# Digital Image Analysis: Fractal Analysis

Fractals are complex and irregular geometric objects whose parts resemble the whole (self-similarity). Fractals are objects with non-linear scaling rules measuring classical topological parameters (e.g. length) depends on the magnification of the object. Ideal fractals are generated mathematically, but fractals or approximate fractals are also found in natural structures such as clouds, snow flakes, river networks, mountain ranges and coastlines.

In contrast to Euclidean geometry with integer topological dimensions, fractal geometry permits fractional dimensions and the mathematical calculation of the fractal dimension D is one of the basic methods to characterize fractals.

The Koch curve (right panel) is initiated with a straight line by replacing the middle third with two lines of the same length. Each line is then replaced with a new line according the same rule. The procedure can be continued with no end and results in a line with infinite lenght. The fractal dimension D is calculated from the length of the line at various scale sizes (from the slope of the regression line of a log-log plot). For the Koch curve the result is D = 1.26.

In software-supported fractal analysis, fractal dimension is often estimated by the box-counting method from the ratio of increasing detail with increasing scale. A second common parameter in fractal analysis is the lacunarity, which is a measure of the structural heterogeneity and is calculated during a box-counting scan.

Fractal analysis has been applied in life sciences to characterize very complex and irregular structures including vascular, pulmonary and neuronal networks as well as bone and tumor pathology. However, it should be mentioned that the clinical significance of fractal analysis is still discussed controversially.

MetaPhysiol offers fractal analysis of heterogenic, irregular and complex cellular objects for further characterization (e.g. angiogenesis) and as quantitative approach to characterize structures which have been rather abstract or descriptive (e.g. cytoskeleton).