# Measuring fractal dimension by box-counting¶

## Theory¶

The term ‘fractal dimension’ was introduced by Benoit Mandelbrot in 1967 to explain self-similarity of a pattern. A fractal dimension is defined as a ratio of the change in detail to the change in scale. It is used as an index that quantifies the complexity of a fractal pattern (https://en.wikipedia.org/wiki/Fractal_dimension).

Famously, fractal dimensions have been used to analyze the length of the British coastline. A coastline’s measured length is observed to change depending on the length of the measuring stick used. In 2-D and 3-D, this notion can be extended to the length of a measuring pixel or voxel, respectively.

### Box Counting Method¶

One way to determine fractal dimension of an image is the box counting method. Boxes of various sizes are laid over the image in a fixed grid pattern. The number of boxes that span the edge of the pattern (i.e. partially 1 and partially 0) are tallied as a function of box size. This count is then used to calculate the fractal dimension (https://en.wikipedia.org/wiki/Box_counting).

## Mathematical Definition¶

The relationship of a pattern’s fractal dimension and its measuring element can be expressed as:

\(N \propto \frac{1}{D^F}\)

\(F = \lim_{D \to 0} \frac{-log N(D)}{log(D)}\)

N: number of boxes of side D that span an edge

D: size of the boxes

F: fractal dimension

## Example¶

A Sierpinski carpet has a known fractal dimension of 1.8928. Performing the box counting method found its fractal dimension as approximately 1.8 ~ 1.9.

First, import the needed packages.

```
[1]:
```

```
import matplotlib.pyplot as plt
import porespy as ps
ps.visualization.set_mpl_style()
```

Generate a sierpinski carpet and visualize.

```
[2]:
```

```
im = ps.generators.sierpinski_foam(4, 5)
plt.imshow(im);
```

Finally, apply the box count function and visualize.

```
[3]:
```

```
data = ps.metrics.boxcount(im)
```

```
[4]:
```

```
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(8, 4))
ax1.set_yscale('log')
ax1.set_xscale('log')
ax1.set_xlabel('box edge length')
ax1.set_ylabel('number of boxes spanning phases')
ax2.set_xlabel('box edge length')
ax2.set_ylabel('slope')
ax2.set_xscale('log')
ax1.plot(data.size, data.count,'-o')
ax2.plot(data.size, data.slope,'-o');
```

The horizontal portion of the `slope vs box edge length`

curve, between \(10^1\) and \(10^2\) is a flat line with a value of approximately 1.9. Beyond a box edge length of \(10^2\) the analysis becomes impacted by the finite image size so the result begins to diverge.