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Measuring fractal dimension and complexity — an alternative approach with an application
Author(s) -
SANDAU K.,
KURZ H.
Publication year - 1997
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.1365-2818.1997.1270685.x
Subject(s) - fractal dimension , box counting , fractal , dimension (graph theory) , mathematics , measure (data warehouse) , fractal analysis , fractal dimension on networks , variance (accounting) , effective dimension , surface (topology) , multifractal system , statistical physics , hausdorff dimension , computer science , geometry , mathematical analysis , combinatorics , data mining , physics , business , accounting
Fractal dimension has often been applied as a parameter of complexity, related to, for example, surface roughness, or for classifying textures or line patterns. Fractal dimension can be estimated statistically, if the pattern is known to be self‐similar. However, the fractal dimension of more general patterns cannot be estimated, even though the concept may be retained to characterize complexity. We here show that the usual statistical methods, e.g. the box counting method, are not appropriate to measure complexity. A recently developed approach, the extended counting method, whose properties are closer to what fractal dimension means, is considered here in more detail. The methods are applied to geometric and to blood vessel patterns. The weak assumptions about the structure, and the lower variance of the estimate, suggest that the extended counting method has beneficial properties for comparing complexity of naturally occurring patterns.