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 |