Measuring Fractal Dimension

[C.K.]

Alright, so here's a question that's been bugging me since I started getting interested in fractals and chaos (I'm still making my way through the fundamentals).  I know that there are many ways to compute a fractal's dimension given it's formula or mode of generation, but there seem to be only a few widely accepted, or even feasible, methods of actually measuring the fractal dimension of a shape.  I say shape since the only method that I'm personally familiar with are the box counting method and specific variations of it, but I know that people have claimed to measure the fractal dimension of three dimensional forms, such as clouds.  So that's, what, cube counting?  In addition, it seems to me that reliably measuring a fractal dimension is quite a daunting task, or even impossible, since the "least count" or resolution of your measure is going to have a drastic effect on the dimension you come up with, this being chaos and all.

Numerically, I guess my questions boil down to the following.
1) What are the ways to measure, reliably, the fractal dimension of shapes or forms?
2) How do you measure the fractal dimension of forms (3D)?

To put this in context, I've recently been interested in studies regarding finding the fractal dimension of grain boundaries in metals to try and come up with some sort of universal measure for observed properties. I have my own theories on this, but all the studies I've seen so far have been for very simple grain structures, such as purely austenitic stainless steel.  I'd be interested in seeing if you could come up with a way to measure the fractal dimension (reliably, again) or more complex structures, such as martensite or bainite.

Here's some images of those structures, for those that aren't familiar.
http://core.materials.ac.uk/repository/doitpoms/miclib/000771.jpg <---Austenitic Stainless Steel
http://www.lassp.cornell.edu/sethna/Tweed/Olson_Owen/1901_Martensite.gif<---Martensite (the black structure)
http://www.lassp.cornell.edu/sethna/Tweed/Olson_Owen/Pick_Maddin_Martensite.gif<---A very dense martensitic structure

DISCLAIMER: I took none of these^^^^^

[cKleinhuis]

hi there, one way to play around with it would be to use the imagej imageprocessing software
http://rsbweb.nih.gov/ij/

it has a fractal dimension calculation plugin:
http://rsb.info.nih.gov/ij/plugins/fraclac/fraclac.html


so, i recently started to write my own fractal dimension calculator using the "box-counting-dimension"
so, the measured value is highly dependent of general conditions, and i advice to use HUGE data sets
for applying, the box counting halves the size of a box, and then counts how many boxes overlap with
a feature you want to analyze, since the method converges to what can be interpreted as fractal
dimension, you want to have many measurements

check these:
http://en.wikipedia.org/wiki/Fractal_dimension
http://en.wikipedia.org/wiki/Fractal_dimension#Estimating_from_real-world_data
http://en.wikipedia.org/wiki/Box_counting

[C.K.]

Thanks for the tip!  I actually had the imagej software already downloaded on my computer, but I wasn't aware of this functionality.  I will definitely try it out!

And for those interested, I found a very good reference for fractal dimension mathematics (at least a primer) and it also references how to find the fractal dimension of forms!  As I understand it, you apply a box-counting method in each 2D plane (xy, xz, yz) and form a lattice from the box counting grid.  How this is specifically used to find the fractal dimension of a 3D shape it doesn't go into, but it certainly did help in satisfying my curiousity!

The link is below:
http://www.wahl.org/fe/HTML_version/link/FE4W/c4.htm
It's authored by a Prof. Bernt Wahl, so credit goes to him for this.

[cKleinhuis]

you can easily extend the box counting algorithm to 3d objects as well, but as always things get complicated in 3d
i was rather disappointed to see that the fractal dimension is really hard to measure, and i would love to have the
ability to assign the correct dimension to a form that is arbitrary rotated/layed out, but hadnt no time to dive deeper
into it....