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A fragmented geometric shape that can be split into parts, each of which approximates a reduced-size copy of the whole, a property which is called self-similarity. Fractals provide the mathematics behind structures in the natural universe—e.g., frost crystals, coastlines, etc.—which cannot be described by the language of euclidean geometry. Fractal analysis is providing new ways to interpret biomedical phenomena. It has been used for classifying histopathology, enzymology, and signal and image compression
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When the minimum size of the box reaches the length of the smallest fault and the correlation coefficient is higher at 0.92, it can be viewed meeting requirements of statistical self-similarity. Therefore, T3x faults in the study area have fractal features.
Zhang, "Retinal image enhancement using robust inverse diffusion equation and self-similarity filtering," PLoS One, vol.
Then, in order to obtain a robust second-order regression function estimation, not only self-similarity property in a single MR image but also intrasimilarity property in adjacent MR slices is employed.
Ferrari (2007) establishes a formal definition of self-similarity, as follows:
The late Walter Freeman found this self-similarity causal in how the brain quickly moved between vastly different states.
Random fractals, as opposed to geometric fractals, do not have true self-similarity in that we do not see the exact same pattern repeated over and over.
The Hurst values were varied from 0 to 1 with step 0.1 to produce a regular diagram, even if values 0 and 1 are less meaningful in the self-similarity domain.
inopina had significantly steeper slopes at all times after fire than predicted by self-similarity models.
Assuming self-similarity between small and large microseismic events we can conclude that the microseismic events generally radiate energy at low frequencies.
The Sierpinski gasket, which has the characteristics of self-similarity, is used to analyse the fractal particle size distribution of real unit system.
Next, we will see that Hausdorff measure indeed determines the dimension of a fractal curve but does not describe its analytic properties, for example, the self-similarity between local and global shapes of a fractal curve.
In Section 2, we quickly cite some preliminary results which show why fBm captures the financial data with long-range dependence and self-similarity. Since fBm is not a semimartingale, the traditional Ito calculus method is not valid here.