## 2 Theoretical Background
### Dic
Although the displacements of the DIC grid points (the centers of the grid subsets \(P(x_{0},y_{0})\) in the Figure 2c) are measured directly, strains require additional mathematical operations, namely numerical differentiation, which can be performed in several different ways in a DIC algorithm: \"subset shape function\" differentiation, displacement field division into triangle finite elements and then their numerical differentiation, derivatives of local analytical equations defined over the selected size of local regions and approximating the displacement field, displacement field spline fit differentiation [38]. The principle of differentiation algorithm of the DIC post processing software VIC-3D from Correlated Solutions [41] used in this study is shown in Figure 3a. A grid of triangles (similar to FE triangular mesh) having the subsets center points of known displacement data at their vertexes is used. The strain distribution over each triangle is calculated from the deformed shape of each triangle FE element based on the strain shape functions defined over each triangular finite element. Because the edge length of each triangle is about several pixels, strain results without additional smoothing techniques would be noisy. Because of this, DIC strain filters which smoothen the strain results over the strain filter area (Figure 3b) are applied. The filter size is a user defined parameter which can be expressed by the number of the DIC step sizes (e.g., filter 5-5 steps sizes, filter 35-35 step sizes, etc.). The DIC user should select between the high noise when using a small size strain filter vs low accuracy when using large strain filter (Figure 3b). The subset size and the distance between two consecutive subset centers denoted as the step size are also user-defined parameters related to images post-processing, see Figure 2c. Although the proper subset size selection guidelines is given in the literature and DIC manuals [6; 37], to make the subset size selection easier for the user, the DIC software  
Figure 3: Strain calculation in the DIC software VIC-3D: (**a**) triangular mesh for strain calculation; (**b**) strain. calculates default subset values according to the speckle size. The general recommendation for the step size selection is about 1/4 of the value of subset size [42]. The step size is related to the spatial resolution (although the subset size and shape function have much higher effect on it) of the DIC measurements because with a smaller step size more DIC data points are taken and the resolution of the results increased, but the computation time and noise increase as well [42]. The proper DIC parameters selection for each step of the method (namely specimen preparation, images recording and images post processing) has an impact on the DIC results accuracy, especially strains. The main DIC parameters affecting the results accuracy are summarized in Table 1. This study deals with the DIC step size (ST) and strain filter size (FS) which together with the subset size (SS) can be combined to a single parameter, called virtual strain gauge (VSG) and expressed as [38]:  
\[\text{VSG}=(\text{FS}-1)\cdot\text{ST}+\text{SS}. \tag{1}\]  
VSG influence on the DIC strain results accuracy and the possibility to improve the strain results accuracy by using the PDDO to differentiate DIC displacement field are analyzed in this study.