# Gradient Correlation Functions in Digital Image Correlation
## 4 Discussion and Conclusions
For example, comparing the final deformation step for \(f/22\) with the \(f/8\) the random error is 0.73 with an intensity correlation value of 0.72 as compared with 0.02-0.04 and 0.90 for the larger aperture. This dramatic effect is caused by the double effect of enlarging the speckle size and enlarging the speckle decorrelation. In comparing the performance of the different algorithms for the same aperture settings the same trends as shown in Figure 3 are found. The random errors for the smaller speckle sizes and larger correlation values are about a factor of two smaller for the combined correlation functions as compared with the pure intensity correlation. The function combining all three pure correlations performs the best and the pure Hessian correlation the worst. The effect decreases for the larger speckle sizes and for lower correlation values. Finally, it was noted that the reliability turned out to be close to unity for all analyzed images with all correlation functions in contrast to the reliability of the simulated patterns. The reason for this discrepancy is however unknown. In conclusion, the performance of seven different correlation functions applied in DIC have been investigated using simulated and experimentally acquired laser speckle patterns. The correlation functions were constructed as combinations of the pure intensity correlation function, the gradient correlation function and the Hessian correlation function, respectively. It was found that the correlation function that was constructed as the product of all three pure correlation functions performed best for the small speckle sizes and large correlation values, respectively, but that the difference between the different functions disappeared as the speckle size increase and the correlation value drops. On average the random error of the combined correlation function was half that of the traditional intensity correlation function within the optimum region. It was also found that for the small speckle sizes, all combined correlation functions involving the Hessian correlation function appear to be more robust against speckle decorrelation down to correlation values of roughly 0.85. The reason for this has not been investigated in detail but a good guess is that the more well-defined peak provided by the Hessian correlation makes its position more defined and less susceptible to noise. This effect disappears for smaller correlation values and for larger speckles. In addition, the monumental dependence of the imaging \(f-\)number on the accuracy of DIC using laser speckles is demonstrated experimentally. This dependence appears for all seven correlation functions and practically dominates the performance of the calculations. While the difference between the most optimum correlation function and the worst is in the order of three, the difference between results using different image apertures may be five times as large. Modern digital detectors with pixel sizes in the order of \(\upmu\)m may therefore open up for a renaissance of laser speckles in experimental mechanics, in particular in situations where a good random pattern is difficult to apply. The authors declare no conflict of interest.