What is the fractal dimension ?

The fractal dimension is the
key quantity in the study of fractal geometry. Fundamental is the concept of
”measurement at scale δ". For each δ we measure an object in a way that ignores
irregularity of size less than δ, and we see how these measurements behave as δ
goes to 0. I has been found that most natural objects satisfy the power
law, which states that the estimated quantity (for example the length of a
coastline ), is proportional to (1/δ)^{D }for some D at small scales δ.
Thus, we can compute its limit, which is called the *fractal dimension*.
For a point set E defined on R^{2},
the fractal dimension of E is defined as:

where N(δ, E) is the smallest number of sets of diameter less than δ that cover E. One usually divides up the space with a mesh of boxes of size δ, and counts the squares occupied by the point set. The fractal dimension computed this way is called box counting dimension.

Examples:

What is the the Multi-fractal Spectrum (MFS) ?

The MFS is the extension of the fractal dimension. It is a vector of fractal dimensions. One defines a point categorization on the image according to some criteria. Then the fractal dimension is computed for every point set from this categorization. In the example below image points are categorized by their intensities. That is, all points within an interval are set to 0 (black) and all others to 1 (white).

D = 1.67 D = 1.49

grass texture points with intensity 100 - 120 points with intensity 80 -100