Measuring fractal dimension and complexity--an alternative approach with an application

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.