The properties mentioned above reflect the complexity of the object or event. This is not to say that fractal analysis is the only means of quantifying complexity. In principal, fractal analysis should improve on the description of morphological features compared to conventional shape parameters. The lack of conclusive results can, in many cases, be explained by the apparent lack of a linguistic base, that is a sound description of fractal theory and its relationship to the associated analysis procedures. In practice any object is represented by a finite data set and the measurement is restricted to a finite magnification range.
Communication becomes meaningful if all involved understand the terms.
That is a transparent linguistic base exists. The definition is an example of communication that requires linguistic literacy as it requires an understanding of what is meant by metric space, Hausdorff dimension and topological dimension. It does not help in the understanding of fractal theory nor how this relates to fractal analysis. It would be more appropriate to point out that the definition applies to theoretical fractals and may be totally useless in practice if the image to be analysed does not reflect, albeit even statistically, a fractal set. In practice, the estimate of D quantifies scale invariance over a limited scaling range and does not indicate whether the image is fractal or not.