Lacunarity

Figure 1. Basic fractal patterns increasing in lacunarity from left to right.
The same images as above, rotated 90°. Whereas the first two images appear essentially the same as they do above, the third looks different from its unrotated original. This feature is captured in measures of lacunarity listed across the top of the figures, as calculated using standard biological imaging box counting software FracLac, Image.

Lacunarity, from the Latin lacuna, meaning "gap" or "lake", is a specialized term in geometry referring to a measure of how patterns, especially fractals, fill space, where patterns having more or larger gaps generally have higher lacunarity. Beyond being an intuitive measure of gappiness, lacunarity can quantify additional features of patterns such as "rotational invariance" and more generally, heterogeneity.[1][2][3] This is illustrated in Figure 1 showing three fractal patterns. When rotated 90°, the first two fairly homogeneous patterns do not appear to change, but the third more heterogeneous figure does change and has correspondingly higher lacunarity. The earliest reference to the term in geometry is usually attributed to Benoit Mandelbrot, who, in 1983 or perhaps as early as 1977, introduced it as, in essence, an adjunct to fractal analysis.[4] Lacunarity analysis is now used to characterize patterns in a wide variety of fields and has application in multifractal analysis[5][6] in particular (see Applications).

  1. ^ Smith, T. G.; Lange, G. D.; Marks, W. B. (1996). "Fractal methods and results in cellular morphology — dimensions, lacunarity and multifractals". Journal of Neuroscience Methods. 69 (2): 123–136. doi:10.1016/S0165-0270(96)00080-5. PMID 8946315. S2CID 20175299.
  2. ^ Plotnick, R. E.; Gardner, R. H.; Hargrove, W. W.; Prestegaard, K.; Perlmutter, M. (1996). "Lacunarity analysis: A general technique for the analysis of spatial patterns". Physical Review E. 53 (5): 5461–8. Bibcode:1996PhRvE..53.5461P. doi:10.1103/physreve.53.5461. PMID 9964879.
  3. ^ Plotnick, R. E.; Gardner, R. H.; O'Neill, R. V. (1993). "Lacunarity indices as measures of landscape texture". Landscape Ecology. 8 (3): 201. doi:10.1007/BF00125351. S2CID 7112365.
  4. ^ Mandelbrot, Benoit (1983). The Fractal Geometry of Nature. ISBN 978-0-7167-1186-5.
  5. ^ Cite error: The named reference Karperien was invoked but never defined (see the help page).
  6. ^ Al-Kadi, O.S.; Watson, D. (2008). "Texture Analysis of Aggressive and non-Aggressive Lung Tumor CE CT Images" (PDF). IEEE Transactions on Biomedical Engineering. 55 (7): 1822–30. doi:10.1109/TBME.2008.919735. PMID 18595800. S2CID 14784161. Archived from the original (PDF) on 2014-04-13. Retrieved 2014-04-10.