with the geometrical structure of growing deterministic fractals. The fractal dimension of cell shape is estimated according to the slope of line between the numbers of boxes and box size in double logarithm coordinates.
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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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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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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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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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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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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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JÄRVENPÄÄ, ESA, MAARIT JÄRVENPÄÄ, ANTTI KÄENMÄKI, HENNA KOIVUSALO, ÖRJAN STENFLO, and VILLE SUOMALA. "Dimensions of random affine code tree fractals." Ergodic Theory and Dynamical Systems 34, no. 3 (January 30, 2013): 854–75. http://dx.doi.org/10.1017/etds.2012.168.
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AbstractWe study the dimension of code tree fractals, a class of fractals generated by a set of iterated function systems. We first consider deterministic affine code tree fractals, extending to the code tree fractal setting the classical result of Falconer and Solomyak on the Hausdorff dimension of self-affine fractals generated by a single iterated function system. We then calculate the almost sure Hausdorff, packing and box counting dimensions of a general class of random affine planar code tree fractals. The set of probability measures describing the randomness includes natural measures in random $V$-variable and homogeneous Markov constructions.
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Nasr, Pedram, Hannah Leung, France-Isabelle Auzanneau, and Michael A. Rogers. "Supramolecular Fractal Growth of Self-Assembled Fibrillar Networks." Gels 7, no. 2 (April 14, 2021): 46. http://dx.doi.org/10.3390/gels7020046.
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Complex morphologies, as is the case in self-assembled fibrillar networks (SAFiNs) of 1,3:2,4-Dibenzylidene sorbitol (DBS), are often characterized by their Fractal dimension and not Euclidean. Self-similarity presents for DBS-polyethylene glycol (PEG) SAFiNs in the Cayley Tree branching pattern, similar box-counting fractal dimensions across length scales, and fractals derived from the Avrami model. Irrespective of the crystallization temperature, fractal values corresponded to limited diffusion aggregation and not ballistic particle–cluster aggregation. Additionally, the fractal dimension of the SAFiN was affected more by changes in solvent viscosity (e.g., PEG200 compared to PEG600) than crystallization temperature. Most surprising was the evidence of Cayley branching not only for the radial fibers within the spherulitic but also on the fiber surfaces.
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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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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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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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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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Liu, Zheng, and Xiao Mei Liu. "Fractal Characteristics of Primary Phase Morphology in Semisolid A356 Alloy." Advanced Materials Research 535-537 (June 2012): 936–40. http://dx.doi.org/10.4028/www.scientific.net/amr.535-537.936.
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Semisolid A356 alloy was prepared by low superheat pouring and slightly electro- magnetic stirring(LSPSES). The fractal dimensions of primary phase morphology in semisolid A356 alloy were researched by the calculating program written to calculate the fractal dimensions of box-counting in the image of primary phase morphology in semisolid A356 alloy. The results indicated that the primary phase morphology in the alloy was characterized by fractal dimension, and the morphology obtained by the different processing parameters had the different fractal dimension. The morphology at the different position of ingot had the different fractal dimensions, which reflected the effect of solidified conditions at different position in the same ingot on the morphology in the alloy. Solidification of the alloy was a course of change in fractal dimension.
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Xu, Jie, and Giusepe Lacidogna. "A Modified Box-Counting Method to Estimate the Fractal Dimensions." Applied Mechanics and Materials 58-60 (June 2011): 1756–61. http://dx.doi.org/10.4028/www.scientific.net/amm.58-60.1756.
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A fractal is a property of self-similarity, each small part of the fractal object is similar to the whole body. The traditional box-counting method (TBCM) to estimate fractal dimension can not reflect the self-similar property of the fractal and leads to two major problems, the border effect and noninteger values of box size. The modified box-counting method (MBCM), proposed in this study, not only eliminate the shortcomings of the TBCM, but also reflects the physical meaning about the self-similar of the fractal. The applications of MBCM shows a good estimation compared with the theoretical ones, which the biggest difference is smaller than 5%.
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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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Maryenko, N. I., and O. Y. Stepanenko. "Fractal dimension of external linear contour of human cerebellum (magnetic resonance imaging study)." Reports of Morphology 27, no. 2 (June 25, 2021): 16–22. http://dx.doi.org/10.31393/morphology-journal-2021-27(2)-03.
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Fractal analysis is a method of mathematical analysis, which provides quantitative assessment of the spatial configuration complexity of the anatomical structures and may be used as a morphometric method. The purpose of the study was to determine the values of the fractal dimension of the outer linear contour of human cerebellum by studying the magnetic resonance images of the brain using the authors’ modification of the caliper method and compare to the values determined using the box counting method. Brain magnetic resonance images of 30 relatively healthy persons aged 18-30 years (15 men and 15 women) were used in the study. T2-weighted digital magnetic resonance images were studied. The midsagittal MR sections of the cerebellar vermis were investigated. The caliper method in the author’s modification was used for fractal analysis. The average value of the fractal dimension of the linear contour of the cerebellum, determined using the caliper method, was 1.513±0.008 (1.432÷1.600). The average value of the fractal dimension of the linear contour of the cerebellum, determined using the box counting method, was 1.530±0.010 (1.427÷1.647). The average value of the fractal dimension of the cerebellar tissue as a whole, determined using the box counting method, was 1.760±0.006 (1.674÷1.837). The values of the fractal dimension of the outer linear contour of the cerebellum, determined using the caliper method and the box counting method were not statistically significantly different. Therefore, both methods can be used for fractal analysis of the linear contour of the cerebellum. Fractal analysis of the outer linear contour of the cerebellum allows to quantify the complexity of the spatial configuration of the outer surface of the cerebellum, which is difficult to estimate using traditional morphometric methods. The data obtained from this study and the methodology of the caliper method of fractal analysis in the author’s modification can be used for morphometric investigations of the human cerebellum in morphological studies, as well as in assessment of cerebellar MR images for diagnostic purposes.
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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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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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XIA, YUXUAN, JIANCHAO CAI, WEI WEI, XIANGYUN HU, XIN WANG, and XINMIN GE. "A NEW METHOD FOR CALCULATING FRACTAL DIMENSIONS OF POROUS MEDIA BASED ON PORE SIZE DISTRIBUTION." Fractals 26, no. 01 (February 2018): 1850006. http://dx.doi.org/10.1142/s0218348x18500068.
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Fractal theory has been widely used in petrophysical properties of porous rocks over several decades and determination of fractal dimensions is always the focus of researches and applications by means of fractal-based methods. In this work, a new method for calculating pore space fractal dimension and tortuosity fractal dimension of porous media is derived based on fractal capillary model assumption. The presented work establishes relationship between fractal dimensions and pore size distribution, which can be directly used to calculate the fractal dimensions. The published pore size distribution data for eight sandstone samples are used to calculate the fractal dimensions and simultaneously compared with prediction results from analytical expression. In addition, the proposed fractal dimension method is also tested through Micro-CT images of three sandstone cores, and are compared with fractal dimensions by box-counting algorithm. The test results also prove a self-similar fractal range in sandstone when excluding smaller pores.
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Hermán, Peter, László Kocsis, and Andras Eke. "Fractal Branching Pattern in the Pial Vasculature in the Cat." Journal of Cerebral Blood Flow & Metabolism 21, no. 6 (June 2001): 741–53. http://dx.doi.org/10.1097/00004647-200106000-00012.
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Arborization pattern was studied in pial vascular networks by treating them as fractals. Rather than applying elaborate taxonomy assembled from measures from individual vessel segments and bifurcations arranged in their branching order, the authors' approach captured the structural details at once in high-resolution digital images processed for the skeleton of the networks. The pial networks appear random and at the same time having structural elements similar to each other when viewed at different scales—a property known as self-similarity revealed by the geometry of fractals. Fractal (capacity) dimension, Dcap, was calculated to evaluate the network's spatial complexity by the box counting method (BCM) and its variant, the extended counting method (XCM). Box counting method and XCM were subject to numerical testing on ideal fractals of known D. The authors found that precision of these fractal methods depends on the fractal character (branching, nonbranching) of the structure they evaluate. Dcap s (group mean ± SD) for the arterial and venous pial networks in the cat (n = 6) are 1.37 ± 0.04, 1.37 ± 0.02 by XCM, and 1.30 ± 0.04, 1.31 ± 0.03 by BCM, respectively. The arterial and venous systems thus appear to be developed according to the same fractal generation rule in the cat.
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JOHNSEN, WILLIAM A., and CHRISTOPHER A. BROWN. "COMPARISON OF SEVERAL METHODS FOR CALCULATING FRACTAL-BASED TOPOGRAPHIC CHARACTERIZATION PARAMETERS." Fractals 02, no. 03 (September 1994): 437–40. http://dx.doi.org/10.1142/s0218348x94000600.
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The objective of this work is to compare fractal-based, topographic characterization parameters calculated by several different fractal analysis methods. Four fractal characterization methods (compass, patchwork, box counting, and 2-point correlation) are systematically applied to five topographic data sets, which encompass a wide range of scale, and the results are compared. The compass and patchwork methods calculate similar values for the fractal dimension and smooth/rough crossover. The box and 2-point correlation methods calculate similar values for the fractal dimension. The compass and patchwork methods are capable of calculating the smooth/rough crossover.
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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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Liu, Huabo, Fanjing Meng, and Shaozhen Hua. "4D Mapping of the Fracture Evolution in a Printed Gypsum-Like Core by Using X-Ray CT Scanning." Advances in Civil Engineering 2021 (April 17, 2021): 1–12. http://dx.doi.org/10.1155/2021/8820828.
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The paper presents the use of micro-X-ray computed tomography (CT) system and associated automatic loading device in visualizing and analyzing the propagation of penny-shaped flaw in gypsum-like 3D printing specimen. During the loading process, a micro-X-ray computed tomography (CT) system was used to scan the specimen with a resolution of 30 × 30 μm2. The volumetric images of specimen were reconstructed based on two-dimensional images. Thus, the propagation of penny-shaped flaw in gypsum-like 3D printing specimen in spatial was observed. The device can record the evolution of the internal penny-shaped flaw by X-ray CT scanning and the evolution of the surface crack by digital radiography at the same time. Fractal analysis was employed to quantify the cracking process. Two- and three-dimensional box-counting methods were applied to analyze slice images and volumetric images, respectively. Comparison between fractal dimensions calculated from two- and three-dimensional box-counting method was carried out. The results show that the fractal dimension increases with the propagation of cracks. Moreover, the common approach to obtain the 3D fractal dimension of a self-similar fractal object by adding one to its corresponding 2D fractal dimension is found to be inappropriate.
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YU, BOMING, L. JAMES LEE, and HANQIANG CAO. "FRACTAL CHARACTERS OF PORE MICROSTRUCTURES OF TEXTILE FABRICS." Fractals 09, no. 02 (June 2001): 155–63. http://dx.doi.org/10.1142/s0218348x01000610.
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It is found that the pore microstructures of textile fabrics, widely used in the manufacture of fiber-reinforced composites, exhibit the fractal characters. The fractal behaviors are described by the proposed analytical method and measured by the box-counting method for the three different types of textile fabrics: plain woven, four-harness, bidirectional-stitched fiberglass mats. The pore area fractal dimension is derived analytically and found to be the function of the porosity and architectural parameters of fabrics. The results indicate that the fractal characters are isotropic although the fabrics are rothotropic in structures. The theoretical predictions by the proposed analytical model are in good agreement with those from the box-counting method, and this verifies the proposed fractal dimension model. The present fractal analysis may have the potential and significance on fractal analysis of transport properties (such as the permeability, dispersion, thermal and mechanical properties) in porous media.
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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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Pizzetti Mariano, Pedro Oscar, and Gabriela Pinho Mallmann. "Parametric process of a box-counting model for evaluation of fractal compositions." arq.urb, no. 31 (August 9, 2021): 114–24. http://dx.doi.org/10.37916/arq.urb.vi31.517.
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his article seeks to create and evaluate a parametric process that allows identifying the D dimension in fractal compositions through a Box-counting tool. The use of this method in a parametric process allows sequential tests that can confirm whether a composition is a fractal structure. The theoretical survey and the development of a parametric process in a visual algorithm were carried out for the development of this research. The tests with the tool took place in linear fractal compositions already known and developed in another research. As a result, it was possible to compare the D dimension of different compositions, made with fractal geometric patterns. In the conclusion, it was possible to observe that the process through a parametric tool was successful in making it possible to evaluate compositions and arrangements in an agile way. A direct relationship was identified between the iterations used and the proportional increase dimension D.
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Yu, Jack C., Ronald L. Wright, Matthew A. Williamson, James P. Braselton, and Martha L. Abell. "A Fractal Analysis of Human Cranial Sutures." Cleft Palate-Craniofacial Journal 40, no. 4 (July 2003): 409–15. http://dx.doi.org/10.1597/1545-1569_2003_040_0409_afaohc_2.0.co_2.
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Objectives Many biological structures are products of repeated iteration functions. As such, they demonstrate characteristic, scale-invariant features. Fractal analysis of these features elucidates the mechanism of their formation. The objectives of this project were to determine whether human cranial sutures demonstrate self-similarity and measure their exponents of similarity (fractal dimensions). Design One hundred three documented human skulls from the Terry Collection of the Smithsonian Institution were used. Their sagittal sutures were digitized and the data converted to bitmap images for analysis using box-counting method of fractal software. Results The log-log plots of the number of boxes containing the sutural pattern, Nr, and the size of the boxes, r, were all linear, indicating that human sagittal sutures possess scale-invariant features and thus are fractals. The linear portion of these log-log plots has limits because of the finite resolution used for data acquisition. The mean box dimension, Db, was 1.29289 ± 0.078457 with a 95% confidence interval of 1.27634 to 1.30944. Conclusions Human sagittal sutures are self-similar and have a fractal dimension of 1.29 by the box-counting method. The significance of these findings includes: sutural morphogenesis can be described as a repeated iteration function, and mathematical models can be constructed to produce self-similar curves with such Db. This elucidates the mechanism of actual pattern formation. Whatever the mechanisms at the cellular and molecular levels, human sagittal suture follows the equation log Nr = 1.29 log 1/r, where Nr is the number of square boxes with sides r that are needed to contain the sutural pattern and r equals the length of the sides of the boxes.
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Stypa, Jörn. "Numerical Estimates of the Fractal Dimension of the Spatial Human Bronchial Tree by Two-Dimensional Box-Counting-Method." Fractals 06, no. 01 (March 1998): 87–93. http://dx.doi.org/10.1142/s0218348x98000109.
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Photographs of the human bronchial trees were obtained from resin casts. After data editing, the fractal dimension of the contour of the bronchial tree was determined by the 2D box-counting-method (2D-BCM). Mathematical theorems about projections and intersections allow us to determine the fractal dimension of the spatial human bronchial tree.
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Liu, Shi Feng, Hui Ping Tang, Xin Yang, and Zhao Hui Zhang. "Fractal Research of Pore-Structure in Porous Titanium Fibers." Materials Science Forum 804 (October 2014): 259–62. http://dx.doi.org/10.4028/www.scientific.net/msf.804.259.
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This paper adopted the vacuum sintering technology to prepare titanium fiber porous material with a three-dimensional spatial network fiber backbone and connectivity pore structure. With the help of fractal geometry theory and scanning and digitizing the image, the fractal research of pore-structure in porous titanium fibers is executed and we studied the influence of adopting the box-counting dimension method to calculate the fractal dimension. Additionally, we determined the quantitative relationship between fractal dimension and the porosity of the porous in titanium fiber, while described the physical meaning of the fractal dimension.
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Zhou, Yuankai, Xue Zuo, and Hua Zhu. "A fractal view on running-in process: taking steel-on-steel tribo-system as an example." Industrial Lubrication and Tribology 71, no. 4 (May 7, 2019): 557–63. http://dx.doi.org/10.1108/ilt-08-2018-0319.
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Purpose Running-in is a transient process prior to steady state and of great importance for mechanical performance. To reveal the fractal behavior in the running-in process, the steel-on-steel friction and wear tests were performed. Design/methodology/approach The friction coefficient, friction temperature, friction noise and vibration were recorded, and the surface profile of lower sample was measured on line. The signals and profiles were characterized by correlation dimension and box-counting dimension, respectively. Findings The signals have the consistent fractal evolvement law, that is, the correlation dimension increases and tends to a stable value. The box-counting dimension of one surface becomes close to that of the other surface. The running-in process can be interpreted as a process in which the fractal dimension of friction signals increases, and the counter surfaces spontaneously adapt to and modify each other to form a spatial ordered structure. Originality/value The results reveal the running-in behavior from a new perspective.
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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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Osman, Daniel, David Newitt, Alice Gies, Thomas Budinger, Vu Hao Truong, Sharmila Majumdar, and John Kinney. "Fractal Based Image Analysis of Human Trabecular Bone using the Box Counting Algorithm." Fractals 06, no. 03 (September 1998): 275–83. http://dx.doi.org/10.1142/s0218348x98000328.
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An image texture measure based on the box counting algorithm is evaluated for its potential to characterize human trabecular bone structure in medical images. Although bone images lack the self-similarity of theoretical fractals, bone images are candidates for characterization using fractal analysis because of their highly complex structure. The fractal based measure, herein called the box counting dimension (BCD), is an effective dimension, and does not imply an underlying fractal geometry. The importance of resolution in quantifying bone characteristics using the BCD is addressed. The relationship of BCD to standard measures of trabecular bone structure is also analyzed. To evaluate the variability of the BCD with change in resolution, the BCD is determined for two sections from each of seven 3D X-ray Tomographic Microscopy (XTM) images of human radius bone specimens, while the resolution is varied using lowpass filtering. An automated method of choosing the range of scales for the fractal analysis curve regression is used. The relationship of BCD to trabecular bone width and spacing is analyzed both for the XTM images and for simulated images representing idealized structures. The range of BCD values is 1.21–1.54. Variation in BCD over a range of resolutions is found to be small compared to the variation in BCD between different bone specimens. Maximum change in BCD over a large range of resolutions (17.60–176 microns per pixel) is 0.08. BCD decreases as space between trabeculae increases. Fractal based texture measures may potentially allow clinical monitoring of changes in bone structure — for example, using Magnetic Resonance Imaging at 150–200 micron resolution.
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Parkinson, Ian, and Nick Fazzalari. "FRACTAL ANALYSIS OF TRABECULAR BONE: A STANDARDISED METHODOLOGY." Image Analysis & Stereology 19, no. 1 (May 3, 2011): 45. http://dx.doi.org/10.5566/ias.v19.p45-49.
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A standardised methodology for the fractal analysis of histological sections of trabecular bone has been established. A modified box counting method has been developed for use on a PC based image analyser (Quantimet 500MC, Leica Cambridge). The effect of image analyser settings, magnification, image orientation and threshold levels, was determined. Also, the range of scale over which trabecular bone is effectively fractal was determined and a method formulated to objectively calculate more than one fractal dimension from the modified Richardson plot. The results show that magnification, image orientation and threshold settings have little effect on the estimate of fractal dimension. Trabecular bone has a lower limit below which it is not fractal (λ<25 μm) and the upper limit is 4250 μm. There are three distinct fractal dimensions for trabecular bone (sectional fractals), with magnitudes greater than 1.0 and less than 2.0. It has been shown that trabecular bone is effectively fractal over a defined range of scale. Also, within this range, there is more than 1 fractal dimension, describing spatial structural entities. Fractal analysis is a model independent method for describing a complex multifaceted structure, which can be adapted for the study of other biological systems. This may be at the cell, tissue or organ level and compliments conventional histomorphometric and stereological techniques.
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BRADLEY, DAVID M., ANDRÉ KHALIL, ROBERT G. NIEMEYER, and ELLIOT OSSANNA. "THE BOX-COUNTING DIMENSION OF PASCAL’S TRIANGLE r mod p." Fractals 26, no. 05 (October 2018): 1850071. http://dx.doi.org/10.1142/s0218348x18500718.
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We consider Pascal’s Triangle [Formula: see text] to be the entries of Pascal’s Triangle that are congruent to [Formula: see text]. Such a representation of Pascal’s Triangle exhibits fractal-like structures. When the Triangle is mapped to a subset of the unit square, we show that such a set is nonempty and exists as a limit of a sequence of coarse approximations. We then show that for any given prime [Formula: see text], any such sequence converges to the same set, regardless of the residue(s) considered. As an obvious consequence, this allows us to conclude that the fractal (box-counting) dimension of this nonempty, compact representation of Pascal’s Triangle [Formula: see text] is independent of [Formula: see text].
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Garafutdinov, Robert, and Sofya Akhunyanova. "ADAPTED BOX-COUNTING METHOD FOR ASSESSMENT OF THE FRACTAL DIMENSION OF FINANCIAL TIME SERIES." Applied Mathematics and Control Sciences, no. 3 (October 5, 2020): 185–218. http://dx.doi.org/10.15593/2499-9873/2020.3.10.
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This paper continues research within the framework of the scientific direction in econophysics at the Department of Information Systems and Mathematical Methods in Economics of the faculty of Economics of PSU. Modeling and prediction of financial time series is quite a perspective area of research, because it allows participants of financial processes to reduce risks and make effective decisions. For example, we could research financial processes with the help of fractal analysis. In the article there is studied and worked out in detail one of the methods of fractal analysis of financial time series – the box-counting method for assessment of the fractal dimension. This method is often used in studies conducted by domestic authors, but the authors do not delve into the characteristics and problems of using the box-counting method for analysis of time series, that means that the answers to the interested questions have not yet been given. The main problem is that, as a rule, the analyzed object in the tasks of applying the box-counting method to time series is a computer image of the plot of series. In the article there is proposed the procedure of adaptation of the box-counting method for assessment of the fractal dimension of time series, the procedure does not require the formation of a computer image of the plot. In the article there is considered following difficulties developed from this adaptation: 1) high sensitivity of the resulting estimation of the dimension to the input parameters of the method (the ratio of the sides of the covered by cells plane with the plot; the used range of lengths of the side of the cell; the number of partitions of the plane into cells); 2) the non-obviousness of choosing the optimal values of these parameters. In the article there are analyzed approaches to the selection of these parameters that were proposed by other authors, and there are determined the most suitable approaches for the adapted box-counting method. Also there are developed unique methods for determining the ratio of the sides of the plane with the plot. In the paper there is written the computer program that implements the developed method, and this program is tested on the generated data. The study obtained the following results. The fact of sensitivity of the adapted box-counting method to input parameters is confirmed, that indicates the high importance of the correct choice of these parameters. According to the study, there is found out inability of the proposed methods of automatic determination the ratio of the sides of the plane in relation to artificial time series. There are obtained the most precise (in a statistical sense) estimates of fractal dimension, those found by means of the adapted box-counting method, with the fixed ratio of the sides 1:1. According to comparing the adapted box-counting method and R/S analysis, there are obtained the most precise estimates by the second method (R/S analysis). Finally in the paper there are formulated the possible directions for further research: 1) comparison of the accuracy of various methods for assessment of the fractal dimension on series of different lengths; 2) comparison of the methods of fractal analysis and p-adic analysis for modeling and prediction of financial time series; 3) determination of the conditions of applicability of various methods; 4) approbation of the developed methods for determining of the ratio of the sides of the plane with the plot on real economic data.
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METZLER, WOLFGANG, and CHOL HUI YUN. "CONSTRUCTION OF FRACTAL INTERPOLATION SURFACES ON RECTANGULAR GRIDS." International Journal of Bifurcation and Chaos 20, no. 12 (December 2010): 4079–86. http://dx.doi.org/10.1142/s0218127410027933.
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We present a general method for generating continuous fractal interpolation surfaces by iterated function systems on an arbitrary data set over rectangular grids and estimate their Box-counting dimension.
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Zhang, Ju, Qingwu Hu, Hongyu Wu, Junying Su, and Pengcheng Zhao. "Application of Fractal Dimension of Terrestrial Laser Point Cloud in Classification of Independent Trees." Fractal and Fractional 5, no. 1 (February 1, 2021): 14. http://dx.doi.org/10.3390/fractalfract5010014.
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Tree precise classification and identification of forest species is a core issue of forestry resource monitoring and ecological effect assessment. In this paper, an independent tree species classification method based on fractal features of terrestrial laser point cloud is proposed. Firstly, the terrestrial laser point cloud data of an independent tree is preprocessed to obtain terrestrial point clouds of independent tree canopy. Secondly, the multi-scale box-counting dimension calculation algorithm of independent tree canopy dense terrestrial laser point cloud is proposed. Furthermore, a robust box-counting algorithm is proposed to improve the stability and accuracy of fractal dimension expression of independent tree point cloud, which implementing gross error elimination based on Random Sample Consensus. Finally, the fractal dimension of a dense terrestrial laser point cloud of independent trees is used to classify different types of independent tree species. Experiments on nine independent trees of three types show that the fractal dimension can be stabilized under large density variations, proving that the fractal features of terrestrial laser point cloud can stably express tree species characteristics, and can be used for accurate classification and recognition of forest species.