Up until now, we investigated two methods to address space resection (DLT, and the Empirical Method). Here, we state the last method that is useful when we have good estimation of initial values. Firstly, we consider the following basic form
By rewriting the equations similar to DLT, we get to the following two simple equations
Now we rewrite them to convert the problem into an optimization problem of root finding type
As we cab see, for each control model point, we have two equations with 6 unknowns, therefore, at-least 3 control points are required to solve this problem. We need to take into consideration, that initial estimations should be of high quality for least-square to work.
To find the initial values of the unknowns in aerial photogrammetry, 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, we first calculate the parameters of a two-dimensional conformal transformation between the image coordinates and the ground coordinates. Then we set the X_0 and Y_0 values of the image equal to the conformal shift and the rotation element κ (rotation around the Z axis) equal to the rotation obtained from the conformal transformation. We set the value of Z_0 to the approximate flight height. Now that we have the unknowns with sufficient accuracy, we can use the least squares in the following way. First, we consider the vector of unknowns as follows
Now we can form the Jacobian matrix A as
Now we take one step of Newton optimization
Note 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 soft vector of dx corrections becomes less than a small threshold value.
One way to validate the results is to use image-residues at control points. Another way is to use the estimation of uncertainty about the unknown parameters based on the observational equations.
Ground Sampling Distance or GSD
Ground sampling distance is the distance between two adjacent pixel centers when they are imaged on the average surface of the ground. In aerial photogrammetry in vertical mode, we can calculate the approximate value of the ground sampling distance along the line with the photo ratio based on the following equation.
We must pay attention that when the image is oblique, the ground sampling distance will not be the same for the entire image.