The Hausdorff-Besicovitch dimension is the mathematical expression of the
dimension of objects. The generalized approach of Hausdorff makes this
definition useful for natual objects.
Take a line, a square and a cube, each with a unary length.
If you divide that length by two, keeping the same space, you finally end up with two segments of half-length, four squares with each one-fourth of the original surface and nine cubes with each one-ninth of the original volume.
Again, dividing the unary length segment by three yields, respectively, 3 lines, 8 squares and 27 cubes.
Summarizing this into a table, we have:
Expressed mathematically, the property of the object, its length, area or volume, noted N, is related to its dimension, noted D, by:
N = rD
Taking the logarithm on both sides of the equation and solving for D, we finally obtain the expression of the Hausdorff-Besicovitch dimension:
D = log (N) / log ( r)
Intuitively, N can be considered as the number of self-similar objects created from the original object when it is divided by r.
Taking the cube as an example, replace N by 27 and r by 3, and you got the dimension of the cube:
D = log (27) / log ( 3) = 3
Replacing the members N and r of the equation by the adequate values and one can compute the Hausdorff-Besicovitch or fractal dimension of any object.
The Box-Counting method is useful when the fractal curve fits the boxes of a simple grid.
To use the Box-Counting method, draw a fractal curve into a square area. Divide that area in small boxes so that the self-similarity drawings fit the size of the small boxes.
As an example, the Squares Curve is drawn on such a grid.
The number of boxes covered by the fractal curve is then determined and the Hausdorff-Besicovitch or fractal dimension is then calculated using the fundamental formula of the box-counting method:
N = 1 / dD
where N: number of boxes covered by the fractal curve
where d: size of the grid box (see below)
where D: fractal dimension of the curve.
This formula relates the fractal dimension with the size of the small boxes that made the overall grid.
Solving for D, we got: dD = 1 / N
Taking the logarithm on both sides gives:
log d * D = - log N
and then D = -log N / log d
To simply the calculations, one can take the side length of the grid as a unit value and define r as the number of small boxes along one side of that grid.
The formula becomes: D = -log N / log (1/r)
which, finally, gives: D = log N / log r
Woops, it's the self-similarity formula showed above !!!
The number of boxes, noted N, is then recorded for different values of r. Plotting log(N) on the y-axis against log(r) on the x-axis should give a straigth line, whose slope is equal to the fractal dimension.