## 2 Materials and Methods
### Relationship between the Width of the Correlation Function and the Out-of-Plane Shift  
In order to analyze the functional relationship \(\Delta\zeta(\sigma_{x}\sigma_{y})\), the laboratory setup of a speckle-photographic measurement system was realized, see Figure 4a.  
Figure 4: Speckle-photographic laboratory experiment: (**a**) laboratory setup with indicated bending of the workpiece sheet over a locating bowel pin (yellow) fixed in the camera system; (**b**) field of the respective local widths of the correlation functions, at a linear axis displacement of 28 mm.  
The measurement object, a sandblasted sheet metal strip (400 mm \(\times\) 50 mm) is illuminated with a collimated widened Omicron laser beam (\(\lambda\) = 405 nm). The illuminated area is observed and imaged with an Imaging Source camera (DMK 21 AU04) through a Pentax 25 mm lens. Camera, laser and the center of the sheet metal strip (in the center of the measurement field) are fixed on a linear axis from PI company, Germany. The outer ends of the sheet metal strip, on the other hand, are fixed to the laboratory bench. Thus, a movement of the linear axis causes a bending of the sheet metal strip (yellow line in Figure 4a) and thus a relative movement between the center and its outer ends.  
Figure 4b shows the global field of the respective locally calculated correlation function widths  
\[\sigma=\sqrt{\sigma_{x}^{2}+\sigma_{y}^{2}}=\sqrt{\frac{1}{-2c_{20}}+\frac{1}{ -2c_{02}}} \tag{15}\]  
for a linear axis displacement of 28 mm in \(\zeta\)-direction with a total measurement field size of 30 mm \(\times\) 40 mm. The parabolic bending around the dowel pin can be seen very well in the center of the image. The width of the correlation function increases significantly from the original 1.3 pixels to about 4 pixels. If looking in each case at the mean value of the width of the cross-correlation function along the \(y\)-axis at the position \(x\) = 0 in Figure 4b, one can see the linear relationship between the respective adjusted displacement of the linear axis and the broadening of the interpolated correlation function \(\overline{\sigma}\) shown in Figure 5. The functional relationship between a changed loading condition with a resulting displacement \(\Delta\zeta\) and the resulting speckle decorrelation or change in the width \(\Delta\sigma\) of the Gaussian fit \(\mathcal{g}\) from Equation (1) is thus linear in nature, i.e.,  
\[\Delta\zeta=\pm B\ \Delta\sigma. \tag{16}\]  
The indefinite sign for the calibration factor \(B\) results intuitively from Figure 2. The shift by \(+\Delta\zeta\) leads to the decorrelation of the speckle pattern to the same extent as a shift by \(-\Delta\zeta\). While the calibration factor can be defined prior to a respective measurement on the basis of a defined displacement, the sign can only be determined via prior knowledge of the observed deformation process. In the case of the bending test, for example, the outer ends of the sheet metal strip will move away from the camera in negative \(\zeta\)-direction, so that negative signs are to be expected. While the in-plane deformation (\(\Delta\zeta\),\(\Delta\psi\)) is linearly related to the displacement (\(\Delta x\),\(\Delta\psi\)) of the maximum of the cross-correlation via the scaling factor \(A\) (Equations (11) and (12)), it is shown by Equation (16) that a linear relationship  
Figure 5: Linear relationship between the width of the correlation function \(\overline{\sigma}\) calculated in the sub-pixel interpolation and the shift of the linear axis in \(\zeta\)-direction.  
also results for the out-of-plane deformation \(\Delta\zeta\) via the factor \(B\) and the change in width of the cross-correlation function \(\Delta\sigma\).