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Table 1 List of fractal dimensions that are most commonly used

From: Fractal frontiers in cardiovascular magnetic resonance: towards clinical implementation

Dimension Synonym Symbol Context Author, Year described
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