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An empirical formulation relating boundary lengths to resolution in specimens showing ‘non‐ideally fractal’ dimensions
Author(s) -
Rigaut J. P.
Publication year - 1984
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 - 10.1111/j.1365-2818.1984.tb00461.x
Subject(s) - fractal , boundary (topology) , fractal dimension , mathematics , range (aeronautics) , plot (graphics) , geometry , dimension (graph theory) , resolution (logic) , object (grammar) , constant (computer programming) , fractal analysis , mathematical analysis , algorithm , combinatorics , statistics , computer science , artificial intelligence , programming language , materials science , composite material
SUMMARY The concept of fractal dimension offers an elegant basis for the understanding of changes in boundary lengths due to different resolutions. However, in many cases, the usual log‐log ‘fractal plot’ does not quite fit the observed data, with most biological structures for instance. An automated boundary following algorithm has been used with an IBAS image analyser on various objects, to evaluate boundary lengths at different magnifications and with a wide range of discrete divider stride lengths on each digitized image. A formula which relates the boundary length to the divider stride length and fits very closely the experimental data has been empirically obtained. Examples are shown with various object boundaries observed by TV‐camera, macroscopically or with a light microscope. At different magnifications, the same constants are obtained for a given object, notably a theoretical maximum boundary length ( B m ) when the divider stride tends towards zero. The formula allows to calculate a variation of the log‐log plot slope which fits very closely the upper convex plots observed when the boundaries do not display an ideally fractal behaviour. Plots derived from the Lineweaver‐Burk and Hill ones (used in enzymology) allow a relatively easy graphic determination of the parameters of the proposed formula, which may offer useful criteria for object shape classification, as demonstrated by a biological example. Instead of a true (constant) fractal dimension, it seems that, in many situations, exists a ‘continuous fractal dimensional transition’, inside which an apparently linear segment in a given range of resolution of the usual log‐log plot could be explained by asymptotic limits at higher and lower resolutions. The ideal situation with a constant fractal dimension over all resolution ranges could be considered as a limit, achieved when the maximum boundary length ( B m ) tends towards infinity.