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Asymptotic fractals in the context of grey‐scale images
Author(s) -
Rigaut Jp,
D. Schoëvaërt-Brossault,
Downs Am,
Gabriel Landini
Publication year - 1998
Publication title -
journal of microscopy
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.569
H-Index - 111
eISSN - 1365-2818
pISSN - 0022-2720
DOI - 10.1046/j.1365-2818.1998.00284.x
Subject(s) - fractal , fractal dimension , mathematics , dilation (metric space) , scaling , context (archaeology) , multifractal system , mandelbrot set , asymptote , scale (ratio) , dimension (graph theory) , mathematical analysis , geometry , statistical physics , pure mathematics , physics , geology , paleontology , quantum mechanics
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.