Network Box Counting Heuristics

2020 ◽  
pp. 145-175
Author(s):  
Eric Rosenberg
Keyword(s):  
Box Counting

3D box-counting algorithm for calculating fractal dimension of cities

2010 ◽  
Vol 30 (8) ◽  
pp. 2070-2072
Author(s):  
Le-shan ZHANG ◽  
Ge CHEN ◽  
Yong HAN ◽  
Tao ZHANG
Keyword(s):  
Fractal Dimension ◽  
Box Counting ◽  
Counting Algorithm

scholarly journals Supramolecular Fractal Growth of Self-Assembled Fibrillar Networks

Gels ◽  
2021 ◽  
Vol 7 (2) ◽  
pp. 46
Author(s):  
Pedram Nasr ◽  
Hannah Leung ◽  
France-Isabelle Auzanneau ◽  
Michael A. Rogers
Keyword(s):  
Fractal Dimension ◽  
Crystallization Temperature ◽  
Cayley Tree ◽  
Fractal Dimensions ◽  
Self Similarity ◽  
Avrami Model ◽  
Cluster Aggregation ◽  
Box Counting ◽  
Self Assembled ◽  
Limited Diffusion

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.

CT Image Analysis on the Meso-Damage of Construction Waste Recycled Brick

2012 ◽  
Vol 588-589 ◽  
pp. 1930-1933
Author(s):  
Guo Song Han ◽  
Hai Yan Yang ◽  
Xin Pei Jiang
Keyword(s):  
Fractal Dimension ◽  
Strain Curve ◽  
Ct Image ◽  
Stress Strain Curve ◽  
Crack Evolution ◽  
Construction Waste ◽  
Box Counting ◽  
Industrial Ct ◽  
Ct Image Analysis ◽  
Box Counting Method

Based on industrial CT technique, Meso-mechanical experiment was conducted on construction waste recycled brick to get the real-time CT image and stress-strain curve of brick during the loading process. Box counting method was used to calculate the fractal dimension of the inner pore transfixion and crack evolution. The results showed that lots of pore in the interfacial transition zone mainly resulted in the damage of the brick. With the increase of stress, the opening through-pore appeared and crack expanded, and the fractal dimension increased.

scholarly journals ASSESSMENT OF URBAN SPRAWL USING SHANNON’S ENTROPY AND FRACTAL ANALYSIS: A CASE STUDY OF ATAKUM, ILKADIM AND CANIK (SAMSUN, TURKEY)

2017 ◽  
Vol 25 (3) ◽  
pp. 264-276 ◽  
Author(s):  
Derya OZTURK
Keyword(s):  
Urban Sprawl ◽  
Fractal Analysis ◽  
Urban Form ◽  
Entropy Method ◽  
Shannon’S Entropy ◽  
Analysis Method ◽  
Maximum Likelihood Classification ◽  
Shannon's Entropy ◽  
Box Counting

Urban sprawl is one of the most important problems in urban development due to its negative environmental and societal impacts. Therefore, the spatial pattern of urban growth should be accurately analyzed and well understood for effective urban planning. This paper focuses on urban sprawl analysis in the Atakum, Ilkadim and Canik districts of Samsun, Turkey. In this study, urban sprawl was examined over a period of 24 years using Shannon's entropy and fractal analysis based on remote sensing and Geographic Information System (GIS). The built-up areas in 1989, 2000 and 2013 were extracted from Landsat TM/ETM+/OLI images using the maximum likelihood classification method, and urban form changes in the 1989–2013 period were investigated. The Shannon's entropy method was used to determine the degree of urban sprawl, and a fractal analysis method based on box counting was used to characterize the urban sprawl. The results show that Atakum, Ilkadim and Canik experienced important changes and have considerable sprawl and complex characteristics now. The study also revealed that there is no monotonic relationship between Shannon's entropy and fractal dimension.

The dimension of self-affine fractals II

1992 ◽  
Vol 111 (1) ◽  
pp. 169-179 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Exact Expression ◽  
Singular Value ◽  
Compact Set ◽  
Value Functions ◽  
Affine Transformations ◽  
Box Counting ◽  
Box Counting Dimension ◽  
Almost All

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.

Outer Cutoff Value for the Box-Counting Method for Fractal Analysis of the Nucleus Using Kirsch Edge Detection

2020 ◽  
pp. 1-8
Author(s):  
Haruhiko Yoshioka ◽  
Kouki Minami ◽  
Hirokazu Odashima ◽  
Keita Miyakawa ◽  
Kayo Horie ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Edge Detection ◽  
Fractal Analysis ◽  
Nuclear Size ◽  
Original Position ◽  
Cytological Diagnosis ◽  
Cutoff Value ◽  
Position Bias ◽  
Box Counting ◽  
Nuclear Rotation

<b><i>Objective:</i></b> The complexity of chromatin (i.e., irregular geometry and distribution) is one of the important factors considered in the cytological diagnosis of cancer. Fractal analysis with Kirsch edge detection is a known technique to detect irregular geometry and distribution in an image. We examined the outer cutoff value for the box-counting (BC) method for fractal analysis of the complexity of chromatin using Kirsch edge detection. <b><i>Materials:</i></b> The following images were used for the analysis: (1) image of the nucleus for Kirsch edge detection measuring 97 × 122 pix (10.7 × 13.4 μm) with a Feret diameter of chromatin mesh (<i>n</i> = 50) measuring 17.3 ± 1.8 pix (1.9 ± 0.5 μm) and chromatin network distance (<i>n</i> = 50) measuring 4.4 ± 1.6 pix (0.49 ± 0.18 μm), and (2) sample images for Kirsch edge detection with varying diameters (10.4, 15.9, and 18.1 μm) and network width of 0.4 μm. <b><i>Methods:</i></b> Three types of bias that can affect the outcomes of fractal analysis in cytological diagnosis were defined. (1) Nuclear position bias: images of 9 different positions generated by shifting the original position of the nucleus in the middle of a 256 × 256 pix (28.1 μm) square frame in 8 compass directions. (2) Nuclear rotation bias: images of 8 different rotations obtained by rotating the original position of the nucleus in 45° increments (0°, 45°, 90°, 135°, 180°, 225°, 270°, and 315°). (3) Nuclear size bias: images of varying size (diameter: 190 pix [10.4 μm], 290 pix [15.9 μm], and 330 pix [18.1 μm]) with the same mesh pattern (network width: 8 pix [0.4 μm]) within a 512 × 512 pix square. Different outer cutoff values for the BC method (256, 128, 64, 32, 16, and 8 pix) were applied for each bias to assess the fractal dimension and to compare the coefficient of variation (CV). <b><i>Results:</i></b> The BC method with the outer cutoff value of 32 pix resulted in the least variation of fractal dimension. Specifically, with the cutoff value of 32 pix, the CV of nuclear position bias, nuclear rotation bias, and nuclear size bias were &#x3c;1% (0.1, 0.4, and 0.3%, respectively), with no significant difference between the position and rotation bias (<i>p</i> = 0.19). Our study suggests that the BC method with the outer cutoff value of 32 pix is suitable for the analysis of the complexity of chromatin with chromatin mesh.

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