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. (read more)