Fractal Growth Models

Author: Audrey Karperien email
History: 2005/11/15: First version
2011/02/11: Fixed a bug with the rendering of rectangles vs ovals and added stacks that show the growth at various stages for comparison.
2012/04/18: Fixed problem with no image appearing on screen; responded to user request to retain arrays while working on a model; added "About" message
2015/01/18: Fixed bugs reported with spiral model and GUI. Buttons now show the selected model. Added requested feature to render a model without always having to configure it. Added requested feature to specify the image size.
2015/01/20b1239: Fixed bug requests so that coefficient files now load properly and on screen progress display runs more smoothly.
Source: Fractal_Growth_Models.jar
Installation: Download Fractal_Growth_Models.jar to the plugins folder, or subfolder, and restart ImageJ.

This plugin draws iterated growth models for multifractal analysis in ImageJ (e.g., using FracLac for ImageJ). These are saveable screen drawings made of points calculated using the basic expressions:
xn+1 = mxnXxpow + nynXypow + o
yn+1 = fxnYxpow + gynYypow + h.

For each model, each equation's coefficients (m,n,o;f,g,h; Xxpow, Xypow, Yxpow, Yypow) are specified in sets that the user can modify. Each of these sets also contains a probability of using the set. This is for cases when more than one set of coefficients is used to calculate each point, chosen according to a likelihood that the user sets.

Each particle calculated and drawn is coloured on a background, and all are the same size. The user can specify to use a single colour or gradient, and can set the gradient, foreground, and background.

The user can change the size of the particles in the models using an option in the first popup.


There are five default models. The user can change the coefficients to vary the basic pattern, or change the number of sets to create a new model.

  • The "Henon" model is specified by one set of coefficients (i.e., the likelihood of using that set is 1).
  • The Henon Map model is a multifractal benchmark model; it is also specified by one set of coefficients. It has a (capacity) fractal dimension that is assumed to be around 1.261 (correlation dimension around 1.25).
  • The "Random" model is similar to the Henon models, but uses two sets of coefficients and probabilities.
  • The "D. Greene Fern" model uses four sets of coefficients. It is from an algorithm by David Greene (Charles Sturt University),
  • The custom model is essentially the same as the fern, with a few changes in coefficients and probabilities.

Using the plugin

  1. To generate a default model, the user selects a model from the menu on the GUI (see the screenshot), then clicks a button at the bottom to render the model or reconfigure prior to rendering. In the second case, various dialogs will appear asking the user to specify some settings.
  2. The user can also save or load files of coefficients using the checkboxes and buttons on the interface.
  3. When configuring, popups appear displaying the coefficients for each set, which the user can modify as desired.
  4. At the bottom of each set, there is a space for the probability that the coefficients above it will be used in generating each next point. The total of all probabilities should not be greater than 1; all sets for values that make the cumulative total greater than 1 are ignored.
  5. Animations There is a setting to generate stacks showing the progression. The stacks can be saved as avis or analyzed individually to quantitate multifractal scaling at different stages.

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