Estimation of fractal dimension and fractal curvatures from digital images

Joint work with Peter Straka and Steffen Winter.
Most of the known methods for estimating fractal dimensions are based on the evaluation of a single geometric characteristic, usually the volume. We propose a method involving the evaluation of several geometric characteristics, namely all the Minkowski functionals (i.e. volume, surface area, Euler characteristic etc.). Motivated by recent results on the limiting behaviour of Minkowski functionals of the parallel sets of self-similar fractals, we use these functionals to estimate the fractal dimension of sets from digital images by regression and time series methods. Simultaneously, we also obtain estimates of the fractal curvatures of these sets, some fractal counterpart of Minkowski functionals, allowing for a finer classification of fractal sets than fractal dimension only.