In this paper, we introduce a new robust procedure to identify material parameters of mechanical models from full-field measurements. It is based on data information coming from the Digital Image Correlation technique. The procedure aims at defining a suitable numerical processing, in terms of model selection and discretization mesh, with respect to information and noise contained in the data. The nature of the procedure is to minimize a cost functional based on the modified Constitutive Relation Error concept, which is made of modeling and measurement terms. Constructing an admissible stress field, verifying the equilibration equation in a full sense, enables one to obtain estimates on both discretization and modeling errors, which can then be compared with measurement noise in order to drive mesh adaptation and model enrichment. In addition, the procedure is coupled with reduced order modeling techniques in order to optimize computation costs. The overall approach is implemented on several numerical experiments with linear or nonlinear material behaviors.