The absolute orientation is to connect two 3D coordinate systems, e.g., a 3D model to a ground coordinate system. The 3D model is usually created from the relative orientation. For absolute orientation, we use a three-dimensional conformal transformation as
In the above equation, we have 7 unknown parameters
1- λ scale
2- three orientations
3- Three transitions
We can use two methods to solve the above equation. If the initial values of the unknowns are not known, we use singular value decomposition approach, and if the initial values are known, we use nonlinear least squares.
1-Solution without knowing the initial values
In this approach, we first estimate the scale by comparing the longest distance between two datasets as
After finding scale, both gravity centers are calculates
Then, two data points are moved to a new coordinate system with Cg as the center and we apply the scale that we calculated on both sets
Now that both datasets are aligned on a common center point and has the same scale, we can say that the following relationship applies between them
In order to estimate the above rotation matrix, one way is to use the SVD approach as
Finally, the rotation matrix is calculated as
The shift is calculated by the following equation
2- When initial values are known
In this case, we use non-linear least square method to solve it. We first form a non-linear system of observational equations as
The above problem is a convex optimization problem of root finding type. In the above equations, we have three equations for each common point, so at least three points are required to solve these equations. To find the initial values of the unknowns in aerial mode, we consider the rotations ω (rotation around the X axis) and ϕ (rotation around the Y axis) to be zero. To find the other unknowns, first, the parameters of a 2D to 2D conformal transformation between the image coordinates of the points and the 2D ground coordinates. Then, we set the X_0 and Y_0 values of the image again equal to the conformal shift and the rotation element κ (rotation around the Z axis) equal to the rotation obtained from the conformal transformation. The scale is also obtained from the parameters of the conformal transformation. We set the value of Z_0 again to the approximate height of the flight. Now that we have the unknowns with sufficient accuracy, we can use the least squares in the following way
Now we can form Jacobian matrix A as
We have now all the ingredients to take a Newton step as
Note above that our optimization problem is of root finding type, and as a result F(x) is equal to zero. The optimization operation continues until the length of the correction vector dx becomes less than a small threshold.
A good way to validate the absolute orientation results is to use residuals in control points. Another way is to use the estimation of uncertainty about the unknown parameters based on the observation equations.