1. Ravid Rodney 066514837 Or Maltabashi 308551332

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

3. 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.

4. History

5. 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.

6. 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!

7. 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

8. Box Counting Method – Fractal Dimension Calculation Ver. 1

9. Box Counting Method – Fractal Dimension Calculation Ver. 2

10. 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.

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

12. Parallelization of the Box Counting method Parallel Pseudo Code:

13. 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:

14. Improvement of Parallelization of the Box Counting method Speedup :

15. 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

16. Questions?