Motion capture has a wide range of applications, including virtual reality, sports, and medicine. Each of these applications requires high accuracy so as to distinguish the pathological deficits from the normal, or to simulate an authentic virtual environment to train surgeons. One of the methods in motion capture is using automated three-dimensional (3D) reconstruction of moving skin markers to determine joint kinematics. A critical factor, however, affecting the accuracy of the kinematic data is the quality of the camera calibration. Errors may affect the accurate determination of joint centres, which, in turn, will affect the calculation of moments and powers at a joint.

The theory and process of deriving 3D positions of retro-reflective markers from several two-dimensional (2D) camera projections have been extensively studied (Brown, 1971; Tsai, 1987; Chen et al., 1994; Zhang, 2000). This was traditionally achieved by utilizing 2D reference points, whose 3D coordinates were then determined in a defined coordinate system. Calibration success, however, is dependent on factors such as calibration procedure, camera setup, and volume effects. A good calibration, therefore, is frequently achieved in a controlled environment such as in a gait laboratory.

In principle, accurate 3D measurements could be achieved in a conventional setting. Previous studies have reported good accuracy when commercially available camera systems were used in a predefined volume (Dorociak and Cuddeford, 1995; Ehara et al., 1997; Richards, 1999; Papic et al., 2004). Liu et al. (2007) further explored the accuracy of an optical system in the 0.5–200 μm range for really small tooth displacements. In these studies, deviations from known distances or angles between fixed markers were determined. This, however, does not reflect what happens in reality. One only has to consider the complicated movements of a multi-segmented human body to know that distances between markers are not constantly fixed while in motion.

The difficulty of calibrating multiple cameras simultaneously is increased if it was performed outdoors or in wide open spaces, where moving markers are too small for a clear and concise reconstruction and calculation of the calibration parameters (Barreto and Daniilidis, 2004). The process of being able to capture movements accurately thus continues to evolve.

In this study, we have developed a procedure for quantifying calibration accuracy of a multi-camera system based on collision detection by using markers that move relatively to each other during the calibration procedure. Saini et al. (2009) had previously utilized this procedure. In their study, the geometry of a mandibular (lower) tooth obtained from computed tomography (CT) and motion capture data of natural chewing movements were used to automatically reconstruct the tooth’s maxillary (upper) counterpart. We have, likewise, used motion capture to track successive collisions of two solid moving objects.

We simulated the recorded movement in Python programing language (https://www.python.org/), and represented the virtual objects in a voxel-based manner. One of the virtual objects served as reference, and the other as test object. In order to account for the fact that solid objects cannot penetrate each other, we progressively eroded the virtual test object by removing voxels every time it collided with the reference object. The volumetric difference between the initial and the final virtual object reflected the degree to which the simulation deviated from reality. This process, therefore, provided a measure of calibration accuracy.

Camera calibration is the process of reconstructing the transformation from points in a world coordinate system to their corresponding points in an image plane. The transformation can be represented by a 3 × 4 matrix T_W2I, which is composed of intrinsic and extrinsic camera parameters. Intrinsic camera parameters (focal length, image center, aspect ratio, and distortion of the lens) characterize the camera’s projection properties. Extrinsic camera parameters specify the orientation and position of the camera in the world coordinate system.

The transformation process is described by the following equation in homogeneous coordinates P_I = (x, y, 1) for image points and P_W = (X, Y, Z, 1) for real-world points: PI=TW2I∗PW       =TC2I∗TW2C∗PW

T_W2I is the product of the 3 × 3 intrinsic T_C2I and the 3 × 4 extrinsic T_W2C calibration matrixes.

In the field of motion analysis, the process of finding the intrinsic parameters (linearization) and the extrinsic parameters (calibration) is performed separately. One popular technique for linearization is proposed by Zhang (2000), which takes into account projection errors by having a camera observe a planar pattern in at least two different orientations. The intrinsic parameters are then optimized for all patterns simultaneously. One technique in determining the extrinsic parameters is the wand calibration method, where a wand comprising two markers at a known fixed distance is waved in the capture volume. The 2D image coordinates attained are then used to calculate the 3D coordinates using bundle adjustment (Triggs et al., 2000).

Since the cameras in our study had already been linearized, we assumed that the intrinsic calibration parameters were known, and focused only on the camera extrinsic matrix (T_W2C). The extrinsic parameter matrix T_W2C = (R|t), which consisted of a 3 × 3 rotation matrix R and a 3 × 1 translation vector t, described the camera position relative to the world coordinate system. Here, we evaluated the accuracy of T_W2C as estimated by the motion capture system during wand calibration.

Based on the three experiments, a mean volumetric difference of 19.12 ± 5.36 cm³ was found (Table 2). This amounts to approximately 1.9% of the volume of C_T. While this percentage might seem small, even tiny changes to camera calibration could affect the 3D reconstruction of markers in space (Datta et al., 2009).

Figure 6 shows a plot of VolDiff_N^(t) (cm³) over time (s) at varying voxel resolutions as observed in Experiment 1. At a voxel resolution of N = 8, the total volumetric difference decreased to 0 cm³. As mentioned, the intersection of Voxel centres between C_R and C_T was required for the removal of a voxel. For very coarse voxel resolutions (as in the case when N = 8), voxels were only removed in the case of a big discrepancy between simulation and reality. After subsequent increases in voxel resolution, we observed that the total volumetric difference converged to 22.13 cm³.

At the beginning of the experiment (0–2 s), the sharp drop was a result of the removal of Type II voxels (Figures 3B and 7) when both cubes were in contact with each other. It would seem that no gaps were found between the cubes in all the experiments, and Type II voxels were not as problematic as had been anticipated. As all the outer voxels had been removed, subsequent increases in the combined volumetric error were due to Type I voxels (Figure 8) being removed at instances when both cubes were rubbing against each other.

For this technique to work, therefore, an appropriate voxel resolution had to be chosen carefully so as to determine the volumetric difference between the initial and final-shaped C_T. In our study, a voxel resolution of N > 64 demonstrated a convergence to the total volumetric difference.

In motion capture systems, the determination of the intrinsic and extrinsic parameters is performed separately. In our study, we evaluated the accuracy of the extrinsic parameter matrix during wand calibration, assuming that the intrinsic parameters were already known. Our proposed collision detection technique could also be utilized when intrinsic parameters are unknown or to be determined simultaneously with the extrinsic parameters. Larger errors, however, will be expected. This is because while it is simpler to use the wand calibration method to simultaneously determine both intrinsic and extrinsic parameters, all the parameters will ultimately become sensitive to the movement of the calibration wand¹. This may, therefore, result in larger calibration errors.

It is clear that there may be other sources of errors affecting the extrinsic calibration parameters, such as camera placement. In our study, average residuals of 0.46–0.75 mm were recorded during calibration, which were within the accepted range of 0.5–1.5 mm as recommended by Qualisys. It is, therefore, apparent that while intrinsic parameters such as lens distortion cannot be completely eradicated, random errors could still be determined and minimized. In a controlled environment such as in the laboratory, high quality calibration could easily be achieved. In measurements performed outdoors and/or where large capture volumes are required, it becomes difficult to consider all the 2D image coordinates simultaneously so as to attain the 3D coordinates required to determine the extrinsic camera parameters (Barreto and Daniilidis, 2004). We are not suggesting that this technique should replace the commonly used calibration procedures (recommended by Qualisys and/or other motion capture systems), but our approach can instead complement these procedures to minimize calibration errors.

This technique could be an additional criterion to the wand calibration method, since this collision detection takes into account the errors that resulted from markers moving close to one another. The wand calibration method includes waving a wand with two markers at a fixed distance in the capture volume. Previous studies performing accuracy tests of several motion capture systems also utilized markers at fixed distances to each other (Dorociak and Cuddeford, 1995; Ehara et al., 1997; Richards, 1999; Papic et al., 2004). Compared to our technique, the use of fixed markers in the previous studies was straightforward and easier to implement. Distances between moving markers, however, are not constantly fixed during capture. The human body, consisting of multi-linkages, is fully capable of performing complicated 3D movements. Our technique, therefore, better replicates actual movements during motion capture.

In conclusion, we have proposed a procedure based on collision detection, which could be used as an indicator of calibration accuracy. This technique can complement current calibration methods to minimize calibration errors when simultaneous calibration of multiple cameras is required.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.