A local connected fractal dimension scan considers the pixel distribution in terms of local environments.

Are you experiencing deep consternation over an image you know must have hidden areas of significance that will be objectively revealed by fractal analysis?

Does the illustration showing my retinal vessels, a binarized version of the image, and a colour-coded scan result automatically highlighting certain areas look like what you need? Then this might be the place to find inner peace.

The basic premise of a local connected fractal dimension (LCFD) scan is that we can identify meaningful local variation in complexity...and even present it visually!

Landini's paper applying
LCFD analysis to the diagnosis of retinal vascular disease is a great reference for the technique. FracLac implements his basic ideas and this page shows you how it all works.

Are you ready to build on your earlier background and learn how to set up and interpret an LCFD analysis?

    Using this Page:
  • Setting Options:
  • General Knowledge:

This page is the practical reference for Local Connected Fractal Dimension or LCFD scans.If you are looking for something similar and think you arrived at the wrong page, try the Sub Scans page.

The page assumes you are familiar with the basics of using FracLac, are grounded in connected sets, know how LCFD scanning methods (illustrated in the image above) differ from regular box counting, and want to learn more about setting the options and interpreting the results from LCFD scans.


To learn about options, click on them on the screenshot, which is an image of the basic options panel that comes up when you select a LCFD scan in FracLac. If you prefer, scroll down to the list.

The Options

Save Files

Select this box to automatically save the graphics and data files from your scan.

Tutorials on automated saving and batch jobs.

Show colour coded image?

Select this box to generate saveable images colour-coded according to the LCFD at each foreground pixel sampled. If you need to know about the colour-coding options in general, click here. See TIPS

Show Df per Pixel Data

Select this option to generate a data file listing the LCFD at each pixel (see also colour coded graphic). As illustrated in the screen shot, this file also includes the y-intercept and other stats describing the LCFD at each pixel.

Show Mass per Pixel Data

Select this option to generate a data file having three matrices of data, showing in each row the coordinates for each pixel sampled and each column the size of the sampling unit, where the contents of the 3 matrices are:

  1. the number of pixels in each concentric box or circle that were in both the connected set and the sampling unit
  2. the number of pixels in each connected set (the same for each size column)
  3. the total number of pixels in each concentric box or circle

Skip Scans covering more than...

This setting sets a margin around the scanning area. The image illustrates a scan using a 50 pixel maximum box size with the area to skip set at 10 pixels and at 50 pixels. Use this setting to avoid sampling edges when that will adversely affect your results. You may need to explore how the maximum box size and the margin interact for different types of images.

Show grids?

Select this option to generate saveable images showing the sampling units that were actually used. See a sample of the graphic that will be generated here.

The Size of the Sampling Element

The options for the size of the boxes or circles used to sample should all be considered together. The most important point to remember is that the option for Type of Series should be set to "Use Odd Series" for LCFD scans, so that the sampling units will rest concentrically on the seed pixel with the same margin around all sides. Using other options may give undesirable results. See also the general explanation of this option.

The list below links to general explanations from regular box counting
Use greater dimension of ROI
Maximum box size in pixels
Maximum box size as a % of ROI
Minimum size in pixels
Sizes per Series

Background Colour

This option determines the colour of pixel that FracLac assesses. Read the global explanation of this option.

Number of pixels to slide (X axis or Y axis)

Type a number, x or y, so that the connected set and LCFD will be calculated for only every xth pixel horizontally and every yth pixel vertically. Setting this greater than 1 speeds up processing but ignores pixels and may ignore many pixels in images that are sparse in the first place. Prior to doing batch jobs in particular, you may want to do a trial run selecting the option to draw the boxes in order to see if the spacing selected will give the results you want.

Use round scan areas

Select this box to use round samples or leave it unselected to use square samples. See an illustration of the difference here.

Maximum Bin

Type a number for the maximum size of bins in the frequency distribution for the LCFD.

Minimum Bin

Type a number for the minimum size of bins in the frequency distribution for the LCFD.

Bin Size

Type a number for the increment between bins to use in the frequency distribution for the LCFD. As an example, if you are interested in the distribution of the fractal dimension over an image in the range of 1.3 to 1.4, then you could set the minimum to 1.29 and the maximum to 1.41, with a bin size or increment of 0.01. Note that this distribution is not the BPD used for lacunarity.Use the DF distribution along with the colour-coding and graphing options.

Print frequency distribution?

Generate a file (see graph and colour-coded image also) of data showing how the values for the LCFD were distributed, using the maximum, minimum, and bin size options. The screen shot shows the contents of the file - a column for the LCFD and one for the frequency of each value.

The distribution by actual pixel in colour coded graphics
A graph of the distribution

Use the DF distribution along with the colour-coding and graphing options.

Graph Distributions

Generates a graph of the distribution, as in the screen shot.

Include Circularity Data

Select this box to calculate but not necessarily draw the convex hull and bounding circle. To see or save these graphics, select the "Draw Hull" and "Draw Circle" options as appropriate. See more on the Discussion and Results pages.

Create a text image?

Select this option to generate a text image of the colour coding results.


DF Distributions

To help determine a suitable range for colour-coding, you can generate a data file as well as a graphics file of how the LCFD was distributed over an image. There are 3 parameters to set for the distribution itself—the minimum, maximum, and bin size or increment between DFs. The default is from 0 to 2 with a binsize of 0.0133; however, these numbers should reflect the numbers you expect to see in the images you are analyzing, and you may wish to do trial runs to determine a particular range. See the colour-coding tip for more information.

Colour Coding

With the colour-coding option selected, you can view the same data set in different ways, adjusting the colours to accentuate areas with a certain range of LCFD values. In other words, the picture of retinal vessels you saw above is not all there is to LCFD scans. This montage shows the results of one scan on one image, with multiple LUTs applied when the scan was done. Each LUT highlights different aspects of the distribution of LCFDs (e.g., compare the right central area of all of the images).

You can colour-code any LCFD scan on screen or as a batch job, and can view the results on screen or save them automatically to a folder.

Prior to scanning batch jobs, you may want to know what range of fractal dimensions your images will show variation over. You can determine this by doing a scan on a single slice on screen, which will show a popup letting you modify the results in real time. This option is not available for batch jobs and stacks with more than 1 slice.The coloured graphic of your original image will appear showing the LCFD at each pixel along with a legend showing what each colour means. A dialog asks if you would like to try another LUT (another set of colours and/or associated codes). You may choose to do it many, many times with one image as, for example, you home in on the areas in the image that trouble you most. See the distribution tip for more on this issue.

Use the Frequency Distribution to help determine the best range.