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 f or investigating self-affine and self-similar objects. First, it enabl es us to extract fractal exponents v(x) and v(y) from one pattern of a n object having any topology. Second, we show that the integral z(x, y ) which completely characterizes a monofractal object provides more in formation on the density correlation properties of the object than jus t the exponents v(x) and v(y). We quantify this additional information by introducing two parameters: delta, characterizing the object's ani sotropy of a nonscaling nature, and kappa characterizing the curvature of the logarithmic correlation integral of the object. We demonstrate that the four parameters: v(x), v(y), delta and kappa provide an effe ctive parametrization of the logarithmic correlation integral of a sel f-affine monofractal object. We give some examples of self-affine obje cts, having the same fractal exponents v(x) and v(y) but different par ameters delta and kappa indicating the differences in the correlation properties of the objects. We demonstrate that even a self-similar obj ect showing isotropic scaling (v(x) = v(y)) may have the non-scaling a nisotropy parameter delta different from zero, which indicates that th e object has an asymmetric integral z(x, y) and, therefore, different correlation properties in different directions. It is shown that the e quality kappa = 0 outlines a class of objects for which the exponents v(x) and v(y) are not defined uniquely. For instance, such objects can be treated as both self-similar and self-affine. If kappa is close to zero, estimation of the exponents v(x) and v(y) may become problemati c. Relationships connecting the exponents v(x), v(y) and fractal dimen sions of the projection and cross section of an object are established .