# Gradient Correlation Functions in Digital Image Correlation
## 2 Theory  
Consider two images \(I_{1}(\mathbf{x}_{1})\) and \(I_{2}(\mathbf{x}_{2})\) registered at the two time instances \(t_{1}\) and \(t_{2}\), respectively, containing approximately the same features. The images are assumed sampled on an \([M,N]\) sized grid with pixel pitch \([p_{y},p_{x}]\) in row and column directions, respectively. It is assumed that the motions of the features between the two images are small wherefore local information can be used to estimate local motion. Apart from the intensities it is assumed that the gradient vector \(\mathbf{G}_{i}(\mathbf{x}_{i})\) and the Hessian matrix \(H_{i}(\mathbf{x}_{i})\) may be generated in each pixel, where \(i=1,2\) for each of the images respectively. These additional fields are generated from application of the in-plane gradient column vector \(\
abla_{\perp}=(\partial/\partial x,\partial/\partial y)^{t}\) as \(\mathbf{G}_{i}(\mathbf{x}_{i})=\
abla_{\perp}I(\mathbf{x}_{i})\) and \(H_{i}(\mathbf{x}_{i})=\
abla_{\perp}I(\mathbf{x}_{i})\
abla_{\perp}^{t}\), respectively. Associated with each pixel are therefore an intensity value, an intensity gradient vector, and an intensity Hessian matrix, respectively. Figure 1 shows as a cropped example the intensity, the magnitude of the gradient vector and the determinant of the Hessian, respectively, of a laser speckle image. It is obvious that these three images contain different information and that they will perform differently in a correlation calculation. While the intensity image presents the distribution in intensity, the gradient image shows where the intensity changes most rapid. In regions with zero image gradient, the intensity correlation is essentially insensitive. The gradient image therefore dictates with what precision the intensity correlation function can be positioned. In addition, the Hessian matrix shows regions with a large intensity curvature. As the determinant can take on negative values, the background appears grayish, but regions close to speckle peaks light up. It is obvious that these peaks appear at the same positions as speckle maxima, but that they are locally more confined. Given these images, two continuous approximations of the intensity distributions can be formed using a nine-node quadratic Hermitian Finite Element. The intensity information in each node is taken from the intensity images and the gradient information required along the edges of the element is taken from the gradient and Hessian matrices, respectively. This approximation allows for a continuous description of image values, image gradients, and image curvatures, respectively, that is used throughout the remaining part of this paper. An intensity correlation is usually expressed as,  
\[CI(\Delta\mathbf{x})=\frac{\sum i_{1}(\mathbf{x}_{1}+\Delta\mathbf{x})i_{2}( \mathbf{x}_{2})}{\sqrt{\sum i_{1}^{2}(\mathbf{x}_{1}+\Delta\mathbf{x})\sum i_{2 }^{2}(\mathbf{x}_{2})}}, \tag{1}\]  
where \(i=I-<I>\) is the zero-mean intensity image, \(\mathbf{x}_{1}\) and \(\mathbf{x}_{2}\) are coordinates associated with the first and second image, respectively, and the summation is taken over all image points considered. The correlation variable, \(\Delta\mathbf{x}\), contains all translations and translation gradients considered in the correlation. The general procedure in image correlation is to pick out a subimage of size \(m\times n\) pixels from \(I_{1}\) and search in \(I_{2}\) for the set \(\Delta\mathbf{x}_{max}\) of \(\Delta\mathbf{x}\) that maximizes the correlation value \(CI(\Delta\mathbf{x})\) in Equation (1). The two translation components \([u,v]\) of \(\Delta\mathbf{x}_{max}\) are then taken as an estimate of the local displacement vector in the region of the chosen subimage. A displacement field is generated from repetition of the procedure for a multitude of different subimages.