1. Procedure for calculating 2D and 3D Fractal Dimension, including some test-case results and potential applications
August 2012
T Murphy

2. 2D BOX-COUNT Each data-point has a ‘bin-coordinate’ for each box-size.
XY point data adjusted such that no zeros occur
Sort on X, assign ‘bin-number’ by box-size (edge) to data points by dividing by edge size.
Unsort back using INDEX field
Repeat 2 & 3 for Y.
Collate data in the form of BIN-COORDINATES:
XY,
X_bin-number:2/ Y_bin-number:2;
X_bin-number:5/ Y_bin-number:5 etc
Bin/edge sizes used are 2, 5, 10, 20, 50, 100, 200, 500
Concatenate X, Y bin-numbers (co-ordinates)
Calculate unique number of co-ordinates.....this is the minimum number of boxes (‘count’) required for each bin/box-size to capture every point in the dataset
Plot Ln ‘Box Size’ vs Ln ‘Count’ and derive Fractal DimensionBOX COUNT
Each data-point has
a ‘bin-coordinate’
for each box-size.

3. Box-count FD for LH image as calculated in ImageJ is 1. 927
Box-count FD for LH image as calculated in ImageJ is Using the method described on previous page, FDBOX COUNT is calculated at also, thereby verifying the method.

4. The Koch curve above has a documented FD of 1. 262
The Koch curve above has a documented FD of Box-count FD for the curve as calculated in ImageJ is Using the method described on previous pages, FDBOX COUNT is calculated at 1.332, again verifying the method.

5. The Sierpinski Triangle above has a documented FD of 1. 5849
The Sierpinski Triangle above has a documented FD of Box-count FD for the curve as calculated in ImageJ is Using the method described on previous pages, FDBOX COUNT is calculated at , differences in the ImageJ value and this method unexplained.

6. Each data-point has a ‘bin-coordinate’ for each cube-size.
3D CUBECOUNT
XYZ point data adjusted such that no zeros occur
Sort on X, assign ‘bin-number’ by cube size (edge) [divide Xco-ord by cube size) to data points.
Unsort back using INDEX field
Repeat 2 & 3 for Y and Z data.
Collate data in the form of BIN-COORDINATES:
XYZ,
X_bin-number:2/ Y_bin-number:2 /Z_bin-number:2;
X_bin-number:5/ Y_bin-number:5 /Z_bin-number:5 etc
Bin/edge sizes used are 2, 5, 10, 20, 50, 100, 200, 500
Concatenate X, Y, Z bin-numbers (co-ordinates) for each datapoint at respective bin-sizes
Calculate unique number of co-ordinates.....this is the minimum number of cubes (‘count’) required for each bin/cube-edge size to capture every point in the dataset
Plot Ln ‘Cube Size’ vs Ln ‘Count’ and derive Fractal DimensionCUBE COUNT. Develop solver process to maximise R2 using 4 consecutive points on graph.
Each data-point has a ‘bin-coordinate’
for each cube-size.

7. 3D model cells simulating cubes, drawn using the bin-coordinate
cubes with side-length = 5
cubes with side-length = 10
cubes with side-length = 20
original data points
3D model cells simulating cubes, drawn using the bin-coordinate
data derived through the CubeCount process.

8. Some test examples used in development of CUBECOUNT

9. Random points (24 points) Initial small test dataset to test concept
and build the CUBECOUNT process

10. Horizontal plane
(10100 points)

11. Dipping plane
45 degrees-> south
(10100 points)

12. Single Chevron
(10100 points)

13. Chevron ripple
(10100 points)

14. Point Cloud – (random Z
between 1 and 51)
(10100 points)

15. Point Cloud – (random Z
between 1 and 101)
(10100 points)

16. Sphere (surface)
(21160 points)

17. Sphere (solid/cloud)
(33184 points)

18. Some early stage examples of application of CUBECOUNT…
Some early stage examples of application of CUBECOUNT…. Analysis of 3D grain shape

19. XrayCT – test particle ‘5’ (9300 points)
FOV =100µm
XrayCT – test particle ‘5’
(9300 points)

20. XrayCT – test particle ‘9’ (33828 points)
FOV =1500µm

21. XrayCT – test particle ‘6’ (28992 points)
FOV =900µm

22. XrayCT – test particle ‘4’ (24919 points)
FOV =1500µm

23. XrayCT – test particle ‘10’ (31515 points)
FOV =1200µm
XrayCT – test particle ‘10’
(31515 points)

24. FOV =1200µm
FOV =1500µm
FOV =900µm
FOV =1500µm
FOV =100µm

25. X, Y, Z co-ords normalised and *1000
All grain surfaces contained within a 1000x1000x1000 space
retains surface irregularity but some deformation of 3D volume

26. XrayCT – test particle ‘5’
(subtract mean and +1000)
(9300 points)

27. Potential Applications (primary cause for development listed first for both methods)
2D Box-count
3D Cube-count
Quantify textural maps output by hyperspectral logging.
Quantify shape-irregularity of particulate material and individual grain shapes as scanned with Xray-CT (affect of geometry change on comminution, flotation recovery etc)
Quantify mineral distributions in MLA mineral maps
Quantify distribution of fault intersections in planned block cave (relationship to cave propagation)
Quantify rock fragment shapes as imaged in draw-points/on conveyor belts/on stockpiles w.r.t. comminution
studies.
Quantify distribution of intersections/block-model cells above a specified cut-off ....use as a parameter for conditional simulation?
Quantify mapped outcrop geometries of alteration/lithology/-mineralisation/faults/joints.