# Gradient Correlation Functions in Digital Image Correlation
## 4 Discussion and Conclusions
An additional positive effect of this is for images that contain sharp edges. Such images are notoriously tricky to analyze using traditional intensity correlation, but with the combined correlations the gradient and Hessian correlations will act as filters, which actually is the motivation for the gradient correlation introduced by [PERSON] and [PERSON] [14]. Secondly, as the relative weight of the gradient and Hessian correlations decrease all three correlation functions behave approximately Gaussian and their combined effect will follow the same general trend. In addition, as they are uncorrelated their combined effect will also be Gaussian, and one gains very little to combine them. In conclusions therefore, combinations that include gradient and Hessian correlations contribute positively for sharp and dense patterns, but their positive effect decrease rapidly for larger features. The reliability presented in the lower right corner of Figure 3 shows a dramatic behavior on speckle size and correlation value. It is seen that the reliability is always unity for highly correlated patterns for all speckle sizes and for all correlation functions. As the correlation drops the reliability starts to drop, in particular for the larger speckle sizes. As a matter of fact, the reliability drops to as low as 0.6 for a speckle size of 7 pixels and a correlation value of 0.7. The significant drop-off in reliability for the larger speckle sizes shown in Figure 3 is to a large extent associated with the pure Hessian correlation and to some extent the combined gradient and Hessian correlation. All other correlation functions are unaffected. The susceptibility of these two correlation functions comes from the magnification of noise caused by numerical differentiation in combination with a small sample size characterized by the quotient \(N/S\), where in this case \(N=32\) and \(S\) is the speckle size. A larger correlation window would significantly improve the reliability. As a matter of fact, it is recommended to keep the quotient \(N/S\) above ten for reliable results in low-correlation images [12]. The reliability is therefore considered manageable for all relevant correlation functions considered. The speckle image shown in Figure 4 shows a typical feature often encountered in practice, that of an uneven illumination. Uneven illumination is often difficult to circumvent as the intensity profile of most illumination sources is uneven. A TEM\({}_{00}\) laser, for example, has a Gaussian beam profile. Unless compensated for, uneven illumination will bias the displacement estimate towards the brighter regions [12]. However not explicitly tested, one can speculate that the gradient and Hessian correlation functions would help to prevent this unwanted effect. In this investigation, however, no significant bias was noted. The pattern shown in Figure 4 is a laser speckle pattern. A laser speckle pattern has the unique quality of providing a random pattern defined by the full spatial bandwidth of the field with unit contrast. Hence, the feature size of the pattern can be controlled by choosing an aperture size optimum for the resolution of the detector. The great disadvantage with laser speckles is that they decorrelate in response to any change in the optical set-up [16], an effect that has prevented widespread use of laser speckles in experimental mechanics. This effect is even more pronounced for the small numerical apertures used with former digital detectors. Modern lines of digital detectors can however be purchased with as small pixels as 1 \(\upmu\)m, which considerably opens up applications with laser speckles in experimental mechanics because of their ease of use. Figure 5 shows the results from application of the seven different correlation functions using the lens \(f-\)numbers \(f/22\), \(f/16\), \(f/11\), and \(f/8\), respectively. A dramatic dependence on lens \(f-\)number is seen for all correlation functions, in fact significantly more dramatic than any effect caused by the correlation functions themselves.