Scale-Specific Multifractal Medical Image Analysis

Figure 3

Results of applying our algorithms to four renderings of mathematical fractals. The entropies in (b), (e), (h), and (k) are in nats, which are the natural units for information and entropy, with base 𝑒 rather than 2: 1 nat ≈ 1.44 bits, and are plotted for 𝛼=0,1,2. All the fractal dimension plots (c), (f), (i), and (l) use 𝜎=0.5 in (6), except for (i) where 𝜎=0.3 for the circles. Horizontal bar indicates Hausdorff dimension 𝐷_𝐻 of each mathematical fractal.

(a) 998 × 1152 pixel rendering of the boundary of a Koch snowflake

(b) Scale dependence of entropy of (a)

(c) Scale dependence of fractal dimension of (a)

(d) 2000 × 2000 pixel rendering of the boundary of a Koch island

(e) Scale dependence of entropy of (d)

(f) Scale dependence of fractal dimension of (d)

(g) 1460 × 1460 pixel rendering of Pascal’s triangle (mod 3)

(h) Scale dependence of entropy of (g)

(i) Scale dependence of fractal dimension of (g)

(j) 2187 × 2187 pixel rendering of the Sierpinski carpet

(k) Scale dependence of entropy of (j)

(l) Scale dependence of fractal dimension of (j).