Euclidean geometry (which involves regular objects like lines, rectangles, circles, boxes, cones,…) fails to describe complex natural objects like clouds, mountains, trees, land coasts, human organs, tissues, cells, proteins, EEG/ECG signals, skin lesions, etc., whose structure is complex and irregular [1, 9]. All of the mentioned object can be identified as natural fractals and analyzed with the tools of fractal analysis. Although the mathematicians tried to define the term fractal, there is still no strict, mathematical definition of the term; since after any attempt for defining it, a fractal object would be found that doesn’t satisfy the definition. Therefore, it seems like that the best definition for fractals is descriptive one, i.e. we refer to the subset of R^n as a fractal if:
- it has a fine structure, noticed even on arbitrarily small scales,
- it is too irregular to be described in traditional geometric language, both locally and globally,
- it often has some form of self-similarity, rigorous or approximate,
- its “fractal dimension” (defined in some way) is greater than its topological dimension,
- in most of the cases, it is defined in a very simple way, perhaps recursively. ([4])
You can see an example of fractal set at Fig.1.
c)
Figure 1: Typical fractal sets: a) Koch Curve, b) Version of Koch curve, c) Cascade.
The main tool from fractal analysis that describes and quantifies both natural and laboratory fractal objects, commonly used in medical image analysis in general, is the fractal dimension. It gauges the degree of complexity (or fragmentation, or irregularity) of the fractals and it is usually a non-integer number. There are many different fractal dimensions, whose definitions are given in [1], [4], [8], all of them having a range of fractal sets for which they are or they are not suitable to be applied, but for sure the most used one (especially in biomedical sciences) is box-counting fractal dimension, because of the simple definition and functionality. The definition of box-counting fractal dimension is as follows. Given Let be the smallest number of boxes of side length needed to cover the fractal set F. We define the box-counting fractal dimension to be the number
