# Gradient Correlation Functions in Digital Image Correlation
## 4 Discussion and Conclusions  
The structure highlighted in the three images in Figure 1 shows the different features that contributes to the different types of correlation functions defined by Equations (1), (3) and (4), respectively. These features are also responsible for the different shapes and drop-offs of the correlation functions shown in Figure 2. It is seen that out of the fundamental correlation functions, both the gradient correlation and to a greater extent the Hessian correlation produces a narrower and more well-defined peak and drops off more rapidly in response to feature degradations. With reference to the general behavior of correlation functions, these two effects should have contradictory effects on the accuracy of the deformation calculation [12]. The sharper peak should make the peak position more well-defined. The decrease in correlation value will on the other hand increase noise. The combined correlation functions show up a similar behavior with the most well-defined correlation peak produced by the multiplication of all three fundamental correlation functions. This is also the correlation function that drops off most rapid. One can also notice that this combined correlation function is basically free from higher order ringing, which indicates that the three fundamental correlation functions basically are uncorrelated. Figure 4: Sketch of the experimental set-up. The set-up consists of a 10 mW He-Ne laser, a monochrome camera from Dalsa equipped with a mikro-Nikkor camera objective with focal length \(f=55\) mm, and a white aluminum plate that can be translated in-plane using a fine-pitch micrometer. The right part of the figure displays one of the acquired speckle images. This image is acquired with an aperture setting of \(f/32\). Figure 5: Random error in displacement as a function of speckle motion using different aperture numbers and analyzed using the seven different correlation functions. All results for the same aperture setting are lumped together. A few interesting observations emerge from the results shown in Figure 3. For the high correlation values, the random errors always show up in the same order with the mixed correlation values the lowest. The difference between the best performing correlation function, which is the combination of all three fundamental functions, and the pure intensity correlation function is roughly a factor of two. The fundamental Hessian correlation performs constantly the worst, which is not surprising as the intensity Hessian is most susceptible to image noise. This disadvantage seems to be cancelled by multiplication with the other, more noise tolerant, correlation functions enhancing its advantage of being sharp. In fact, all mixed correlations involving the Hessian correlation performs well. One can also notice that the intensity correlation follows the general behavior \(\sqrt{(1-\gamma)/\gamma}\), where \(\gamma\) is the intensity correlation value for all speckle sizes, while the other correlation functions do not. In fact, for the smallest speckle sizes the mixed correlation functions involving the Hessian correlation seem to be more robust to decorrelation and show up an opposite curvature as compared with the pure intensity correlation. This effect seems to be valid up to a correlation of roughly 0.85 where these functions turns up and approach the other functions. At this point the combined correlation function is approximately three times more accurate as compared to the pure intensity correlation. This effect is not as pronounced for larger speckle sizes and for speckle sizes in the range 5-7 pixels all correlation functions perform approximately the same. By this one can notice two things. Firstly, for small speckle sizes both the gradient image and the Hessian image becomes more pronounced and will dominate in areas where they generate large numbers. Hence, their positive feature of being sharp is pronounced. As the features grow larger their relative weight decrease.