Noise dependency of algorithms for calculating fractal dimensions in digital images

Fractal properties of real world objects are commonly examined in digital images. Digital images are discrete representations of objects or scenes and are unavoidably contaminated with noise disturbing the representation of the captured objects. We evaluate the noise dependency of frequently applied algorithms for the calculation of the fractal dimension in digital images. Three mathematically defined fractals (Koch Curve, Sierpinski Gasket, Menger Carpet), representative for low, middle and high values of the fractal dimension, together with an experimentally obtained fractal structure were contaminated with well-defined levels of artificial noise. The Box-Counting Dimension, the Correlation Dimension and the rather unknown Tug-of-War Dimension were calculated for the data sets in order to estimate the fractal dimensionality under the presence of accumulated noise. We found that noise has a significant influence on the computed fractal dimensions (relative increases up to 20%) and that the influence is sensitive to the applied algorithm and the space filling characteristics of the investigated fractal structures. The similarities of the effect of noise on experimental and artificial fractals confirm the reliability of the obtained results.