4 L. Bang-Jian et al. Then the updated displacement uu can be derived by substituting formulas (1.3) and (1.4) into (1.1): uu D 2uCe.u/ CeŒ u 2e.u/ 2 : (1.5) According to the quasi odd function feature of the system error and formula (1.2), the updated displacement uu and formula (1.5) can be written as: uu DuC e.u/ eŒuC2e.u/ 2 : (1.6) From the above derivation, the system error of the updated displacement can be written as: e.uu/ D e.u/ eŒuC2e.u/ 2 : (1.7) When e(u) is small and generally this situation is tenable with suitable interpolation method, u C2e(u) will be close to u and then e(uu) will be smaller than e(u). That is to say, the system error will be reduced by the proposed strategy mathematically. 1.3 Experiments The open access speckle images (https://sem.org/dic-challenge/) will be used to verify the proposed strategy too. Measurement accuracy will be investigated in this part. The proposed strategy is tested with the open access speckle images (https://sem.org/dic-challenge/). The open access speckle image is selected as reference image. And one deformed image set is obtained by translating. The deformed image set contains 11 deformed images which are generated by Fourier shift theorem with displacement from 0.5 to 0.5 pixels and the displacement interval is 0.1 pixels. In the experiments, the area of interest (AOI) is 101 101 pixels. The subset size is 25 25 pixels. The subset based local DIC with first order shape function and bi-cubic interpolation is utilized. The system error which is calculated by the mean bias error is displayed in Fig. 1.3. As showed in Fig. 1.3, the system error is reduced for different bi-cubic interpolations by the proposed strategy. In addition, the result of 2 2 bi-cubic interpolation is worse than others, which can be explained that 2 2 bi-cubic interpolation is similar to bi-linear interpolation which is low order interpolation method [26]. Here it should be noted that the reduced system error is not equal to zero for many other factors induced system error such as shape function [27] and correlation criterion [28]. Fig. 1.3 System error reduction for different bi-cubic interpolation methods (update denotes the proposed strategy)
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