# Bisector-Based Tracking of In Plane Subpixel Translations and Rotations
## 1 Introduction  
In the last years, computer vision systems have been demonstrated to be an efficient tool to accurately track movements and vibrations, mainly in those applications where contact measurement is inadequate. The majority of applications rely on displacement tracking, which is also a key parameter in structures in order to evaluate their security under loading. Local displacements can be described through cross-correlation, which allows an investigator to obtain the relative position between two identical images [1]. This correlation can be applied on specific targets as well as on distinct object parts. The advantages of cross-correlation are widely known. Unfortunately, it also has some disadvantages. Among them, because of its linear formulation, it has a strong dependence on the scene luminance. Also, it has a limited spatial accuracy, and, additionally, it only stands for displacement tracking but not for object rotations. The first issue has been addressed through normalized cross-correlation [2], and the second through smart interpolation of the scene or the correlation peak itself [3]. Solving the third problem is not so easy, and it is necessary to usually make use of a mathematical transformation of both the scene and the target prior to the cross-correlation operation [4; 5; 6]. Measuring rotations and torsions is of great importance for completely describing the movements and forces acting on a structure [7]. Traditionally, this movement is described indirectly through localdisplacements by using contact methods. The measured shifts may be then decomposed in order to separate real displacement from torsions and rotations, thus introducing calculation noise and a level of uncertainty into the movement determination. Additionally, contact measurements are adequate in controlled environments and small parts, but the cost-benefit ratio in real structures is really high, thus impeding its use in this case. In our opinion, it would be of great interest to design a noninvasive method capable of tracking object displacements and rotations simultaneously, even in big elements located far away from the measuring point. In the published literature, one can find different methods used to image rotation measurement, such as phase-encoded grayscale image correlation [8], which is used to measure rotation at a constant interval of 3\({}^{\circ}\) using holographic storage; double phase-encoded correlation [9], which provides results with absolute errors of 0.22 degrees and around 0.12 px in the determination of the rotation and the displacement, respectively; log polar correlation [5; 6], which is able to detect translation, rotation, shear, and scale changes with an angle resolution of 0.7 degrees for a 512 \(\times\) 512 log-polar representation [10]; Hilbert wavelet transformation [11], which allows an investigator to obtain out-of-plane rotations at steps of 15 degrees; and holographic optical correlator methods [12], which are able to measure the rotation of the image as accurately as 0.05 degrees. However, in spite of the variety of approaches to the problem, we find that some of them [8; 9] can be sensitive to certain kinds of noise that make them not suitable for some practical applications, and others [6; 11] may require computationally intensive calculations or are of limited application because of low computing accuracy [12]. On the other hand, the position and orientation of an object can also be obtained by calculating some object features, like the centroid and second-order moments of the image's intensity distribution [13]. However, these techniques can impose constraints on the target morphology or on scene illumination in order to facilitate the detection of these features. In certain cases, these constraints cannot be met due to the nature of the system or the limitations of the experiment, and, additionally, if the selected features are too restrictive, they might be inadequate for tracking complex motions.