On the fractal dimension of self-affine profiles

Abstract

One-dimensional profiles f(x) can be characterized by a Minkowski-Bouligand dimension D and by a scale-dependent generalized roughness W(f, epsilon ). This roughness can be defined as the dispersion around a chosen fit to f(x) in an epsilon -scale. It is shown that D=limepsilon to 0(2-In W(f, epsilon )/In epsilon ) holds for profiles nowhere differentiable. This establishes a close connection between the roughness and the fractal dimension and proves that D=2-H for self-affine profiles (H is the roughness or Hurst exponent). Two numerical algorithms based on the roughness, one around the local average (f(x))epsilon (usual roughness) and the other around the local RMS straight line (a generalized roughness), are discussed. The estimates of D for standard self-affine profiles are reliable and robust, especially for the last method.