9.5 Coordinate Transformation

The technique of coordinate transformation is useful for geometric correction with ground control points (GCP). The key points are contained in the following two selections.

a. Selection of transform formula
Depending on the geometric distortions, the order of polynomials will be determined. Usually a maximum of a third order polynomials will be sufficient for existing remote sensing images, such as LANDSAT. Table 9.5.1 shows the examples of available formulas.
b. Selection of ground control points
The number and distribution of ground control points will influence the accuracy of the geometric correction. The number of GCP's should be more than the number of unknown parameters as shown in Table 1, because the errors will be adjusted by the least square method.

The distribution of GCP's should be random, but almost equally spaced including corner areas. About ten to twenty points which are clearly identified both on the image and the map should be selected depending on the order of the selected formula or the number of unknown parameters. Figure 9.5.1 shows the comparison of accuracy with respect to number and distribution of GCP's. The accuracy of geometric correction is usually represented by the standard deviation (RMS), in pixel units, in the image plane as follows.

u : standard deviation in pixel number
v: standard deviation in line number
where

u={ui-f(xi,yi)}/n
v={vi-g(xi,yi)}/n

(ui, vi) : image coordinates of the i th ground control point
(xi , yi) : map coordinates of the i th ground control point
f ( xi , yi) : coordinate transformation from map coordinates to pixel number
g ( xi , yi) : coordinate transformation from map coordinate to line number

The accuracy should be usually within +- one pixel. If the error is larger than the requirement, the coordinates on the image or map should be rechecked, otherwise the formula should be reselected.


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