# fractal dimension

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## fractal dimension

(mathematics)
A common type of fractal dimension is the Hausdorff-Besicovich Dimension, but there are several different ways of computing fractal dimension. Fractal dimension can be calculated by taking the limit of the quotient of the log change in object size and the log change in measurement scale, as the measurement scale approaches zero. The differences come in what is exactly meant by "object size" and what is meant by "measurement scale" and how to get an average number out of many different parts of a geometrical object. Fractal dimensions quantify the static *geometry* of an object.

For example, consider a straight line. Now blow up the line by a factor of two. The line is now twice as long as before. Log 2 / Log 2 = 1, corresponding to dimension 1. Consider a square. Now blow up the square by a factor of two. The square is now 4 times as large as before (i.e. 4 original squares can be placed on the original square). Log 4 / log 2 = 2, corresponding to dimension 2 for the square. Consider a snowflake curve formed by repeatedly replacing ___ with _/\_, where each of the 4 new lines is 1/3 the length of the old line. Blowing up the snowflake curve by a factor of 3 results in a snowflake curve 4 times as large (one of the old snowflake curves can be placed on each of the 4 segments _/\_). Log 4 / log 3 = 1.261... Since the dimension 1.261 is larger than the dimension 1 of the lines making up the curve, the snowflake curve is a fractal. [sci.fractals FAQ].
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(3) Both the real and imaginary parts of the complex wave number increase with the tortuosity fractal dimension. The real part increases with the frequency and decreases with the pore area fractal dimension; the imaginary part decreases with the frequency and the pore area fractal dimension and finally keeps a constant.
Figure 2: Retinal image processed by SIVA and used for analysis of vascular caliber, fractal dimension, tortuosity, length-diameter ratio, and branching coefficients.
Fractal dimension (i.e., box dimension) is determined as the slope of the log-log relationship between N and r.
The covering divider method is applied in order to extract fractal dimension of the earth surface (Dsurf) directly for estimating surface roughness of the earth topography through geographic information system (GIS) approaches.
Feature extraction based on LMD and fractal dimension for EGG signals
Thereby, this study aimed to analyze the repair of skeletal muscle tissue, using a combination of cryotherapy and therapeutic ultrasound treatments, through analysis of body and muscle mass, measure of the smallest diameter of the muscle fibers and quantification of collagen through fractal dimension.
Through the use of the data from this lab study and a formula applied, called "fractal dimension," researchers discovered a negative relationship between the fractal dimension of pupil dilation and a person's workload, showing the researchers that pupil dilation could be used to indicate the mental workload of a person in a multitasking environment.
After many experiments with different methods, we decided to apply the fractal dimension calculation method for the time series available in the well-known program Benoa (BENOIT, version 1.31), which proved to be very good in the calculations used earlier by different authors.
The collage theorem can make the fractal dimension of an image based on fractal interpolation unchanged[23,24].

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