Fractal Dimension and Lacunarity

Author: A. Karperien, Charles Sturt University, Australia
History: 2002/08/26: First version
2002/09/26: Calculates circularity
2003/12/17: Major update
2004/06/30: Updated
2005/03/23: Updated to version 2.0aF
2005/11/24: Updated to version 2.3
2006/03/17: Updated to version 2.3j
2006/05/22: Updated to version 2.4b
2006/11/20: Updated to version 2.4e
Source:Contained in Frac_Lac.jar, which can be opened using a ZIP utility
Requires:Java 1.5
Installation: Download Frac_Lac.jar to the plugins folder, or subfolder, restart ImageJ, then run the plugin using the Plugins/Fractal Analysis/FracLac command.
Description: FracLac quantitates complexity in patterns represented in digital images. It analyzes binary images (e.g., contours of biological structures, fractals, or threshholded textures) and grayscale images and returns data, graphs, and other images for:
  • box counting fractal dimensions (DBs),
  • multifractal analysis,
  • and
  • lacunarity (λ).
  • FracLac calculates one measure of the DB for binary images using the count of boxes containing pixels, and mass related DBs for binary and grayscale images using the difference in either the density or intensity of pixels over an image. For multifractal scans, FracLac returns an array of DQ values over a range of arbitrary values for Q set by the user. It also provides τ, α, and the multifractal function (α). For lacunarity, FracLac returns measures of λ based on the data gathered during either overlapping or nonoverlapping box counting.

  • FracLac also calculates and generates images for morphometrics based on the convex hull and bounding circle enclosing the foreground pixels of binary images.
  • Data can be viewed and saved as a summary report, a data analysis file, and raw pixel count data.

FracLac scans images using a shifting grid algorithm that can do multiple scans from different locations on each image, and uses either a nonoverlapping or overlapping sliding box method, depending on user choices. The overlapping method can be exhaustive or not, depending on user choices.

Analyses may be done on entire images, rois, or subareas of either to show variation in the DB and λ over the image. The user can choose to view and save a colour-coded graphic showing variation in the DB over an image. Subareas are determined according to the user's choice, using ImageJ's ParticleAnalyzer, a square grid sized according to the user's choice, or randomly selected blocks.

FracLac works on binary images (black pixels on a white background, or white pixels on a black background) and grayscale images. For a binary analysis (e.g., of cellular outlines), images must be thresholded prior to analysis to ensure that only the pixels of interest are assessed. For grayscale scans, images and rois are padded with a value that is ignored by the counting method.

A User's Guide and Help are available by contacting the author at Charles Sturt University through email or online at

A 38 page PDF manual is available for download.

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