Fractal Dimension and Lacunarity

Author: Audrey Karperien (akarpe01@postoffice.csu.edu.au)
This plugin, an ongoing project, was initially developed as part of a master's at Charles Sturt University with input from Dr. H. Jelinek and Dr. Alastair Buchan, and code from Thomas R. Roy, University of Alberta. It continues to develop in response to the many suggestions from users and includes inputs from the ImageJ, neuroscience, engineering, programming, etc. communities. For information on how to cite the software and work published using it, see the online guide.
Source:Contained in Frac_Lac.jar, which can be opened using a ZIP utility
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: Some patterns - such as contours of biological cells, other biological structures, fractals, and some textures - are difficult to describe using Euclidean measures like diameter or length, but can be quantitatively assessed using measures of complexity. FracLac quantitates such difficult to describe details of morphology represented in binary or grayscale digital images, using mass and box counting fractal dimensions (DBs), multifractal analysis data, lacunarity (λ) data, and some other morphometrics. FracLac also includes MicroMod2015 for ImageJ, for modelling branching structures with fractal features.
Support: Limited inline help is available (2014) through a window that comes up when a user selects any scan type. The help includes a single page search, help hints on hover, and other features to make the software accessible to the non-expert and explain features in context. Hovering over a component on the set up dialog shows hints, and right clicking on them loads a page of instructions for the component. This inline help complements the User's Guide available online. Contact the team for assistance, suggestions, and bug reports.

A help archive that can be unzipped into the plugins folder where FracLac will access it, is available for download. The 38 page PDF manual for previous versions is still available for download.

Features:

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 an overlapping sliding box method. The overlapping method can be exhaustive or not, depending on user choices. FracLac also scans images using an optional rotation function that analyzes an image from several angles and summarizes the results. FracLac automatically generates different types of series of box sizes (e.g., power or linear). In addition, local dimension scans may be done on subareas of either images or rois to show local variation in the DB and λ. These subareas are determined according to the user's choice, using ImageJ's ParticleAnalyzer, a square grid sized according to the user's choice, randomly selected blocks, or individual pixels. Users can view or save the results, and can reapply LUT's to colour coded images or generated text images (see Adjusting LUTs).

Use: Before images can be analyzed, FracLac has to be set up. This is usually simply a matter of clicking a button for the type of scan you want to do, then selecting the defaults in the relevant options panel that appears, or making modifications if required.

Once FracLac has been set up, the type of scan appears at the bottom of the FracLac panel, and buttons to perform scans are enabled, as shown on the screen shot below. Using FracLac, one can do fractal, multifractal, and lacunarity analyses on:

  1. entire open images,
  2. rois,
  3. stacks, and
  4. multiple files selected from directories.
You can use FracLac to investigate and compare fractal dimensions using the Graphic Options Panel. The panel appears when you select "Show Colour Coded Graphics" on any of the set up panels. Use the panel to change the colour coding and transparency values to show variation in the fractal dimension over an image. The images below were generated by doing a default grayscale box count scan on the top images using the Roi Manager button on FracLac. Then, the fractal dimension limits were adjusted with real-time feedback on the panel to recolour the scanned images to highlight differences in the 3 cells (middle row) that were not evident from the first scan (bottom row). For the image on the right, the scan revealed areas with notably different complexity (green circles) within a cell. The panel has a Help button that explains how to use it.
Outputs: The information returned by the various types of scans is summarized below:
  1. Standard Box Count
    • DBs for binary images, based on multiple origins for each image or roi, including a minimum cover DB, slope-corrected DB, and average DB all based on the count of boxes containing pixels at different box sizes and grid locations
    • Mass dimensions for each image or roi for binary and grayscale images using the difference in either the density or intensity of pixels in boxes of different sizes placed over an image.
  2. Subareas Scan
    • Local fractal dimensions. These are based on
      1. subareas (e.g., systematically or randomly chosen blocks or else particles) or
      2. pixels and their surrounding environments, which can be set to be round or square
  3. Local Connected Fractal Dimensions.
    • These are based on the round or square environment surrounding each pixel but limited to the connected set around that pixel.
      [cs]
      The screen shot shown here illustrates the panel that comes up when the button for a DLC or "Local Connected Fractal Dimension" scan is clicked. The panel as shown is set up to use a circular sampling element by selecting the box marked "oval". At the bottom of the panel are options to save generated files or view them on the screen. In the centre is the help window,
      • [panel]
  4. Multifractal spectra.
    • 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 ƒ(α).
  5. Lacunarity data
    • FracLac returns measures of λ based on data gathered during
      • overlapping scans (sliding box lacunarity) or
      • nonoverlapping (regular box counting)
  6. Other morphometrics, including measures of density, span ratio, circularity, and the convex hull.

Data Files

The measures noted above are reported in data files accompanied by various statistics for the data, including the y-intercept, standard deviation, and coefficient of variation, for instance. Reports generated can be viewed and saved as Results Windows or automatically saved as .txt files (tab-delimited;UTF-8 encoded) to be opened in a spreadsheet. These include:

  • summary reports
  • data files
  • raw pixel counts

    Graphics Files

    Some measures are represented in graphics that can be viewed and saved or automatically saved. These include:

  • colour coded images for local dimensions, as illustrated in the image below
    [coded]
  • text images for colour coded images
  • convex hull and bounding circle images
  • graphs (e.g., multifractal spectra, as shown below, and regression statistics)
    [mf]

    • Limitations:FracLac works on binary images (black pixels on a white background, or white pixels on a black background) and on grayscale images or grayscale images that have been converted to RGB. For a binary analysis (e.g., branching patterns or single pixel wide contours of cellular outlines), the signal is black or white pixels and the noise is the opposite, white or black pixels, thus 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. Grayscale scans are suited to textures as opposed to contours, and assess every pixel as signal, so the region of interest must be selected carefully. To ensure non-significant parts of grayscale images are not assessed, the image can be converted to RGB and the non-significant areas selected with the wand function or filled with a non-gray value, as is illustrated in the image.

    [texture] You should assess your analysis against benchmark images and be aware of how the settings can affect the result. For instance, FracLac will determine the fractal dimension of a 400 x 400 pixel square as a perfect 1.000 (not fractal at all but the correct value) if the software is set to use a scaled series with a scaling factor of 2, a minimum box size of 25, and a maximum box size of 50%. But for most images, the point of the analysis is to determine a value for the fractal dimension without knowing the scaling ahead of time. See the online help for more about this.

    References:

    For details of the calculations, see the FracLac javadoc, which can be generated using the FracLac source code. For more information, the references below are the main ones used in defining the calculations:

    1. Smith, TG Jr., Lange GD and Marks, WB, "Fractal Methods and Results in Cellular Morphology", J. Neurosci. Methods, 69:1123-126, 1996.
    2. Fernandez et al., "Are neurons multifractals?". J. Neurosci. Methods, 89:151–157, 1999
    3. Plotnick, RE; Gardner, RH; and O'Neil, RV. "Lacunarity indices as measures of landscape texture", Landscape Ecology 8(3):201-211, 1993
    4. Iannaccone, PM and Khokha, M. Fractal Geometry in Biological Systems Springer-Verlag. 1996
    5. Costa and Cesar, Shape Analysis and Classification, CRC Press, 2001
    6. Chhabra, A and Jensen, RV "Direct Determination of the ƒ(α) singularity spectrum", Phys. Rev . Lett. 62: 1327, 1989.
    7. Posadas, AND; Gimenez, D; Bittelli, M; Vaz, CMP; and Flury, M "Multifractal Characterization of Soil Particle-Size Distributions", Soil Sci. Soc. Am. J. 65:1361–1367 2001
    8. Landini,G; Murray, PI; and Misson, GP. "Local Connected Fractal Dimensions and Lacunarity Analyses of 60° Fluorescein Angiograms", Investigative Ophthalmology & Visual Science, December 1995, Vol. 36, No. 13
    9. Chaudhuri, B. and Sarkar, N. "Texture segmentation using fractal dimension", Pattern Analysis and Machine Intelligence, IEEE Transactions on 1995, 17 (1):72-77
    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
    2007/03/13: Major update to version 2.5
    2007/03/27: Fixes bug that caused no printed and saved results for some scans (2.5a)
    2007/04/04: Fixes bug that sometimes prevnted maximum epsilon from changing (2.5b)
    2007/07/17: Fixes bugs that prevented box masses from being printed for sliding box lacunarity scans and prevented batches from being processed (2.5c)
    2007/08/01: Fixes bug that prevented user from knowing if the number of grid orientations was changed or not; implements user suggestions for multifractal optimization and local connected fractal dimension; help file is now html (2.5d)
    2011/02/03: Fixed a couple of bugs with multifractal analysis
    2012/04/05: Fixed encoding of autosaved text files so Greek symbols are retained; fixed bug so autosaved files are saved in a date and time stamped folder (2.5g)
    2102/04/15: release 2.5i; Fixed saved results file problem so filtered data are saved in separate files; fixed display problem so images are not displayed while being processed; changed grid location algorithm to use a series of grid positions in response to user requests; fixed bug so images having inverted LUTs are filled with the right colour.
    2012/04/15: release 2.5i.2; fixed bug so subscans can be done using the particle analyzer again; fixed bug so subscans don't hang on stacks
    2012/04/18: 2.5i.3 fixed bug that caused grid position counter to reset with very small box sizes; fixed problem with getting irregular rois
    2012/04/30: 2.5k fixed bugs with particle analyzer on inverting LUTs; fixed random mass sampling bug that caused a hang on stacks
    2012/05/03: 2.5m fixed bugs with lcfd stacks
    2012/05/08: 2.5p fixed bug with regression lines not showing; added feature to let user choose less slices per stack so grid images don't crash
    2012/05/13: 2.5r added feature to work on grayscale images with "holes"
    2012/06/21: 2.5v fixed bug with multifractal scanner autochanging the number of scans; added mass vs distance feature; updated colour coded image outputs
    2012/11/02: 2.5w fixed bug with multifractal scanner to allow user to override minimum box size; changed default for edge effects; fixed grayscale bug with RGB images
    2013/28/02: 2013Feb needs jdk 1.7; more compact UI; fixed bug with roi coordinates; updated output files to meet user requests to compare different fractal dimensions in one file and images in another
    2013/25/04: 2013Apr jar recompiled for compatibility with jdk 1.6
    2013/19/07: 2013Jul Fixed bug causing some data files to be empty in batch mode; updated parts of javadoc;
    2013/24/11: 2013Nov Added feature to visualize changes in graphics showing the variation over an image; updated parts of javadoc
    2013/04/12: 2013Dec Fixed bug with grayscale scans crashing; fixed batch jobs for subscans; fixed ParticleAnalyzer crashing issue;
    2014/22/Mar: 2014Marb765 Added inline help window; fixed bug with colour-coding on screen; fixed bug with incorrect release number appearing on frame
    2014/26/Mar: 2014Marb768 fixed bug for loading and saving settings files; changed encoding to utf-8 so symbols are encoded properly in text files
    2014/03/Apr: 2014Aprb800 fixed a bug that prevented the user from changing the increment between exponents for multifractal analysis
    2014/14/Apr: 2014Aprb804 recompiled to work under IJ 1.48u (in the previous release, the FracLac panel for setting up scans was not appearing in IJ 1.48u)
    2014/17/Apr: 2014Aprb1012 fixed problems with the set up frame not appearing and sliders on the set up frame not working
    2015/07/Jan: 2015Janb1292 Fixed problems with colour-coding dialog; added wave data processing functions.
    2015/12/Jan: version 2015Janb2594 Fixed multifractal analysis ƒ(αQ) graphs; updated some icons.
    2015/15/Jan: version 2015Janb2702 In response to a user request, it now generates an additional file with multifractal data separated according to datatype for comparing images.
    2015/17/Jan: version 2015Janb3079 Updated help file for multifractals. Fixed problems with q-inc and q-array maker that were preventing multifractal apertures from being drawn. Included data file for multifractal analysis that describes the multifractal scaling to complement the multifractal spectra data files. Multifractal spectra plots now save as a stack for unoptimized samples.
    2015/02/Feb: version 2015Febb4135 Fixed bugs in multifractal optimizer and data files so now the data set prints when the option for both optimizing and showing all is selected; fixed bug where plots were not saving for some filenames; included slice labels in data files if available as per user request.
    2015/18/Feb: version 2015Febb5699 Fixed new bugs in multifractal optimizer and data files as per user request so individual file name is included on the plot. Fixed colour-coding bug so grayscale "replace foreground pixels" option works. Fixed random sample bug that made this option crash on grayscale scans.
    2015/27/Feb: version 2015Febb5810 Fixed multifractal ƒ(αQ) options dialogue so now the user can choose to print or not print optimizing data on the graph.
    2015/08/March: version 2015Marb6206 Fixed "OK" button; fixed slider bugs; fixed problem with inverted LUTs in batch mode so they are now processed the same as in screen mode.
    2015/07/April: version 2015Apra4308 Added rotation function and integrated MicroMod for ImageJ.
    2015/06/July: Fixed bug with loading multiple custom files in MicroMod for ImageJ; fixed bug with rotations in results file.
    2015/15/Sept.: Fixed bugs in MicroMod and multifractal rotator.

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