FracLac v2.4e is a plugin for ImageJ that analyzes digital images for specialized measures of the box counting fractal dimension, the {@link FLAnalyzer.FLMain#doMF generalized dimension spectrum}, different types of lacunarity, and other {@link FLAnalyzer.CircStats#measureHullandCircle morphometrics}. The essential algorithm of the plugin is outlined in the javadoc for the {@link FLAnalyzer.FLMain FracLac} class and the class that loads it, {@link GUI.FL_}.

montage of biological cells created with ImageJ and analyzed with FracLac using a local dimension scan with the subarea size set to correspond to the sizeof images making up the montage. Colours show different fractal dimensions of each cell contour.

Use it to measure difficult to describe geometrical forms where the details of design are as important as gross morphology.

Running the Plugin

FracLac runs under ImageJ. The software is packaged as .jar file. To install it, place the current FracLac jar file in the plugins directory of ImageJ and restart ImageJ. Run FracLac through the "Plugins: Fractal Analysis" menu. Before the first scan, the user must select either a multifractal, regular box count, subscan, or sliding box lacunarity scan from the buttons on the FracLac panel. For subsequent analyses, the user can click an analysis button to reuse settings, or a setup button to change them.

Use

FracLac is suitable for analyzing images of biological cells and textures, for instance. It has been tested extensively on control images, including various quadric fractals, Koch fractals, Sierpinski and Menger fat fractals, multifractals (Henon Maps), diffusion limited aggregates, and Euclidean forms, with typically low deviations from expected and high r2 values for the fractal dimension. It works on binary files (black pixels on a white background, or white pixels on a black background), so images must be thresholded prior to analysis to ensure that only the pixels of interest are assessed. It also works on grayscale images (2006).

Inputs:

Outputs:

Needs:

Help and User's Guides

The FracLac code is extensively documented. In addtion, there are several resources available: a small online instructions page that loads with the plugin, an online guide , a User's Instruction Guide, a full pdf instruction Manual , and a zipped html help file. The author also welcomes bug reports and feature requests (see below).

History and Development

FracLac was conceived and developed by Audrey Karperien at Charles Sturt University in 2002-2004 as part of a master's project supervised by Dr. Herbert Jelinek, Neuroscience Department, School of Community Health, Charles Sturt University, Australia. FracLac is available for download from the ImageJ website or as the most recent beta file from the FracLac page. The intent was to develop a practical tool for biological cell morphology analysis, addressing several inherent problems associated with box counting. The box counting algorithm was based originally on ImageJ's function and an NIH Image plugin by Dr. Jelinek. The convex hull algorithm was provided by Thomas R. Roy, University of Alberta, Canada. The basic techniques represented can be found in the references listed at the end of this page.

Features

Box Counting with Various Sampling Methods

Calculations: The Box Counting Fractal Dimension Measures Complexity

Multifractal Spectra and Generalized Dimensions

  1. FracLac delivers distributions of {@link FLAnalyzer.FLsetup#GetSubScanInputs local} dimensions over an image, including colour coding showing how the distribution of complexity changes with the scale at which it is assessed (in addition to {@link FLAnalyzer.FLMain#MultifractalDataProcessor multifractal} spectra)
  2. {@link FLAnalyzer.FLMain#doMF Multifractal} spectra data and graphs: DQ, τ, α, ƒ(α)
  3. customized sampling over {@link FLAnalyzer.FLMain#SetFourCorners different orientations} and {@link FLAnalyzer.FLVars#GridPositions shifted positions} as well as {@link FLUtilities.FLutil#SmoothedArray data smoothing} {@link FLUtilities.FLutil#minMassCover and minimizing}
  4. pseudorandom {@link FLAnalyzer.FLVars#RandomMassSample mass} sampling (for example, manipulable with respect to {@link FLAnalyzer.FLVars#FACTOR pixels sampled}, and so on)

Calculations for multifractal data

Lacunarity (Λ)

Calculations: LACUNARITY measures heterogeneity

  1. Λall grid positions = ∑λ/Nλ
  2. λgrid position=∑λε /Nλε
  3. Slope Λ=-lim[ln λε/ln ε] found as the slope of the {@link FLAnalyzer.FracStats#PowerRegression log-log regression line}
  4. λε = 1+(σ/μ)2, where σ = the standard deviation of the pixel distribution at ε, and μ = the mean
  5. Fλ & FΛ = calculated using foreground pixels only
  6. Eλ & EΛ = calculated using all image pixels
  7. BPDλ & BPDΛ = calculated using a {@link FLAnalyzer.FLDataProcessor#BinnedProbabilityLacunarity binned probability distribution}
  8. All of the above are based on different sampling:

Other Morphometrics

FracLac delivers various morphometrics based on the foreground pixels of a binary image:
  • FracLac calculates the span ratio, the vertical ({@link FLAnalyzer.FLVars#MaxOverMinRadii bottom} and {@link FLAnalyzer.FLVars#MaxRadius top}) and the horizontal ( {@link FLAnalyzer.FLVars#CVRadii left} and {@link FLAnalyzer.FLVars#MeanRadii right}) axes, and the perimeter and area of the {@link FLAnalyzer.CircStats#measureHullandCircle convex hull} enclosing the image. It also delivers the minimum bounding circle and a measure of {@link FLAnalyzer.FLVars#circularity} based on the convex hull.
  • Working Structure:

    Types of Scans

    Images

    Nonoverlapping vs Overlapping

    This program is free software distributed in the same way that ImageJ is.

    Related Documentation

    @see
  • T.G. Smith, Jr., G.D. Lange and W.B. Marks, Fractal Methods and Results in Cellular Morphology, J. Neurosci. Methods, 69:1123-126, 1996.
  • @see
  • E. Fernandez et al., Are neurons multifractals?, J. Neurosci. Methods, 89:151–157, 1999
  • @see
  • R.E. Plotnick, R.H. Gardner, and R.V. O'Neil, Lacunarity indices as measures of landscape texture, in Landscape Ecology 8(3):201-211, 1993
  • @see
  • Innaconne, Geometry in Biological Systems.
  • @see
  • Costa and Cesar, Shape Analysis and Classification, CRC Press, 2001
  • @see
  • A. Chhabra and R.V. Jensen, Direct Determination of the ƒ(α) singularity spectrum in Phys. Rev . Lett. 62: 1327, 1989.
  • @see
  • A. N. D. Posadas, D. GimeŽnez, M. Bittelli, C. M. P. Vaz, and M. Flury, Multifractal Characterization of Soil Particle-Size Distributions, Soil Sci. Soc. Am. J. 65:1361–1367 2001
  • IDE: NetBeans IDE 4.1

    @version 2.4e FracLac_ 2001-May 2006 @version 1.37g ImageJ @author Audrey Karperien, MSc, Charles Sturt University (akarpe01@postoffice.csu.edu.au) @author Thomas R. Roy, BS (comp. science), University of Alberta, Canada (convex hull) @author ImageJ code and borrowed code from Wayne Rasband of the NIH @author H.F. Jelinek, PhD, Charles Sturt University, Australia (idea developer) @since jdk 1.5.0_04