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Fractal Dimensions Analysis and Morphological Investigation of Nanomedicine by Machine-Learning Methods
Published in Omari V. Mukbaniani, Tamara N. Tatrishvili, Marc J. M. Abadie, Science and Technology of Polymers and Advanced Materials, 2019
In this research, box-counting dimension method has been used. To calculate the box-counting dimension, the picture is placed on a grid. The x-axis of the grid is r where r = 1/(width of the grid). For example, if the grid is 240 blocks tall by 120 blocks wide, r = 1/120. Then, count the number of blocks that the picture touches. Label this number N, then, resize the grid and repeat the process. Plot the values found on a graph where the x-axis is the log (r), and the y-axis is the log (N). Draw in the line of best fit and find the slope. The box-counting dimension measure is equal to the slope of that line. Figure 25.9 (a) shows application of Box-counting dimension method on Koch curve and Figure 25.9 (b) illustrate linear diagram of fractal dimension for Koch curve.
Why Economics Has Not Accomplished What Physics Has?
Published in Christos H. Skiadas, Charilaos Skiadas, Handbook of Applications of Chaos Theory, 2017
In the literature, there are many methods * for calculating the fractal dimension (Hausdorff dimension, the box-counting dimension, the information dimension, and the correlation dimension), which nevertheless do not provide equivalent measures (Hentschel and Procaccia, 1983). Among these different algorithms, the correlation dimension proposed by Grassberger and Procaccia (1983a), based on phase space reconstructions of the process to estimate,† has the advantage of being straightforward and quickly implemented.
Autocorrelation Function, Mutual Information, and Correlation Dimension
Published in Nicholas Stergiou, Nonlinear Analysis for Human Movement Variability, 2018
We can use similarity dimension to calculate the dimension of fractal objects that have exact and obvious similarity, like the aforementioned examples. But, we need a more general method that works when the scaling is not immediately apparent, or when the scaling factor is different for different directions. A more general method for calculating the dimension of a fractal object is capacity dimension, sometimes called box-counting dimension (Mandelbrot 1977).
Detection and classification of pavement damages using wavelet scattering transform, fractal dimension by box-counting method and machine learning algorithms
Published in Road Materials and Pavement Design, 2023
Lizette Tello-Cifuentes, Johannio Marulanda, Peter Thomson
To estimate the fractal dimension, the is plotted against the and a linear relation is adjusted using the least squares method; the fractal dimension is defined by the slope of the straight line (Figure 5a). When there is more than one interval of the set of dimension , it is necessary to determine the scale intervals where the relation is linear; that is, the cut-off points must be determined and these points define several fractal dimensions, generating a local dimension in each of these intervals (Figure 5b) (Farhidzadeh et al., 2013; Li & Qi, 2007; Rezaie et al., 2020). Figure 6 shows the box-counting method to estimate the fractal dimension of an image.
Assessment of check dams’ role in flood hazard mapping in a semi-arid environment
Published in Geomatics, Natural Hazards and Risk, 2019
Mehdi Sepehri, Ali Reza Ildoromi, Hossein Malekinezhad, Afshin Ghahramani, Mohammad Reza Ekhtesasi, Chen Cao, Mahboobeh Kiani-Harchegani
In this article, the box counting method of Fractalyse 2.4.1 was applied to assess the drainage network and determine its fractal dimension. This method is similar to the environmental measurement method applied in the above example to calculate the length of the British coastline. The authors of this study placed all the drainage sub-catchments of the study area on a gridded plate of specific dimensions and then proceeded to count the grids (N) in which the drainage network was located. Similarly, the same procedure was repeated for other grids with different sizes (r). It should go without saying that decreasing the grid size results in an increase in the number of grids in which the drainage network is available. In the box counting method, a linear equation is obtained by placing log (N) and log (r) on the y and x-axes, respectively, whose slope is equals to the fractal dimension. The fractal dimension ranges from 1 to 2, with 1 denoting features that are linear and non-branching. The fractal dimension approaches 2 by branching the drainage network or other features. Given the increase of flood hazard rate by increasing the fractal dimension values of the drainage network, Eq. (1) was used for fuzzy scoring of this sub-index (Figure 5(a)) (Ildoromi et al. 2019).
Integration of SPOT-5 and ASTER satellite data for structural tracing and hydrothermal alteration mineral mapping: implications for Cu–Au prospecting
Published in International Journal of Image and Data Fusion, 2018
Reyhaneh Ahmadirouhani, Mohammad-Hassan Karimpour, Behnam Rahimi, Azadeh Malekzadeh-Shafaroudi, Amin Beiranvand Pour, Biswajeet Pradhan
For fractal analysis in this study, the presence or absence of a fractal model for fractures network in the Bajestan area, Box-Counting algorithm was implemented. It is one of the most applicable methods for fractal analysis (Mandelbrot 1983, Hirata 1989, Turcotte 1992, Cello 1997, Ram and Roy 2005, Fagereng 2011, Liu et al. 2015). The Box-Counting is used to measure 2-D fractal dimensions. This analysis is sensitive to a change in fractal dimension with scale (Blenkinsop 1994, Blenkinsop and Sanderson 1999, Park et al. 2010). In this method, grids with square boxes of side dimensions (d) are superimposed on the fracture map and the numbers of boxes containing fractures and Nd are counted. The process is repeated for a range of values of d. On a log–log plot of Nd and d, a fractal fracture pattern produces a straight line with slope –D in the following Equation (1):