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Fractal Dimension Calculation Using the “Box Counting

Method”

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 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 computer-

constructed 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.

• or

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:

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

Questions?