## 1 Introduction  
Developed in early 1980s, digital image correlation (DIC) system is an optical experimental method based on digital image processing and numerical computing [1, 2, 3] to measure full field displacement and strain on almost any solid material surface. The results of DIC are readily comparable with finite element (FE) results [4] or strain gauges [5]. Since the beginning of the DIC system development, its working principles and errors estimation has always been an interesting area of research. [PERSON] et al. [1] and [PERSON] et al. [3] described the DIC system working principles and [PERSON] [6], [PERSON] et al. [7], [PERSON] et al. [8] identified possible error sources which can cause displacement and strain measurement errors. While the full field displacement can be directly calculated based on comparing subsets in images taken from deformed and undeformed object surface, the full field strain evaluation requires additional post-processing of the discrete displacement data points, namely numerical differentiation, which is based on interpolation functions of the discrete displacement field applied over subset areas of the entirely full field area, which subsets are defined according to an user-specified size and step periodicity. Numerical differentiation in mathematical theory is considered as an \"unstable and risky operation\" [9]. First of all, differentiation-based methods face with deficiencies for problems involving discrete/discontinuous fields. Further, finite difference approaches are sensitive to noise. [PERSON] et al. [10] demonstrated that if the measurement error of displacements is \(\pm\)0.02 px and the DIC grid step is 5 pixels, then the error of strain calculation by forward difference is 8000 \(\mu\varepsilon\). An error of this magnitude will hide the underlying strain information and the strain result will be worthless in most cases. The problem of strain measurement for a plate with circular hole under tension from the 2D DIC challenge 1.0 Sample 12 [11] was analyzed by several authors [12; 13; 14] (Figure 1). The exact strain solution for this problem is not given and can be only predicted by comparing with FE simulation or other numerical/experimental methods. It is visible from Figure 1 that increasing the DIC step size leads to a smoother strain profile but decreases accuracy due to the averaging effect of a larger DIC virtual strain gauge [12; 13; 14]. The strain results lead to the necessity to find the trade-off between accuracy and smoothness. Large strain gradients cannot be captured if too much smoothing is introduced, leading to poor accuracy. Conversely, resolving steep gradients leads to a highly oscillatory solution [15; 16]. In order to overcome this issue, several improved differentiation techniques in DIC were offered. [PERSON] et al. [17] used modified Hermite finite element approach to ensure the smoothness and continuity of the strain field. A new nonlocal strain calculation method for DIC systems, based on a Kernel function built up from tensor products of functions that integrate to zero over domain, was proposed by [PERSON] et al. [12]. The method is effective for problems with step gradients and when the displacement data is noisy. This specially applies to the fracture experiments [18; 19] due to traditional DIC differentiation algorithms difficulties obtaining crack paths. In addition to this, other approaches were proposed to solve the differentiation issue in discontinuous DIC field. [PERSON] et al. [20] developed a two-step X-DIC to minimize the errors in computing discontinuous full-field displacements. [PERSON] et al. [21], [PERSON] and [PERSON] [22] used mesh adaptations for measuring discontinuous fields in specimens with cracks. Similar technique based on both computed displacement and strain correlation was introduced by [PERSON] et al. [23]. Methods relying on a subset split strategies measuring the \"displacement jumps\" due to discontinuity and displacements and strains around the discontinuity were also developed [24; 25].