  Discrete rearranging patterns include cellular patterns, for instance liquid
foams, biological tissues, grains in polycrystals; assemblies of particles such
as beads, granular materials, colloids, molecules, atoms; and interconnected
networks. Such a pattern can be described as a list of links between
neighbouring sites. Performing statistics on the links between neighbouring
sites yields average quantities (hereafter "tools") as the result of direct
measurements on images. These descriptive tools are flexible and suitable for
various problems where quantitative measurements are required, whether in two
or in three dimensions. Here, we present a coherent set of robust tools, in
three steps. First, we revisit the definitions of three existing tools based on
the texture matrix. Second, thanks to their more general definition, we embed
these three tools in a self-consistent formalism, which includes three
additional ones. Third, we show that the six tools together provide a direct
correspondence between a small scale, where they quantify the discrete
pattern's local distortion and rearrangements, and a large scale, where they
help describe a material as a continuous medium. This enables to formulate
elastic, plastic, fluid behaviours in a common, self-consistent modelling using
continuous mechanics. Experiments, simulations and models can be expressed in
the same language and directly compared. As an example, a companion paper
(Marmottant, Raufaste and Graner, joint paper) provides an application to foam
plasticity.
