Fractal Dimensions - a review and study on sound captured through skin vibrations

Chapter 5. Invariants of the motion [book - analysis of observed chaotic data - Abarbanel H.D.I.]


Chapter 2/pg 21 - Fractal Dimension: A Quantitative Measure of Self_Similarity and Scaling
[book - Fractal Physiology - James Bassingthwaighte, Larry Liebovitch, Bruce West]

Self-Similarity Dimension-
describes how many new pieces geometrically similar to the whole object are observed as the resolution is made finer. If we cahnge the scale by a factor F, and we find that there are N pieices similar to the original, then the self-similarity dimension D(self-similarity) is given by
N = F ^ D(self-similarity)
This can be a fractional dimension - fractals (fragments) which usually takes in non-integer values and which therefore lies between Euclidean Dimensions.

Hausdorff-Besicovitch Dimension and Capacity Dimension-
Necessary to find fractional dimensions of irregular shaped objects. these two dimensions are technically similar.
This is explained in terms of number of balls required to cover all the points within a space. In 1-D (ball is a line), 2-D (circle), 3-D (sphere), and so on. Consider the radius of the ball to be r/ let N(r) be minimum number of balls of size r needed to cover the object. The capacity dimension Dcap tells us how the number of balls needed to cover the object changes as the size of the balls is decreased.
Dcap = lim (r -> 0) log N(r) / log (1/r)

If the capacity dimension is determined by using balls that are contiguous non overlapping boxes of a rectangular coordinate grid, then the procedure is called as "BOX COUNTING".