# Comparative Analysis of Discrete Subtraction and Cross-Correlation for Subpixel Object Tracking
## 1 Introduction  
Accurate tracking of objects in an image scene is of utmost importance in physics and engineering. Cross-correlation is the most frequently used method for establishing the similarity of two images and tracking changes with time, including relative locations. Although other methods are proposed in the bibliography, e.g., centroid calculation [1] or object recognition and characterization [2], cross-correlation presents clear advantages over these: on the one side, it is very robust with signals corrupted with Gaussian noise, which is the most typical case in images [3]. On the other side, correlation does not require a priori knowledge or characterization of a target since it is based on direct comparison, thus widening its applicability. Additionally, it can be expressed in terms of the Fast Fourier Transform, thus allowing efficient computation algorithms, which may be very convenient for large image sequences. Cross-correlation has been successfully applied in many fields like Geotechnics [4], Structural Engineering [5], Material Sciences [6], or Particle Image Velocimetry [7]. However, many of these applications require refined correlation results since they are based on tracking displacements or deformations smaller than the maximum system resolution, thus requiring the application of subpixel methods. In the literature, we can find several strategies for achieving subpixel accuracy with cross-correlation. The simplest approach consists of interpolating the object and the targetto obtain intermediate matching positions [8]. The method is easy and straight, although it is computationally inefficient and not accurate for very small displacements. A better alternative consists of interpolating neither the scene nor the target, but the cross-correlation peak instead. Under this premise, a small neighborhood around the correlation peak is fitted to a known continuous and derivable function, and the peak position is recalculated with the fitting function [9]. This approach is commonly used with pseudo-speckle patterns applied to solid materials to measure local deformations [10], although the technique can also be successfully used with natural textures [11]. The method is computationally efficient, and, with proper election of the fitting area and the interpolant function, the resolution of the cross-correlation can be increased by more than two orders of magnitude. Independent of the implemented method, the use of interpolation in the cross-correlation method introduces a systematic error in the target position determination called \"peak-locking error\". This error consists of a bias in the peak location with a deviation towards the closest integer [12]. This effect has been largely analyzed, and different proposals have been made in order to minimize its effect, although none of the proposals have been totally effective in canceling the effect [9]. Recently, a new method for tracking objects with subpixel accuracy was proposed [13]. The accuracy of this method does not rely on spatial interpolation but on tracking the luminance changes that a moving object provokes in the sensor. A similar effect was used in [14] in order to establish a limit to the subpixel accuracy that is achievable with a sequence of images. In [15; 16], the authors propose using these changes to obtain the vibration frequency of an object in a scene, but none of these proposals were able to quantify the amount and direction of the shifting produced. In brief, the method introduced by [PERSON] et al. [13] proposes the calculation of the difference between an image and its displaced version, and this difference is used to derive two parameters that are associated with horizontal and vertical shifting. Assuming a small movement, these parameters are directly proportional to the actual shifting that has occurred. The discrete subtraction method (DSM) has been proposed for the estimation of subpixel motion with its advantages of accuracy, simplicity, and low computational cost compared with correlation.