Fractal analysis—Box counting method

Describes the way in which non-normalized objects/structures fill space. Fractal analysis helps solve the issue of quantification for both euclidean and fractal objects by identifying a connection between the value of a measured quantity and the scale at which the measurement is taken.

Fractal objects vs. Euclidean objects

A Euclidean object is something that measures to the same quantity independently of the scale at which it is viewed.

For fractal objects, quantification varies depending on the scale. A fractal object reveals more and more detail as the magnification of the view is increased. As smaller and smaller rulers are used to measure the length or other features of a fractal object, more length or other quantity is uncovered.

How fractal analysis works

Neurolucida Explorer uses the box counting method that consists of nested cubes. Any cube can be divided into 8 smaller cubes, that can in turn be divided into 8 smaller cubes, and so on, ad infinitum.

At each stage of the subdivision, a count is made of the number of cubes that contain a part of the object. The number of cubes grows exponentially.

If the object is fractal, the number of cubes containing a part of the object should also grow exponentially.

The log base 8 is taken of the number of cubes and the number of cubes that contain a part of the object. A plot of these values reveals a straight line for fractal objects and a curve that plunges to zero for Euclidean objects. The slope of the line is the fractal measure of the object. This fractal dimension, as it is called, reveals how the object increases in detail as the magnification is increased.

 

Reference

Panico, J. and Sterling, P. (1995). Retinal Neurons and Vessels Are Not Fractal But Space-Filling. Journal of Comparative Neurology, 361:479-490. (This article does not use our method of fractal analysis, but contains a description of the method that we use.)