Hausdorff (Box-Counting) Fractal Dimension

Returns the Haussdorf fractal dimension of an object represented by a binary image.


Updated 18 Dec 2013

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Returns the Haussdorf fractal dimension D of an object represented by the binary image I. Nonzero pixels belong to an object and 0 pixels constitute the background.


1 - Pad the image with background pixels so that its dimensions are a power of 2.
2 - Set the box size 'e' to the size of the image.
3 - Compute N(e), which corresponds to the number of boxes of size 'e' which contains at least one object pixel.
4 - If e > 1 then e = e / 2 and repeat step 3.
5 - Compute the points log(N(e)) x log(1/e) and use the least squares method to fit a line to the points.
6 - The returned Haussdorf fractal dimension D is the slope of the line.

In this blog post I show how this code can be used to compute the fractal dimension:

Cite As

Alceu Costa (2023). Hausdorff (Box-Counting) Fractal Dimension (, MATLAB Central File Exchange. Retrieved .

MATLAB Release Compatibility
Created with R2009a
Compatible with any release
Platform Compatibility
Windows macOS Linux

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Version Published Release Notes

Updated the description to include a link to a blog post explaining how the code can be used.

Corrected a typo in the file title .