## 2 The One-Shot Projection Moire: Model and Experimental Setup
### One-Shot Projection Moire Measurements and IMPM Variant  
According to [31, 35, 36], the 3D shape of a surface (and hence the 3D deformed configuration of a loaded body) can be fully reconstructed using a projection moire setup with four projectors and one camera whose optical axis coincides with the axis of symmetry of each pair of projectors in the corresponding epipolar planes. Two projectors limit the horizontal direction (i.e., the X-axis of the global reference system) and project vertical lines while the other two projectors limit the vertical direction (i.e., the Y-axis of the global reference system) and project horizontal lines. Each system of lines forms vertical or horizontal fringes that are modulated by the deformed surface, thus providing the X-Z and Y-Z surface profiles, respectively. However, only two projectors are needed if cross-gratings are projected onto the surface to be contoured. In view of this, a two-projectors, one-camera arrangement where cross-gratings are projected onto the specimen was adopted in this study. Figure 4a shows the one-shot projection moire optical setup used here. The moire setup is schematically represented in Figure 4b. The gratings were projected by standard LTPR3W/R led projectors (\(\lambda=630\) nm), symmetrically placed with respect to the direction of observation; each projector projected a glass slide containing a 254 um pitch cross-grating (i.e., p\({}_{0}\) = 100 lines/in), and the projection cone had an aperture of 18\({}^{\circ}\). The two symmetrical projectors realize the condition of projection from infinity. The distance of the CCD camera from the analyzed object and the camera's objective are selected so as also to have viewing from infinity. The distance between the baseline limited by the exit pupils of the two projectors and the reference plane was d = 675 mm, while baseline length was t = 930 mm; distance between camera and reference plane was r = 475 mm; illumination angle made by each projection direction and viewing direction was 0 = 34.6\({}^{\circ}\) (see nomenclature in Figure 4b). The selected angle of illumination allowed fringe quality to be optimized. Symmetry of projectors with respect to the optical axis of the camera was checked with the following method: We first place two small mirrors in correspondence with the centers of the membrane and camera inlet lens. The light beams coming from the two projectors are reflected from the mirror placed on the membrane towards the mirror placed on the camera inlet lens which in turn reflects back two spots towards the membrane mirror. When the four spots coincide, the projectors-camera system is lined up and the one-shot projection moire optical setup is ready for use. Magnification of optical setup used for membrane inflation experiments was 11.974, representing the ratio between the pitch of projected lines by each single projector and the corresponding nominal projected pitch p\({}_{0}\)/cos\(\theta\). The sensitivity of the moire setup to out-of-plane displacements \(\Delta\)s\({}_{\text{w}}\) was 2.678 mm, while sensitivity to in-plane displacements \(\Delta\)s\({}_{\text{u-v}}\) was 1.888 mm. The natural rubber membrane was inflated by a pneumatic loading system, which progressively increased pressure from 1.07 to 2.33 kPa with the intermediate pressure loads of 1.33, 1.60 and 1.87 kPa. The specimen was clamped onto the pressurization chamber by an O-ring seal. The images relative to reference state (i.e., undeformed) and deformed configurations relative to each pressure level were recorded by a standard CCD camera (Nikon Coolpix 990) with a 2048 \(\times\) 1536 pixel matrix. The camera objective was equipped with a Nikkor 8-24 mm optical zoom (3\(\times\)) or a 4\(\times\) digital zoom. The calibrated pixel size was 78.1 um. A ruler was placed on the reference plane coinciding with the undeformed membrane, and numbers of pixels between pairs of adjacent dots werecounted. This operation was repeated over the whole field of view.