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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
Consistent with Falconer [196], fractals should, in addition to being nowhere differentiable and able to have a fractal dimension, be only generally characterized by a gestalt of the succeeding features self-similarity, fine or detailed structure at arbitrarily small scales, Irregularity locally and globally, and simple and “perhaps recursive”. In this regard, self-similarity involves qualitative self-similarity, for instance, as in a time series. On the contrary, statistical self-similarity duplicates a pattern stochastically (Fig. 11.4). In this view, numerical or statistical measures are preserved across scales, for instance, the coastline of Britain whose length has to be measured on a map using a measuring instrument that has a specific step length. Moreover, exact self-similarity: identical at all scales, for instance, Koch snowflake (Fig. 11.5), in which its segment is scaled and repeated as neatly as the repeated unit that defines fractals. Conversely, quasi self-similarity, which approximates the similar shape at diverse scales; may enclose small duplicates of the entire fractal in distorted and degenerate forms; for instance, the Mandelbrot set’s satellites. In this sense, they are approximations of the entire set, but not exact copies. Finally, multifractal scaling is one feature of self-similarity which is characterized by more than one fractal dimension or scaling rule.
Methods of Digital Analysis and Interpretation
Published in Victor Raizer, Optical Remote Sensing of Ocean Hydrodynamics, 2019
Many researches use multifractal analysis for exploring experimental data (e.g., time series) showing strong multiscale spatiotemporal variability. Such data are mostly associated with nonstationary and/or nonuniformity of geophysical processes involving turbulence and energy transfer. In remote sensing, multifractal analysis actually provides segmentation and classification of multivariate data related to strongly nonlinear mixing dynamical processes. Typical example is wave breaking phenomena at high wind. Multifractal analysis allows computing a spectrum or a set of fractal dimensions to characterize and predict the behaviour of dynamic system at different timescale frames. However, many algorithms exist to evaluate this spectrum and numerical differences between the methods and results can appear.
Chaos in Space
Published in Pier Luigi Gentili, Untangling Complex Systems, 2018
Most of the time series that we encounter in the economy, biology, geology, and engineering (Tang et al. 2015) display many singularities, which appear as step-like or cusp-like features. Such singularities may be treated as fractals having different dimensions α and weights f(α). For instance, it has been demonstrated that heart rate fluctuations of healthy individuals are multifractal, whereas congestive heart failure, which is a life-threatening condition, leads to a loss of multifractality (Ivanov et al. 1999). In economy, the classical financial theories conceive stock prices as moving according to a random walk. Information digested by the market is the motor of the random movement of the prices. Since traditional finance theory assumes that the arrival of news is random, and no one knows if it will be good or bad news, then, the prices move randomly. However, because economies of the industrialized countries tend to grow, on average, economists model the randomness of news as having a slight bias toward good news. This model is referred to as the theory of random walk with drift. This theory cannot account for the sudden substantial price changes and spikes that are frequent in specific periods. According to Mandelbrot and Hundson (2004), markets are like roiling seas. Multifractals can describe their turbulence. Multifractal simulations of the market activity provide estimates of the probability of what the market might do and allow one to prepare for inevitable changes.
Fractal analysis of muddy submarine channel slope instability from sub-bottom profile images
Published in Marine Georesources & Geotechnology, 2022
Multifractal analysis, or multi-scale fractal analysis, is a further development of fractal theory. Multifractal analysis is primarily concerned with measuring and characterizing local features using a scaling function, which is usually expressed by a singular probability distribution (Man et al. 2019). The purpose of multifractal analysis is to provide different models for different scales of measurement and to capture variations in fractal dimensions, especially when describing irregular and difficult-to-analyze images (Yu and Qi 2008). A spectral function is used to describe fractal parameters obtained as different scales, as well as local variability and non-uniform characteristics. The varying interval is divided into different scales during the process of calculation. To begin multifractal analysis, all intervals are divided into N units of scale ε. The probability function of each unit is pi (ε). Partition functions (q, ε) are then constructed using the following formula (Chhabra and Jensen 1989): where (q, ε) is the q-order probability of the ith subinterval; q is a real number; ε is the spatial resolution. The multifractal singularity exponent α and its corresponding spectral function f (α) were calculated for − 10 ≤ q ≤ 10 using the following formulas:
Discrimination analysis of coal and gangue using multifractal properties of optical texture
Published in International Journal of Coal Preparation and Utilization, 2022
Chengcai Fu, Fengli Lu, Guoying Zhang
The simple fractal dimension method only establishes the relationship between the number of boxes and their sizes, without considering the pixel intensity fluctuations in each box (Danila et al. 2018). Multifractals theory is a quantitative tool to describe the mass distribution in irregular fractal space. Roughly speaking, a multifractal can be regarded as a kind of complex fractal structure divided into many regions, each fractal structure has a different weight (Xu and Chen 2014). Multifractal spectrum is used to describe the evolution process of probability distribution of fractal structure and measure the homogeneity and complexity of regions with similar scaling indices.