# Gradient Correlation Functions in Digital Image Correlation
## 2 Theory
The random error, \(e\), with which \([u,v]\) can be determined has been shown to vary as [12],  
\[e=k\frac{S^{2}}{M}\sqrt{\frac{1-\gamma}{\gamma}}, \tag{2}\]  
where \(k\) is an algorithm dependent constant of order unity, \(S\) is the average feature size, \(M=[m,n]\) is the correlation window width and \(\gamma\) is the intensity correlation value. The quotient \(M/S\) defines in principle the number of independent contributions to the correlation and relates also to the reliability of the calculation while the additional \(S\) defines the curvature of the correlation peak and hence its susceptibility to random noise. The amount of random noise is specified by the correlation value \(\gamma\). As with Equation (1), a gradient correlation may be expressed as,  
\[CG(\Delta\mathbf{x})=\frac{\sum\mathbf{G}_{1}(\mathbf{x}_{1}+\Delta\mathbf{x} )\cdot\mathbf{G}_{2}(\mathbf{x}_{2})}{\sqrt{\sum|\mathbf{G}_{1}(\mathbf{x}_{1 }+\Delta\mathbf{x})|^{2}\sum|\mathbf{G}_{2}(\mathbf{x}_{2})|^{2}}}, \tag{3}\]  
Figure 1: Images generated from the acquired set of intensity values. The upper left image shows the intensity distribution, the upper right image shows the magnitude of the calculated gradient vector, and the lower left image shows the determinant of the calculated Hessian matrix. where \(\mathbf{G}=\partial I/\partial x\hat{\mathbf{x}}+\partial I/\partial y\hat{\mathbf{y}}\) is the local image gradient vector, and \(\hat{\mathbf{x}}\) and \(\hat{\mathbf{y}}\) are orthogonal coordinate axes, respectively. Finally, a Hessian correlation function may be formulated as,  
\[CH(\Delta\mathbf{x})=\frac{\sum\left[\mathbf{v}_{11}(\mathbf{x}_{1}+\Delta \mathbf{x})\cdot\mathbf{v}_{21}(\mathbf{x}_{2})+\mathbf{v}_{12}(\mathbf{x}_{ 1}+\Delta\mathbf{x})\cdot\mathbf{v}_{22}(\mathbf{x}_{2})\right]}{\sqrt{\sum \left[|\mathbf{v}_{11}|^{2}+|\mathbf{v}_{12}|^{2}\right]\sum\left[|\mathbf{v }_{21}|^{2}+|\mathbf{v}_{22}|^{2}\right]}}, \tag{4}\]  
where in the denominator special reference to \(\mathbf{x}_{1}+\Delta\mathbf{x}\) and \(\mathbf{x}_{2}\) are omitted for ease of reading. In Equation (4), \(\mathbf{v}_{11}\) and \(\mathbf{v}_{12}\) are the two orthogonal principal vectors associated with the local intensity Hessian of image \(I_{1}\). The corresponding vectors for \(I_{2}\) are expressed as \(\mathbf{v}_{21}\) and \(\mathbf{v}_{22}\), respectively. The three correlation functions in Equations (1), (3) and (4) are all normalized between \([-1,1]\); however, their correlation features differ significantly. For example, as both the gradient correlation and the Hessian correlation involves vectors they are expected to drop off more quickly to a change in feature structure. Because of the normalization, the three correlation functions in Equations (1), (3) and (4) can be combined to produce the additional four correlation functions:  
\[CIG(\Delta\mathbf{x}) =CI(\Delta\mathbf{x})CG(\Delta\mathbf{x}), \tag{5}\] \[CIH(\Delta\mathbf{x}) =CI(\Delta\mathbf{x})CH(\Delta\mathbf{x}),\] (6) \[CGH(\Delta\mathbf{x}) =CG(\Delta\mathbf{x})CH(\Delta\mathbf{x}),\] (7) \[CIGH(\Delta\mathbf{x}) =CI(\Delta\mathbf{x})CG(\Delta\mathbf{x})CH(\Delta\mathbf{x}). \tag{8}\]  
Hence, in total seven different correlation functions can be used to estimate the local deformation between the two set of images. Under what conditions either of them is preferable is investigated in the coming sections. Figure 2 shows a comparison of the correlation properties between the seven correlation functions. The left image shows the width of the auto-correlation functions produced by setting \(I_{2}=I_{1}\) in Equations (1) and (3)-(8), respectively, as a function of displacement \(\Delta\mathbf{x}=u\). It is seen that all additional correlation functions are narrower