The standard DLT equations ( of DLT Method) actually consist of 10 independent unknown parameters (xo, yo, zo, uo, vo, du, dv and 3 Eulerian angles), not 11, since the principal distance (d) and the scale factors (lu & lv) are mutually dependent and reduces to 2 independent parameters, du & dv.
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 non-linear constraint to the system to solve this problem. In the standard 11-parameter 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 non-orthogonal transformation from the object-space reference frame to the image-plane 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 ,  &  of DLT Method:
 clearly shows the non-linear relationship among the 11 DLT parameters. Now, the question is how to add this non-linear constraint to the linear system of the DLT. An iterative approach may be used:
Now, let's take a look at Hatze's non-linear MDLT model. From  &  of DLT Method:
Note in  that the right-hand side no longer contains uo & vo. Rather, they are in the left-hand side in the form of x. In other words, the right-hand side of  contains only 8 independent parameters (xo, yo, zo, du, dv and three Euler angles) while the left-hand side includes 7 (uo, vo, & L12 to L16). 15 independent parameters are involved in the non-linear MDLT model.
From  and  of DLT Method:
Three Eulerian angles can be computed from the direction cosines. From  to  and  of DLT Method:
 shows the dependencies among parameters N1 - N11.  can be used to discard N1, N2 & N5 from the model to assure the orthogonality.
To solve  for the 15 independent parameters (uo, vo, N3 - N4, N6 - N11, N3 - N4, N6 - N11 and L12 - L16), a non-linear approach is required.
Hatze, H. (1988). High-precision three-dimensional photogrammetric calibration and object space reconstruction using a modified DLT-approach. J. Biomechanics 21, 533-538.
© Young-Hoo Kwon, 1998-