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

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.

Box-count FD for LH image as calculated in ImageJ is 1. 927 Box-count FD for LH image as calculated in ImageJ is 1.927. Using the method described on previous page, FDBOX COUNT is calculated at 1.927 also, thereby verifying the method.

The Koch curve above has a documented FD of 1. 262 The Koch curve above has a documented FD of 1.262. Box-count FD for the curve as calculated in ImageJ is 1.333. Using the method described on previous pages, FDBOX COUNT is calculated at 1.332, again verifying the method.

The Sierpinski Triangle above has a documented FD of 1. 5849 The Sierpinski Triangle above has a documented FD of 1.5849. Box-count FD for the curve as calculated in ImageJ is 1.5961. Using the method described on previous pages, FDBOX COUNT is calculated at 1.5514, differences in the ImageJ value and this method unexplained.

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.

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.

Some test examples used in development of CUBECOUNT

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

Horizontal plane (10100 points)

Dipping plane 45 degrees-> south (10100 points)

Single Chevron (10100 points)

Chevron ripple (10100 points)

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

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

Sphere (surface) (21160 points)

Sphere (solid/cloud) (33184 points)

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

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

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

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

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

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

FOV =1200µm FOV =1500µm FOV =900µm FOV =1500µm FOV =100µm

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

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

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.