Measurement of spatial correlation functions using image processing techniques

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Abstract
A procedure for using digital image processing techniques to measure the spatial correlation functions of composite heterogeneous materials is presented. Methods for eliminating undesirable biases and warping in digitized photographs are discussed. Fourier transform methods and array processor techniques for calculating the spatial correlation functions are treated. By introducing a minimal set of lattice‐commensurate triangles, a method of sorting and storing the values of three‐point correlation functions in a compact one‐dimensional array is developed. Examples are presented at each stage of the analysis using synthetic photographs of cross sections of a model random material (the penetrable sphere model) for which the analytical form of the spatial correlations functions is known. Although results depend somewhat on magnification and on relative volume fraction, it is found that photographs digitized with 512×512 pixels generally have sufficiently good statistics for most practical purposes. To illustrate the use of the correlation functions, bounds on conductivity for the penetrable sphere model are calculated with a general numerical scheme developed for treating the singular three‐dimensional integrals which must be evaluated.