Adjustment Computations Adjustment Computations: Spatial Data Analysis

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.

 

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