Box-Counting & Theory
Lacunarity (Mandelbrot 1983):
Λ(r) = [E(M²)] / [E(M)]² = 1 + CV(M)²
M = mass in gliding box of size r
Gliding box algorithm:
Slide r×r box over pattern, measure M
Λ(r) = ⟨M²⟩/⟨M⟩² for each r
Λ→1: homogeneous; Λ≫1: lacunar/gappy
Box-counting dim: N(r) ∝ r^(−D_f)
Grid size—
Fill fraction—
Λ at r=10—
Box-counting D_f—