Title: Measuring Fractal Dimension Post by: C.K. on June 20, 2013, 01:45:03 AM 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 (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 (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 (http://www.lassp.cornell.edu/sethna/Tweed/Olson_Owen/Pick_Maddin_Martensite.gif)<---A very dense martensitic structure DISCLAIMER: I took none of these^^^^^ Title: Re: Measuring Fractal Dimension Post by: cKleinhuis on June 20, 2013, 02:37:26 AM 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 Title: Re: Measuring Fractal Dimension Post by: C.K. on June 22, 2013, 02:09:50 AM 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 (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. Title: Re: Measuring Fractal Dimension Post by: cKleinhuis on June 22, 2013, 12:47:53 PM 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.... |