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Linear MDLT Non-linear MDLT References and Related Literature

Linear MDLT

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 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 [8], [19] & [20] of DLT Method:

   [1]

[1] 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:

1. Compute the 11 DLT parameters using the conventional DLT in the first iteration.

2. From the second iteration, remove one parameter (for example, L₁) by using the value obtained from the previous iteration and reduce the system to 10 parameters (L₂ - L₁₁). Solve the system for the 10 parameters. Compute the parameter removed earlier (L₁) based on the 10 estimated parameters using [1].

3. Repeat Step 2 until a stable (converged) set of solution is obtained.

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Non-linear MDLT

Now, let's take a look at Hatze's non-linear MDLT model. From [6] & [11] of DLT Method:

   [2]

or

   [3]

where

   [4]

Note in [2] that the right-hand side no longer contains u_o & v_o. Rather, they are in the left-hand side in the form of x. In other words, the right-hand side of [2] contains only 8 independent parameters (x_o, y_o, z_o, d_u, d_v and three Euler angles) while the left-hand side includes 7 (u_o, v_o, & L₁₂ to L₁₆). 15 independent parameters are involved in the non-linear MDLT model.

From [4] and [20] of DLT Method:

   [5]

   [6]

   [7]

Three Eulerian angles can be computed from the direction cosines. From [4] to [7] and [20] of DLT Method:

    [8]

[8] shows the dependencies among parameters N₁ - N₁₁. [8] can be used to discard N₁, N₂ & N₅ from the model to assure the orthogonality.

To solve [3] for the 15 independent parameters (u_o, v_o, N₃ - N₄, N₆ - N₁₁, N₃ - N₄, N₆ - N₁₁ and L₁₂ - L₁₆), a non-linear approach is required.

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References & Related Literature

Hatze, H. (1988). High-precision three-dimensional photogrammetric calibration and object space reconstruction using a modified DLT-approach. J. Biomechanics 21, 533-538.

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© Young-Hoo Kwon, 1998-