
The standard DLT equations ([7] of DLT Method) actually consist of 10 independent unknown parameters (x_{o}, y_{o}, z_{o}, u_{o}, v_{o}, d_{u}, d_{v} and 3 Eulerian angles), not 11, since the principal distance (d) and the scale factors (l_{u} & l_{v}) are mutually dependent and reduces to 2 independent parameters, d_{u} & d_{v}. The main problem in the DLT method arises here. The DLT equations consist of 11 DLT parameters even though the system has only 10 independent unknown factors. In other words, one of the DLT parameters must be redundant and we need to add a nonlinear constraint to the system to solve this problem. In the standard 11parameter DLT, one computes 11 parameters independently using the least sqaure method. In this process the dependency among the 10 independent factors is impared resulting a nonorthogonal transformation from the objectspace reference frame to the imageplane reference frame. Hatze (1988) reported a method called Modified DLT (MDLT) in this line. In MDLT, one of the parameter is expressed in terms of the other 10 parameters. From [8], [19] & [20] of DLT Method:
[1] clearly shows the nonlinear relationship among the 11 DLT parameters. Now, the question is how to add this nonlinear constraint to the linear system of the DLT. An iterative approach may be used:
Now, let's take a look at Hatze's nonlinear MDLT model. From [6] & [11] of DLT Method:
or
where
Note in [2] that the righthand side no longer contains u_{o} & v_{o}. Rather, they are in the lefthand side in the form of x. In other words, the righthand side of [2] contains only 8 independent parameters (x_{o}, y_{o}, z_{o}, d_{u}, d_{v} and three Euler angles) while the lefthand side includes 7 (u_{o}, v_{o}, & L_{12} to L_{16}). 15 independent parameters are involved in the nonlinear MDLT model. From [4] and [20] of DLT Method:
Three Eulerian angles can be computed from the direction cosines. From [4] to [7] and [20] of DLT Method:
[8] shows the dependencies among parameters N_{1}  N_{11}. [8] can be used to discard N_{1}, N_{2} & N_{5} from the model to assure the orthogonality. To solve [3] for the 15 independent parameters (u_{o}, v_{o}, N_{3}  N_{4}, N_{6}  N_{11}, N_{3}  N_{4}, N_{6}  N_{11} and L_{12}  L_{16}), a nonlinear approach is required.
References & Related Literature Hatze, H. (1988). Highprecision threedimensional photogrammetric calibration and object space reconstruction using a modified DLTapproach. J. Biomechanics 21, 533538. 
© YoungHoo Kwon, 1998 