  By using the most sensitive two-point correlation functions introduced to
date, we reconstruct the microstructures of two-phase random media with
heretofore unattained accuracy. Such media arise in a host of contexts,
including porous and composite media, ecological structures, biological media,
and astrophysical structures. The aforementioned correlation functions are
special cases of the so-called {\it canonical} $n$-point correlation function
$H_n$ and generalize the ones that have been recently employed to advance our
ability to reconstruct complex microstructures [Y. Jiao, F. H. Stillinger, and
S. Torquato, Proc. Nat. Acad. Sci. {\bf 106}, 17634 (2009)]. The use of these
generalized correlation functions is tantamount to dilating or eroding a
reference phase of the target medium and incorporating the additional
topological information of the modified media, thereby providing more accurate
reconstructions of percolating, filamentary, and other topologically complex
microstructures. We apply our methods to a multiply-connected "donut" medium
and a dilute distribution of "cracks" (a set of essentially zero measure)
demonstrating improved accuracy in both cases with implications for
higher-dimensional and biconnected two-phase systems. The high information
content of the generalized two-point correlation functions suggests that it
would be profitable to explore their use to characterize the structural and
physical properties of not only random media, but also molecular systems,
including structural glasses.
