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Outlines
• What is Fractal?
• History
• Fractal dimension
• Box Counting Method
• Fractal dimension Calculations:
Version 1
Version 2
• Parallelization of the Box Counting method

What is Fractal?
• The word fractal comes from the Latin “Fract-” which means broken.
• Fractal is a geometric pattern that is repeated at ever smallest scales to produce irregular shapes and surface that cannot be represented by classical geometry.
• Fractals are used especially in computer modeling of irregular patterns and structure in nature.

History
•
The mathematics behind fractals began to take shape in the
17 𝑡ℎ century when Gottfried Leibniz pondered recursive self-similarity.
•
In 1872 Katl Weierstrass presented the first definition of a function with a graph that would today be considered fractal.
• In 1915, Sierpiński constructed his famous triangle fractal.
•
In March 1918, Felix Hausdorff expanded the definition of
“fractal dimension“.
•
In 1975, Mandelbort coined the word “fractal” and illustrated his mathematical definition with striking computerconstructed visualizations.

Fractal Dimension
• There are two definitions of fractal dimension:
● Fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern changes with the scale at which it is measured.
● Fractal dimension is a measure of the space-filling capacity of a pattern that tells how a fractal scales differently than the space it is embedded in.
• A fractal dimension does not have to be an integer.

Box Counting Method
• The method of Box Counting is a way of gathering data from a complex pattern or image by breaking it up into smaller and smaller pieces, and analyzing each piece separately.
• The essence of the process has been compared to zooming in or out using optical or computer based methods to examine how observations of detail change with scale.
• This method is embarrassingly parallel!

Box Counting Method – Fractal
Dimension Calculation Ver. 1
• In order to calculate fractal’s dimension you begin by covering the fractal image area with different grid sizes. Then count the number of grid blocks containing part of the fractal in them.
◦ S=1/3
◦ S=1/12

Box Counting Method – Fractal
Dimension Calculation Ver. 1
• After counting a sufficient amount of grid sizes we calculate the
Fractal dimension using the formula:
• In the previous example, Where n() is the number of grid blocks containing a part of the fractal and 1/S is the grid scale.
• 𝐷 𝑔𝑟𝑖𝑑
1
3 𝑥
1
12
= 𝑙𝑜𝑔105−𝑙𝑜𝑔18 𝑙𝑜𝑔12−𝑙𝑜𝑔3
≈ 1.27
or 𝐷 𝑔𝑟𝑖𝑑
1
3 𝑥
1
6
= 𝑙𝑜𝑔48−𝑙𝑜𝑔18 𝑙𝑜𝑔6−𝑙𝑜𝑔3
≈ 1.19
Which if we cont. to more measurements will all average to 1.26 which is the calculated dimension of the Koch Fractal.

Box Counting Method – Fractal
Dimension Calculation Ver. 2
• This method is slightly different from Ver. 1 in that it only calculates the smallest grid size, then uses the same formula with the biggest grid available (S=1 ,N=1) which leads us to this final formula:
𝐷 𝑔𝑟𝑖𝑑_𝑠𝑚𝑎𝑙𝑙𝑒𝑠𝑡
= 𝑙𝑜𝑔𝑁 − 𝑙𝑜𝑔1 log
1 𝑠
− 𝑙𝑜𝑔1
= 𝑙𝑜𝑔𝑁 log
1 𝑠

Parallelization of the Box Counting method
• For each version there is a different parallelization method from our curriculum we can implement:
• For version 1, we’d use a Condor Approach:
• Then another process will take all the measurements, put them in the formula and average the results.

Parallelization of the Box Counting method
• For Version 2, we will parallelize using MPI or OpenMP :
• Serial Pseudo Code:

Parallelization of the Box Counting method
• Parallel Pseudo Code:

Improvement of Parallelization of the
Box Counting method
• Yu-Chang and Kuo-Tai [1] tested an improved Parallel box counting algorithm using version 2 with these results:

Improvement of Parallelization of the
Box Counting method
• Speedup
:

Bibliography
• [1] A Parallel Differential Box-Counting Algorithm Applied to
Hyperspectral Image Classification : Yu-Chang Tzeng, Kuo-Tai Fan, and Kun-Shan Chen
• [2] Using the Fractal Dimension to Cluster Datasets : Daniel
Barbar ´a & Ping Chen