## 1 Introduction
This becomes necessary in the case of moire, as measured displacement components are taken in the deformed configuration of the object while displacements computed by finite element models are referred to the undeformed configuration of the specimen. As is clear, the error functional \(\Omega\) is not correctly defined and the inverse problem is ill-posed if reference systems of measured and computed displacements do not match. It is hence of paramount importance to use an optical setup that can measure the 3D displacement vector by recording just one image of the deformed object. This avoids the need for matching reference systems relative to the different displacement components, an inherently complicated process especially when one has to relate in-plane displacements with out-of-plane displacements. Another critical issue in mechanical characterization of materials is the quantity of data matched in the optimization problem, which may depend on the number of control points as well as on the number of displacement components involved in the definition of the error functional \(\Omega\). It would be good to compare all displacement components \(\{u(x,y,z),v(x,y,z),w(x,y,z)\}\), but optical setups may be sensitive only to some of them. Using an optical setup able to perform a one-shot measurement of the 3D displacement vector certainly improves the efficiency of the identification process because (i) the number of experimental tests entailed by the characterization is reduced and the whole identification process becomes faster, (ii) the displacement vector is determined with respect to just one scale, and (iii) numerical noise and nonconvexity of the optimization process caused by any difference in scale between displacement components are significantly reduced. Projection moire can provide information on the 3D displacement components of a deformed body. In fact, displacement components are the difference of coordinates between corresponding points of an object's surface measured with respect to a reference plane before and after the object itself deforms. Strains are determined by differentiating displacements, and stresses are determined from a constitutive model. Measuring the deformed shape of an object is equivalent to contouring the surface of the deformed object. The contouring problem is a very complicated topic, and this explains the variety of approaches proposed in the literature. [PERSON] et al. [35; 36] pointed out that cross-gratings must be projected onto the specimen to determine the local curvature of the object'ssurface and consequently to precisely reconstruct the 3D shape of the object. Orthogonal lines give information on the first and second derivatives of the function defining the analyzed surface, which is treated as a tensor entity similar to a strain tensor. In summary, the mechanical characterization of soft hyperelastic membranes brings at least four major challenges: (i) dealing with the inherently nonlinear behavior of the material, which may be amplified by structural inhomogeneity; (ii) measuring the whole 3D displacement field with just one optical setup and performing a single experiment; (iii) making sure that the inverse problem is well posed; and (iv) solving the inverse problem (1) with an efficient, robust and computationally cheap optimization algorithm. The above challenges fully motivate the study documented in this article, which presents a novel hybrid framework for mechanical characterization of soft membranes combining biaxial tests, moire measurements and metaheuristic optimization. A one-shot projection moire setup including two projectors that project cross-gratings onto the tested specimen and one camera is utilized. The two projectors are symmetrically disposed with respect to the optical axis of the camera, thus realizing double illumination with projection from infinity.