How much resolution is enough?: Influence of downscaling the pixel resolution of digital images on the generalised dimensions

Fractals are very often illustrated by digital images, this is a consequence of the widespread availability of electronic and computational equipment. These images are always limited by their pixel resolution. Subsequently the determination of dimensions for such images is restricted practically because for their calculation the range of scales is not infinite. Downscaling the pixel resolution of images to get images with reduced resolution worsens the achievable absolute computational results but surprisingly the changes in these values and especially the changes of the courses of the calculated generalised dimensions give very promising results. This is due to the qualitative nature of these changes with respect to pixel downscaling. This opens the possibility of determining the quality of the estimation of the fractal dimensions of any image with any particular pixel resolution. Furthermore the pattern of changes itself caused by downscaling is different between the individual fractals. Several geometrical fractals (Sierpinski gasket, a fern, a Menger gasket and a modified Menger gasket with circles) as well as an example of a measured fractal, the digital images of cancer invasion studies in vitro were investigated to study the influence of downscaling the pixel resolution of digital images on the estimation of fractal dimensions.