# Gradient Correlation Functions in Digital Image Correlation
## 1 Introduction  
Digital image correlation (DIC), or Particle Image Velocimetry (PIV), has since its introduction in the 1980s evolved into one of the most versatile and widespread techniques in experimental mechanics [1; 2; 3; 4; 5; 6]. Recent examples are found in diverse scientific fields such as biomechanics [7], infrastructure [8], material science [9], composite structures [10], and microfluidistics [11], but the technique is not restricted to these scientific fields. A quick search on a popular search engine lists over 50,000 contributions out of which 16,000 are published throughout the last four years. The great versatility of the technique comes from its simplicity and flexibility, scalability in both space and time, the fact that it is non-intrusive and that it produces deformation fields. Furthermore, the deformation fields can be generated as Lagrangian fields (typically used in DIC) or Eulerian fields (common in PIV), depending on which images that are compared. In general, the technique requires a unique feature to be present in the plane considered. In the absence of natural features, a pattern needs to be added, usually using spray (DIC) or by adding small particles to a flow (PIV). The general approach is then to follow the features in between successive frames using a model of the deformation field, which most often is performed locally involving a limited number of pixels, but global approaches have been demonstrated [6]. At the core of this calculation is a numerical optimization routine whose underlying function most often is defined as an intensity cross-covariance or as a sum of squared intensity differences. The performance of the two approaches differs only in details. It has been shown that the random error in the deformation calculation roughly scales with average feature size, subimage width and correlation value [12]. The quotient between subimage width and feature size defines in principle the number of independent contributions to the correlation and relates also to the reliability of the calculation. In addition, the feature size 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. For a laser speckle pattern, the decay of the correlation value is almost completely dominated by speckle decorrelation, while for a white-light pattern algorithm dependent features suchas precision in interpolation becomes important. This is the reason feature sizes slightly larger than the sampling limit often are preferred with these patterns. Calculated image gradients are used frequently in DIC algorithms as a means of interpolation and as an aid to improve the performance of the optimization routine. [PERSON] et al. has recently published an extensive review of current algorithms using image gradients for DIC calculations [13]. The conclusions were that image gradients speed up search algorithms considerably and the most effective gradient vector is formulated from a weighted blend of image gradients from both images. However, these image gradients are used to speed up the search for the most probable set of correlation parameters. The underlying function is still an intensity correlation. On the other hand, feature detection based on image gradients are frequently used in computer-vision applications. For example [PERSON] and [PERSON] has published a paper on the use of a normalized gradient correlation function for detection of changes in a video stream [14]. The motivation for the use of gradient correlation in their application was mainly that the detection becomes more robust against varying background intensity. Such correlation functions have to my knowledge not yet been explored in connection with DIC. The purpose of this paper is to investigate the performance of correlation functions based on derivatives of intensity images as a function of feature size and image degradation.