Fourier transform fractal structures

Equation 14.29 defines the density correlation function C(r), where p(f) is the density of material at position r, and the brackets represent an ensemble average. In Equation 14.30, A is a normalization constant, D is the fractal dimension of the object, and d is the spatial dimension. Also in Equation 14.30 are the limits of scale invariance, a at the smaller scale defined by the primary or monomeric particle size, and at the larger end of the scale h(rl ) is the cutoff function that governs how the density autocorrelation function (not the density itself) is terminated at the perimeter of the aggregate near the length scale As the structure factor of scattered radiation is the Fourier transform of the density autocorrelation function. Equation 14.30 is important in the development below. 

The algorithms most fi equently used for calculation of fractal coefficients from the AFM results are [58] Fourier spectrum integral method, surface-perimeter method, structural function method and variable method. To determine the surface dimension by the Fourier spectrum integral method it is necessary to obtain the picture of the surface 2D FFT generating amplitude and time of the matrix (more detail s are given in paper [58]. Assuming the surface function as f(x,y), the Fourier transform in two-dimensional space can be expressed as [58] ... 

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