  Two-phase random textures abound in a host of contexts, porous and composite
media, ecological structures, biological media and astrophysical structures.
Questions surrounding the spatial structure of such textures continue to pose
many theoretical challenges. For example, can two-point correlation functions
be identified that can be both manageably measured and yet reflect nontrivial
higher-order structural information about the textures? We present a novel
solution to this question by probing the information content of the widest
class of different types of two-point functions examined to date using inverse
"reconstruction" techniques. This enables us to show that a superior descriptor
is the two-point cluster function $C_2({\bf r})$, which is sensitive to
topological {\it connectedness} information. We demonstrate the utility of
$C_2({\bf r})$ by accurately reconstructing textures drawn from materials
science, cosmology and granular media, among other examples. Our work suggests
an entirely new theoretical pathway to predict the bulk physical properties of
random textures, and also has important ramifications for atomic and molecular
systems.
