(11 parameter approach)
As we saw in the previous section, there are several different ways to address space resection. Perhaps the most practical method is to use the equations of the collinearity using the non-linear least squares method. In this way, knowing the initial values of the unknowns, we use Newton’s approach to estimate the unknowns. Therefore, we have to calculate partial derivatives.
In DLT method, unlike the above approach, there is no need to know the initial values. In addition, some elements of interior orientation could also be calculated. To understand the DLT method, let’s first consider these equations in a simple form.
- Simple form in pinhole coordinate system (unknowns elements=exterior orientation parameters)
With rewriting the equations we find the following simple equations
One solution is to employ 3D-2D projective transformation. It could be said that the aim of DLT is to rewrite the above equations in this form, so we try to do so
Now, divide both nominator and denominator to the last element to get the standard form
Therefore, the coefficients of the standard form will be
and the standard 3D-2D projective transformation is
As we saw in previous chapters, to solve the above equations we rewrite them as
Now that we have observational equations, we form A matrix as
We should notice that in a case that the calibration body doesn’t have a volume, these equations become singular. To solve it, we use the standard least-square method.
Now we can easily extract the main unknowns from the secondary unknowns. First, we calculate the rotation matrix with a degree of uncertainty, then we can also calculate the unknown scale of the rotation matrix, and finally, the position of the camera is calculated directly from these equations or indirectly by using the equations of the collinearity.
An interesting fact about these equations is that spatial intersection is possible using secondary variables even without extracting the main variables. In photogrammetric calculations, this operation is also called simultaneous resection and intersection. Now that we understand the simpler method, let’s examine the complete form of the DLT equations, taking into account all possible unknowns.
- Complete form in pixel coordinate system (Unknowns=Exterior orientation parameters, focal length, principal point coordinates)
Here, we want to use DLT in a form where image points are in a undistorted or distorted image coordinate system in pixel unit, so we have
Therefore, we can rewrite the equations as
If we move the principal point coordinates to the right, we have
Similar to previous case, we divide all elements to the last element of denominator
We also combine all the unknown coefficients
The result of last two steps is
Therefore, in complete form, we have the following coefficients
Now we repeat the whole process for variable y
Move P.P to the right
The denominator coefficient that we had previously defined
Combining all the unknowns
So, the coefficients are in the form
Finally, the complete list of coefficients from the above equations are
3- Extracting camera parameters from above coefficients
Fist we consider the coefficients that have some correlation to the optical center
By rewriting this equation, we see that other coefficients appear in it
In a similar way
From Gamma equation we had
Therefore, three equations are forms for the optical center as
The Gamma coefficient could be easily calculated as
Now we try to find a useful formula to solve principal point
We already know that columns and rows of an orientation matrix are orthogonal, so
In a similar way
Now we calculate the focal length
and finally, we calculate the orientation matrix’s elements
Other elements are calculated in a similar way. In the end, we find the following structure for the rotation matrix
