An interesting variation of the DLT algorithm is the double-plane method originally used by Drenk et al. (1999). This method is based on the 2-D DLT algorithm and involves two parallel control planes (Figure 1).
Point M shown in Figure 1 is the marker. One can observe this marker from a camera so that the line of sight (perspective line) passes through the two control planes. Points P1 and P2 shown in the figure are the projections of M to the control planes. By placing a set of control points on both control planes, one calibrate the camera and compute the 2-dimensional coordinates of points P1 and P2: [y1, z1] and [y2, z2]. See DLT Method for the details of the 2-D DLT method. The x-coordinates of the two planes are already known: x1 and x2.
From the collinearity among the three points, one can obtain
Since one camera provides 2 equations, as shown in , at least 2 cameras are required to compute the x-, y- and z-coordinate of the marker (Figure 2). Expanding  for n cameras:
Use the least-square approach to solve . See Least-Square Method for the details.
Figure 3 shows an underwater motion analysis setting with two parallel control point grids. Due to the light refraction at the water-air interface plane (precisely speaking, water-glass-air interface), both P1 and P2, projections of the marker to the control planes, map to point I on the image plane through refraction point R. As a result, one obtains a deformed image. See Kwon (1999a & b) for the properties and behavior of the refraction error in underwater motion analysis.
As an effort to reduce the error due to refraction, Drenk et al. (1999) identified the closest 4 control points surrounding the projected marker in each plane and used only these 4 points in calibration and subsequent reconstruction to compute the 2-D coordinates of the projected marker. Kwon & Lindley (2000) generalized this method, under the umbrella of the localized-calibration approach, by allowing grouping of the control points. One can define a set of control point groups and the calibration/reconstruction program detects the closest control point group to which the projected marker belongs in each control plane. The control areas defined by the groups can be either overlapped or distinct (non-overlapped). This approach provides flexibility and more accurate calibration & reconstruction results than the conventional 3-D DLT method (Kwon & Lindley, 2000). Starting from version 3.0, Kwon3D supports the localized-calibration/reconstruction approach and the double-plane method for underwater motion analysis.
Drenk, V., Hildebrand, F., Kindler, M., & Kliche, D. (1999). A 3D video technique for analysis of swimming in a flume. In R.H. Sanders, & B.J. Gibson (eds.), Scientific Proceedings of the XVII International Symposium on Biomechanics in Sports (pp. 361-364). Perth, Australia: Edith-Cowan University.
Kwon, Y.-H. (1999a). A camera calibration algorithm for the underwater motion analysis. In R.H. Sanders, & B.J. Gibson (Eds.), Scientific Proceedings of the XVII International Symposium on Biomechanics in Sports (pp. 257-260). Perth, Australia: Edith-Cowan University.
Kwon, Y.-H. (1999b). Object plane deformation due to refraction in two-dimensional underwater motion analysis. Journal of applied Biomechanics, 15, 396-403.
Kwon, Y.-H., & Lindley, S.L. (2000). Applicability of the localized-calibration methods in underwater motion analysis. Submitted to the XVIII International Symposium on Biomechanics in Sports.
© Young-Hoo Kwon, 1998-