  The versatile physical properties of heterogeneous materials are intimately
related to their complex microstructures, which can be statistically
characterized and modelled using various spatial correlation functions
containing key structural features of the material's phases. An important
related problem is to inversely reconstruct the material microstructure from
limited morphological information contained in the correlation functions. Here,
we present in details a generalized lattice-point (GLP) method based on the
lattice-gas model of heterogeneous materials that efficiently computes a
specific correlation function by updating the corresponding function associated
with a slightly different microstructure. This allows one to incorporate the
widest class of lower-order correlation functions utilized to date, including
those encoding topological connectedness information and interface statistics,
into the Yeong-Torquato stochastic reconstruction procedure, and thus enables
one to obtain much more accurate renditions of virtual material microstructure,
to determine the information content of various correlation functions and to
select the most sensitive microstructural descriptors for the material of
interest. The utility of our GLP method is illustrated by modelling and
reconstructing a wide spectrum of random heterogeneous materials, including
"clustered" RSA disks, a metal-ceramic composite, a two-dimensional slice of a
Fontainebleau sandstone and a binary laser-speckle pattern, among other
examples.
