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In this paper, we detail and complete the existing characterizations of thedecimation of the Ising model on $\Z^2$ in the generalized Gibbs context. Wefirst recall a few features of the Dobrushin program of restoration ofGibbsianness and present the construction of global specifications consistentwith the extremal decimated measures. We use them to consider theserenormalized measures as almost Gibbsian measures and to precise its convex setof DLR measures. We also recall the weakly Gibbsian description and complete itusing a potential that admits a quenched correlation decay, i.e. a well-definedconfiguration-dependent length beyond which this potential decaysexponentially. We use these results to incorporate these decimated measures inthe new framework of parsimonious random fields that has been recentlydeveloped to investigate probability aspects related to neurosciences.

Parsimonious Description of Generalized Gibbs Measures : Decimation of the 2d-Ising Model