Fractal Dimension Estimation

1.1 Preliminaries

Mathematically, any metric space has a characteristic number associated with it called dimension, the most frequently used is the so-called topological or Euclidean dimension. The usual geometrical figures have integer Euclidean dimensions. Thus, points, segments, surfaces and volumes have dimensions 0, 1, 2 and 3, respectively.

But what for the fractals objects, it is more complicated. For an example, the coastline is an extremely irregular line in such way that it would seem to have a surface, it is thus not really a line with a dimension 1, nor completely a surface with dimension 2 but, an object whose dimension is between 1 and 2. In the same way, we can meet fractals whose dimension ranges between 0 and 1 (Like the Cantor set which will be seen later) and between 2 and 3 (surface which tends to fill out a volume), etc. So, fractals have dimensions which are not integer but fractional numbers, called fractal dimension.

In the classical geometry, an important characteristic of objects whose dimensions are integer is that any curve generated by these elements contours has finite length. Indeed, if we have to measure a straight line of 1 m long with a rule of 20 cm, the number of times that one can apply the rule to the line is 5. If a rule of 10 cm is used, the number of application of the rule will be 10 times, for a rule of 5 cm, the number will be 20 times and so on. If we multiply the rule length used by the number of its utilization we will find the value 1 m for any rule used.

This result if it is true for the traditional geometry objects, it is not valid for the fractals objects. Indeed, let us use the same way to measure a fractal curve,

with a rule of 20 cm, the measured length will be underestimated but with a rule of 10 cm, the result will be more exact. More the rule used is short more the measure will be precise. Thus, the length of a fractal curve depends on the rule used for the measurement: the smaller it is, the more large length is found.

It is the conclusion reached by Mandelbrot when he tried to measure the length of the coast of Britain (Mandelbrot 1967). He found that the measured length depends on the scale of measurement: the smaller the increment of measurement, the longer the measured length becomes.

Thus, fractal shapes cannot be measured with a single characteristic length, be­cause of the repeated pattern we continuously discover at different scale levels.

This growth of the length follows a power law found empirically by Richardson and quoted by Benoit Mandelbrot in his 1967 paper (Richardson 1961)

L (n) x П а (2.10)

where L is the length of the coast, n is the length of the step used, the exponent а represents the fractal dimension of the coast.

Other main property of fractals is the self-similarity. This characteristic means that an object is composed of sub-units and sub-sub-units on multiple levels that re­semble the structure of the whole object. So fractal shapes do not change even when observed under different scale, this nature is also called scale-invariance. Mathe­matically, this property should hold on all scales. However, in the real world the self-similarity is only observed over some scales the objects are then statistically self-similar or self-affine.

Updated: July 30, 2015 — 9:29 pm