## 2 Materials and Methods
### Three-Dimensional Speckle Photography in Out-of-Plane Direction  
For the initially targeted objective of an extended understanding of the manufacturing process, the acquisition of information about the 3D deformation field is essential. Speckle photographic approaches are in principle also suitable for this purpose, since the recorded speckle patterns basically represent sections through the three-dimensional structure of the  
Figure 1: Evaluation principle of speckle photography for the measurement of the two-dimensional in-plane deformation. wave field. Figure 2a shows an example of the measured three-dimensional structure of the speckle pattern, for a \(\zeta\)-shift of the measurement plane. The cross-correlation for pure in-plane displacements \(\Delta\xi\), \(\Delta\psi\) is shown in Figure 2b. Deformations and displacements of the surface in the out-of-plane direction (\(\Delta\zeta\)), however, lead to a decorrelation of the speckle patterns, which means a reduced amplitude of the cross-correlation maximum (compare Figure 2b,c). This decorrelation was previously considered as a disturbance of the measurement effect and was minimized by using higher measurement rates. The approach of Fricke-Begemann uses the decorrelation and derives the out-of-plane deformation of the measurement surface via an elaborate shape analysis of the power spectrum of the cross-correlated speckle patterns [17]. A maximum likelihood estimation is first used to determine the tilt angles of the surface points and based on this, comprehensive three-dimensional deformation measurements are finally reconstructed. However, laboratory investigations of simultaneous tilts and in-plane displacements of the measurement surface illustrate the problem that arises in separating tilt and in-plane displacement in this form of evaluation, see Figure 3.  
Figure 3: Power spectra of two speckle patterns for tilted or additional displaced measurement surfaces. Figure 2: Three-dimensional speckle pattern measured by shifting the camera and illumination system in the \(\zeta\)-axis (**a**) and the calculated cross-correlation function of two evaluation windows shifted against each other in the \(y\)-direction (**b**) or in the \(\zeta\)-direction (**c**). Especially in the range of very small in-plane displacements (<1 pixel), as they are predominantly present in the in-process load measurements of manufacturing processes, the differences are hardly recognizable. While in-plane displacements in the cross-correlation (see Figure 1) are clearly recognizable by the displacement of the maximum without significantly influencing the shape of the curve, in the evaluation via the power spectrum both the in-plane displacement and the out-of-plane movement affect the shape of the curve. Even though the power spectrum is very sensitive to the tilts, complex fit and filter algorithms are required to separate the two components. First approaches for the analysis of a two-axis motion, consisting of a tilt in the \(\xi\)-axis and a displacement in \(\psi\)-direction, resulted in computation times in the double-digit minute range for the smallest evaluation windows. With regard to the parallelization of the evaluation algorithms and the goal of a fast in-process measurement, an alternative approach is therefore pursued. The cross-correlation allows for separating the in-plane and out-of-plane displacement information by evaluating the shift of the correlation maximum (see Figure 2b) and the broadening of the normalized cross-correlation function (see Figure 2c), respectively. The shape or the width of the correlation function is already evaluated in the context of the performed sub-pixel interpolation. Inherently hidden in the time-optimized interpolation is the width \(\sigma\) of the approximated Gaussian function \(\sigma(\hat{x},\hat{y})\). It can be considered as a value for the speckle decorrelation and thus for the out-of-plane deformation.