Coarse-graining procedure to generate and analyze heterogeneous materials: Theory

We present a theoretical framework to generate and statistically characterize the microstructure of coarse-grained random two-phase heterogeneous materials. The structures are produced by convolving the intensity function f(x) of a source image with a kernel K(x) to yield a new smoothed intensity F(x) and then using the coarse-grained image which results from taking a cut through the surface F(x) at F0 (''islands within lakes''). By varying F0 and the properties of the kernel K, one can generate a wide class of intricate microstructures. We provide a general means, which heretofore had been lacking, of representing and computing the correlation functions that statistically characterize samples of arbitrary size of such coarse-grained models. To illustrate our formalism we obtain results for specific examples of the source intensity f(x) and the kernel K. We also show how one can use this procedure to generate media consisting of distinct particles in a matrix of another material. The applicability of this study to bulk properties of heterogeneous materials and to image analysis in general is discussed.