Multifractal spectrum

Box Mass

Let N(r) be the total number of boxes (non overlapping) of size length r covering the object. Let M_i(r) be the mass within the i'th box. Define Z(q,r) as follows.

	N(r)
Z(q,r) =		[ Mi(r) ]q
	i=1

The multifractal dimension D(q) is given by

q ranges from -infinity (concentrates on less dense regions) and +infinity (concentrates on dense regions).

For q = 0 this reduces to standard box dimension.

Mass radius

Consider all circles of radius r that have their center on the object. Let M_i(r) be the mass within the i'th circle, and the total number of circles of radius r is N(r). Then Z(q,r) is defined as

	N(r)
Z(q,r) =		[ Mi(r) ]q N(r)
	i=1

The multifractal dimension D(q) is given by

Examples

Gasket Fractal dimension: log(3) / log(2) = 1.585
Snowflake Fractal dimension: log(4) / log(3) = 1.262
Weed
Line Dimension = 1
Solid black Dimension: 2
MM D-infinity = log(17)/log(5) = 1.760 Dinfinity = log(17/4)/log(5/2) = 1.570
NN
DLA
Random 1000x1000 pixels 100000 random points. Uniformly distributed on both axes.
Random 1000x1000 pixels 500000 random points. Uniformly distributed on both axes.

References

Multifractal Characterization of Soil Particle-Size Distributions
A.N.D. Posadas, D. Gimenez, M. Bittelli, C.M.P. Vaz, and M. Flury
Soil Sci. Soc. Am. J. 65:1361–1367 (2001)

Direct Determination of the f(alpha) Singularity Spectrum
A. Chhabra and R.V. Jensen
Physical Review Letters, 62, March 1989, #12

Multifractal features of random walks on random fractals
A. Bunde, S. Havlin, H.E. Roman
Physical Reviewm A 42, 6274 (1990)

Multifractal phenonema in physics and chemistry
H.E. Stanley and P. Meakin
Nature, 335, 405 (1988)

Measuring the strangeness of strange attractors
P. Grassberger and I. Procaccia
Physica, Amsterdam, 9D, 189 (1983)

Scaling in financial prices: IV Multifractal Concentration
M.B. Mandelbrot
Quantitative Finance, Vol 1, 641-649 (2001)

Are Neurons multifractals
E. Fernandez, J.A. Bolen, et al
Journal of Neuroscience Methods, 89, 151-157 (1999)

Physical mechanisms underlying neurite outgrowth: a quantitative analysis of neuronal shape.
Caserta, F., et al.
Physical Review Letters, 1990. 64(1): p. 95-9