Fernerkundung I (Digitale Bildverarbeitung) - Friedrich-Schiller ...

• There is less chance of geometric distortion in areas of no GCPs. The number of
required GCPs depends on the order of the polynomial. Table 1 lists the minimum
required number of GCPs.
In practice, more than the required number of GCPs should be collected, so that any
GCPs which contain significant positional errors on either the georeferenced map or the
uncorrected image will have their errors reduced by averaging.
The result of a first order transformation depends on the number of GCPs:
• One GCP will produce a translation for only X and Y.
• Two GCPs will produce a translation and a scaling change for X and Y if the pixel
geometry is not linear in the X or Y dimension. If it is linear, (that is, the two have
the same X or Y coordinate, producing a scaling factor of zero), only a translation will
be produced. If a scale change is produced, this setup may be used to produce a ip in
the X and/or Y dimension.
• Three or more GCPs produce a translation, scale change, and/ or rotation for X and Y
(a full first order transformation).
Required GCPs
Order
7 2nd
11 3rd
16 4th
22 5th
Table: Minimum Number of Ground Control Points
Required GCPs Order
It is important to note that while a higher order polynomial will result in a more accurate
fit in the immediate vicinity of the GCPs, it may introduce new significant errors in those
parts of the image away from the GCPs. Therefore, worse errors may be introduced into
the imagery than were to be corrected.
Polynomial Equations:
The following equations show the polynomials used for orders one, two, and three.
Orders four and five are extrapolations from these with four and five order terms added.
X’ and y’ are the coordinates in the uncorrected image generated from the georeferenced
matrix system (x, y) coordinates. The current polynomial equation being used can
1st order: x' = a(0) + a(1)x + a(2)y
2nd order: x' = b(0) + b(1)x + b(2)y + b(3)xy + b(4)x^2 + b(5)y^2
3rd order: x' = c(0) + c(1)x + c(2)y + c(3)xy + c(4)x^2 + c(5)y^2 + c(6)x^2y +
c(7)xy^2 + c(8)x^3 + c(9)y^3
1st order: y' = d(0) + d(1)x + d(2)y
2nd order: y' = g(0) + g(1)x + g(2)y + g(3)xy + g(4)x^2 + g(5)y^2
3rd order: y' = h(0) + h(1)x + h(2)y + h(3)xy + h(4)x^2 + h(5)y^2 + h(6)x^2y +
h(7)xy^2 + h(8)x^3 + h(9)y^3
Table 2: Polynomial Equations
Order Equation