Fractal dimension of a polygonal line.

Question

Given an ordered set of points in the unit square, what are the most elegant ways to estimate the fractal dimension of the curve? By "elegant" I mean without resorting to drawing the object and doing box counting...

Answer 1

There are many different fractal dimensions. The trouble is that most (all?) of them ultimately result in a finite set of line segments having dimension 1.

The divider dimension $D$ is defined for curves by measuring the total length $L(t)$ at various yard-stick lengths $t$, then reading $1 - D$ from the slope of a log-log plot.

For a review of practical techniques for "real-world" type data, including a section on the divider dimension for curves, perhaps see:

"A Review of Methods Used to Determine the Fractal Dimension of Linear Features" Brian Klinkenberg (Mathematical Geology, Vol. 26, No. 1, 1994)

An in-depth review of the more commonly applied methods used in the determination of the fractal dimension of one-dimensional curves is presented. Many often conflicting opinions about the different methods have been collected and are contrasted with each other. In addition, several little known but potentially useful techniques are also reviewed. General recommendations which should be considered whenever applying any method are made.

KEY WORDS: fractal measurement, divider method, box counting, spectral techniques, variogram.