Asymptotic fractals in the context of grey-scale images

The estimation of the fractal dimension in the case of concave log-log Richardson-Mandelbrot plots can be obtained by using asymptotic fractal equations. We demonstrate here, under asymptotic fractal conditions, that additional derivations making use of the Minkowski dilation in grey-scales lead to two asymptotes, one having a slope of 1 and the other a slope of D(T) - D + 1 (where D(T) is the topological dimension and D the fractal dimension). The resulting equation offers important advantages. It allows: (i) evaluation of scaling properties of a grey-scale image; (ii) estimation of D without any iteration and (iii) generation of texture and heterogeneity models. We concentrate here on the first two possibilities. Images from cultured cells in studies of cytoskeleton intermediate filaments and kinetic deformability of endothelial cells were used as examples.