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Fractal Algorithm for Discrimination Between Oil Spill and Look-Alike
Published in Maged Marghany, Automatic Detection Algorithms of Oil Spill in Radar Images, 2019
The fractal word is derived from the Latin word fractus signifying “irregular segments”. Mandelbrot was the pioneer scientist, who defined and explored the fractal in 1977. In mathematics, a fractal is a subset of a Euclidean dimension for which the Hausdorff dimension (Fig. 11.1) rigorously surpasses the topological space. Fractals develop to exhibit almost the identical at exclusive levels, as is illustrated here in the successively small magnifications of the Mandelbrot set (Fig. 11.2) [187–190]. Because of this, fractals are encountered ubiquitously in nature. Fractals exhibit similar patterns of an increasing number of small scales referred to as self-similarity [188], additionally recognized as increasing symmetry or unfolding symmetry; If this replication is precisely identical at each and every scale, as in the Menger sponge [190], it is known as affine self-similar. Nonetheless, scientists disagree on the exact definition of the fractal. In this sense, the majority agrees with the rudimentary concepts of self-similarity and the infrequent correlation fractals have with space they are implanted in [187],189,191,192].
Fractals and chaos
Published in A. W. Jayawardena, Environmental and Hydrological Systems Modelling, 2013
Hausdorff dimension, sometimes known as Hausdorff–Besicovitch dimension, was introduced by Felix Hausdorff in 1918 with computational details provided by Abram Samoilovitch Besicovitch. Suppose the linear size of an object residing in Euclidean dimension D is reduced by 1r in each spatial direction, its measure (length, area, or volume) would increase to rD times the original. For example, in a one-dimensional case of a line, and if r = 1 (there is no reduction of the length of line), the number of self-similar objects, N = rD remain unchanged at 1 for one, two, and three dimensions; if r = 2, N = rD becomes 2 for one dimension, 4 for two dimensions, and 8 for three dimensions; if r = 3, N = rD becomes 3 for one dimension, 9 for two dimensions, and 27 for three dimensions. When the logarithms of the relationship N = rD are taken, the topological dimension D can be expressed as () D=lnN(r)ln(r)
Autocorrelation Function, Mutual Information, and Correlation Dimension
Published in Nicholas Stergiou, Nonlinear Analysis for Human Movement Variability, 2018
Therefore, for a line, ln(3)/ln(3) = 1. This definition of the dimension is called the Hausdorff dimension (Mandelbrot 1977). In Figure 8.29, we can see that this definition matches our intuition for a line, a square, and a cube.
An improved method for mapping tidal waterways based on remotely sensed waterlines: A case study in the Yellow River Delta, China
Published in Marine Georesources & Geotechnology, 2020
Xiang Yu, Chao Zhan, Mengquan Wu, Xueli Niu, Xueyu Zhang, Qing Wang, Buli Cui
We adopted three indices, length, area and fractal dimension, to evaluate the spatial variation of tidal waterways from 1985 to 2009. Attribute tables of vectorized tidal waterways provided data on length and area. Then, fractal dimension was calculated based on length and area. Fractal dimension of the landscape pattern reflected the complexity of the landscape and the degree of spatial stability. The higher the value of the fractal dimension, the more complex the geometry and the stronger the naturalness. That was, the smaller the fractal dimension, the greater the artificial interference. Thus, the fractal dimension was generally adopted in describing landscape distribution and variation (Cámara, Gómez-Miguel, & Martín 2015; Dombrádi et al. 2007), and in calculating landscape vulnerability index (Qiu et al. 2007). Former studies had defined the fractal dimension as Hausdorff dimension (Bowen 1979), Minkowski dimension (Marana et al. 1999), Box dimension (Sarkar & Chaudhuri 1994), Similarity dimension (Toyoki & Honda 1985) or Spectral dimension (Meakin & Stanley 1983). Minkowski dimension and Box dimension are the two most common methods (He 1996). Thus, we adopted Box dimension here (Eq. 4). where a was the side length of the net grid and N(a) denoted the number of grids containing tidal waterway segments, A was a coefficient to be determined and D was defined as box dimension. In this study, side length a was artificially set as 1000, 2000, 3000, 5000 and 8000 m, respectively. Fishnets were generated based on the given side lengths. N(a) was subsequently generated using Tabulate Area tool of Spatial Analysis module in ArcGIS. The fractal dimension, which was defined as the slope of the equation, was generated based on datasets of log (a) and log N(a) for 1985, 1990, 2000 and 2009, respectively (see Figure 7).