Study of self-similar and self-affine objects using logarithmic correlation integral

We suggest a logarithmic correlation integral z(x,y) as a good tool for investigating self-affine and self-similar objects. First, it enables us to extract fractal exponents vx and vy from one pattern of an object having any topology. Second, we show that the integral z(x,y) which completely characterizes a monofractal object provides more information on the density correlation properties of the object than just the exponents vx and vy. We quantify this additional information by introducing two parameters: delta , characterizing the object's anisotropy of a nonscaling nature, and K characterizing the curvature of the logarithmic correlation integral of the object. We demonstrate that the four parameters: vx, vy, delta and kappa provide an effective parametrization of the logarithmic correlation integral of a self-affine monofractal object. We give some examples of self-affine objects, having the same fractal exponents vx and vy but different parameters delta and kappa indicating the differences in the correlation properties of the objects. We demonstrate that even a self-similar object showing isotropic scaling (vx=vy) may have the non-scaling anisotropy parameter delta different from zero, which indicates that the object has an asymmetric integral z(x,y) and, therefore, different correlation properties in different directions. It is shown that the equality kappa =0 outlines a class of objects for which the exponents v, and v, are not defined uniquely. For instance, such objects can be treated as both self-similar and self-affine. If K is close to zero, estimation of the exponents vx and vy may become problematic. Relationships connecting the exponents vx, vy and fractal dimensions of the projection and cross section of an object are established.