Fourier transform fractal dimension

For the Fourier transform fractal dimension, Dyt, a two-dimensional microscopy image is considered a discrete function, F(x,y), where x and v are the coordinates of the object pixels in the horizontal and vertical direction. The two-dimensional discrete Fourier transform is applied to transform the two-dimensional image to its... 

The particle-counting fractal dimension, Dj, was not sensitive to crystal shape, size, AF or the distribution orderliness. It was found that Df was affected by the radial distribution pattern of the fat crystals as shown in Figure 17.28. The simulation results were found to be consistent with experiments (Litwinenko et al. 2002 Tang and Marangoni 2006). Devalues close to 2 indicated more homogenously distributed fat crystals. It is important to note, the values of Dfm y exceed the dimensionality of the embedding space. This is not the case for the box-counting dimension or the Fourier transform fractal dimension. 

The Fourier transform fractal dimension Dyi decreases almost linearly with the increasing radius of the crystals, and increases slightly when the AF increased, but no clear trend was evident among different distribution order at all AF. Relative to the effects of crystal size, the effects of the distribution order and AF of the crystals on ZlpT were much less significant. 

Besides the molecular probe method using gas adsorption,107 162 recently, the TEM image analysis method163"167 has been applied to evaluate the surface fractal dimension of porous materials. The most attractive fact in this method is that the pores in different size ranges can be extracted from the TEM images which include contributions from many different pore sizes by the inverse fast Fourier transform (FFT) operation by selecting the specific frequency range.165 167... 

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] ... 

The fractal dimension may be observed experimentally by light or neutron scattering [8]. The scattered intensity S( is the Fourier transform of the pair correlation function... 

Equation (6.10) was checked directly, using polymers with different masses. It was also tested using scattering experiments, by measuring the Fourier transform S(q) of the pair correlation function. As above Eq. (6.7), one can show that the scattered intensity is related to the wave vector q by the fractal dimension. n d = 3 one finds, using Eq. (6.10), that... 

The subscript f is used to indicate that this is the fractal dimension computed from the Fourier transform. The value of p computed for these data thus corresponds to a fractal dimension of about 1.45. Fractal dimension estimates of the phE data were also made using Mandelbrot s box dimension and resulted in essentially the same values as D. ... 

The rapid fluctuations of phE during the fracture of many materials suggest that deterministic chaos is a feature of the phE process. The autocorrelation function and conditional probability distributions are consistent with this initial impression. Fourier transform and fractal box dimension analyses show the phE to be fractal in nature, which is quite suggestive of chaos. The strongest evidence for chaos results from an analysis of the correlation dimension of the attractor associated with the epoxy data. A clearly nonintegral dimension of about 3.2 is found. 

For a wave transfer momentum q, the intensity I(m,q) scattered by a cluster of mass m and characteristic size R(m) is proportional to the Fourier transform of its pair correlation function - (r /r )g(r/R(m)) -where D is the fractal dimension, R(m)... 

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