Table 1 List of fractal dimensions that are most commonly used

Fractal

D

Generic term first introduced by Mandelbrot

Mandelbrot, 1983

Hausdorff Hausdorff-Beisicovitch

D _H

Uses image coverage by a number of countable spheres; widely used in pure mathematics but less suitable for use with natural fractals

Hausdorff, 1919

Beisicovitch, 1935

Mandelbrot, 1983

Falconer, 1990

Gulick, 1992

Minkowski-Bouligand Kolmogorov

D _M

Uses circle sweep like for D _H ; easier to evaluate than D _H ; outputs usually greater than or equal to D _H

Mandelbrot, 1983

Smith, 1989

Schroeder, 1991

Calliper Perimeter-stepping, Divider, Richardson, Compass

D _C

Calculates the fractal complexity of a simple continuous perimeter

Richardson, 1961

Mandelbrot, 1967/83

Falconer, 1990

Smith, 1989

Peitgen, 1992

Box-counting Capacity

D _B

Uses a grid method to measure the fractal complexity of 2D and 3D noncontiguous outlines commonly encountered in biological structures

Mandelbrot, 1983

Falconer, 1990

Gulick, 1992

Peitgen, 1992

Mass-radius

D _MR

Typically used in the context of clusters and networks; can be applied to surfaces and biological objects

Caserta, 1990

Jelinek, 1998

Lyapunov

D _L

Used for measuring the dimension of strange attractors in time series analysis.

Gulick, 1992

Packing

D _P

Uses dense packing by disjoint balls of differing small radii.

Falconer, 1990

Local connected set

D _LC

Variant of box-counting applied to binary images where they are sampled pixel by pixel according to the local connectedness of each pixel

Landini, 1995

Packing

D _P

Uses dense packing by disjoint balls of differing small radii.

Falconer, 1990

Grayscale box-counting Differential box-counting Fourier Higuchi’s

D _BC

Does not require image segmentation; suitable for being performed in an unsupervised manner and most amenable to automation.

Sarkar, 1994

Azemin, 2011

Higuchi, 1988