SECTION 18.6
Two-Dimensional Projective Coordinate Transformation
Introduction
The two dimensional projective coordinate transformation projects points in one coordinate system onto another using eight parameters. It is described in Section 18.6 of the book, Adjustment Computations: Spatial Data Analysis. The transformation requires a minimum of 3 points known in xy coordinates in both coordinate systems. This worksheet demonstrates this problem. The data file is read and parsed into input tables, matrices computed, and the least squares solution determined.
Library file
Parsing the Data File
Read data file
Parse control data
Parse observed data
Enter initial approximations for unknowns
where a1 = U0, b1 = U1, c1 = U2,
a2 = U3, b2 = U4, c2 = U5,
a3 = U6 b3= U7
Assuming that a3 and b3 are close to 0, the affine transformation can be used to approximate the other parameters.
Create routine to build W
Create routine to build J and K
Since solution is nonlinear solve
system of equations and iterate.
Repeat until 10 iterations or
convergence of unknown parameters.
Compute J and K matrices
Form normal equations
Compute solution
Add corrections to initial approximations
Compute residuals
Check for convergence
By size of correction
Compute reference variance
Check for small change in
reference variance.
Assign results to standard parameters
Build results
Adjustment Results
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Transform points and propagate errors
Note that point 8 is an extrapolation of the control
since it lies outside their region. Thus it has very large standard deviations.