Fractal image analysis

The first detailed book to describe the practice and theory of stereology was assembled by two Americans, DeHoff and Rhines (1968) both these men were famous practitioners in their day. There has been a steady stream of books since then a fine, concise and very clear overview is that by Exner (1996). In the last few years, a specialised form of microstructural analysis, entirely dependent on computerised image analysis, has emerged - fractal analysis, a form of measurement of roughness in two or three dimensions. Most of the voluminous literature of fractals, initiated by a mathematician, Benoit Mandelbrot at IBM, is irrelevant to materials science, but there is a sub-parepisteme of fractal analysis which relates the fractal dimension to fracture toughness one example of this has been analysed, together with an explanation of the meaning of fractal dimension , by Cahn (1989). 

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... 

Quevedo, R., Lopez, G.C., Aguilera, J.M., and Cadoche, L. 2002. Description of food surface and microstructural changes using fractal image texture analysis. J. Food Eng. 53, 361-371. 

The structure of this review is as follows in Section II, the activation and templating methods for preparing the porous carbons are briefly summarized. Section III surveys the structural characteristics of the porous carbons by using gas adsorption method. In Section IV, the molecular probe method and the image analysis method for quantitative characterization of the pore surface irregularity and the size distribution irregularity based upon the fractal theory are discussed in detail. Section V is devoted to... 

Several simple relations have been proposed for the determination of the fractal dimensions from the results of such experiments as gas molecular probe method, transmission electron microscopy (TEM), small-angle X-ray scattering, neutron scattering, and laser light scattering.63,66,113 116 Among those techniques, gas molecular probe method and image analysis method have been widely used for the calculation of the surface... 

On the other hand, for the microporous carbons with pore size distribution (PSD) with pore fractality, the pore fractal dimensions56,59,62 which represent the size distribution irregularity can be theoretically calculated by non-linear fitting of experimental adsorption isotherm with Dubinin-Astakhov (D-A) equation in consideration of PSD with pore fractality.143"149 The image analysis method54,151"153 has proven to be also effective for the estimation of the surface fractal dimension of the porous materials using perimeter-area method.154"159... 

Kaye BH (1986) Image analysis procedures for characterising the fractal dimension of fine particles. Proc Part Technol Conf Nurnberg... 

The problem of transfer across a fractal surface has been considered in the electrochemical behavior of rough and porous carbon electrodes [36]. The fractal dimension can be determined from nitrogen gas adsorption data, from transmission electron microscopy (TEM), and nanoscopy image analysis. 

For imprinted mesoporous carbons, the overall fractal dimension, determined from gas adsorption data, indicate that these materials are composed of two groups of pores. The surface fractal dimension of the carbonization-induced pores surface and that of the silica-imprinted pores surface has been obtained from TEM image analysis [38]. 

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