Page 202 - Digital Analysis of Remotely Sensed Imagery

Image Geometric Rectification 167

5.5.2 Minimum Number of GCPs
The minimum number of GCPs required to carry out image georefer-
encing is governed by the order of transformation, which is related to
the number of transformation coefficients to be resolved in Eq. (5.16).
In a first-order transformation, there are six coefficients to be deter-
mined. Since each GCP enables two transformation equations (one
for E and another for N) to be established, the minimum number of
GCPs required is equal to three. Accordingly, twelve coefficients are
needed in a second-order transformation. They are solvable with a
minimum of six GCPs. The relationship between the minimum num-
ber of GCPs required N and the order of polynomial equations t is
min
generally expressed as
N = (t + 1) × (t + 2)/2 (5.17)
min
The minimum number of GCPs needed for performing an image
transformation up to the fifth order has been calculated according to
the above relationship (Table 5.2). It increases almost exponentially
with the transformation order.
The transformation coefficients are uniquely determined if only
the minimum number of GCPs is used. In reality, however, the actual
number of GCPs selected usually exceeds the bare minimum shown in
Table 5.2, even in areas that do not offer many quality GCP candidates.
The extra GCPs can be treated as check points through which any errors
or inaccuracy in identifying GCPs in the image, in reading their coordi-
nates from topographic maps, and in entering the coordinates into the
computer are revealed. It is quite common to misidentify GCPs on a
topographic map or on an image. Pinpointing the points on the remote
sensing imagery is subject to inaccuracy. The position can be missed by
a few pixels easily on a coarse resolution image if the GCPs are not suf-
ficiently distinct. The extra GCPs provide the flexibility and luxury of
deleting those that have been poorly identified or inaccurately

Order of Minimum No. of
Transformation GCPs Required
1 3
2 6
3 10
4 15
5 21
6 28

TABLE 5.2 Order of Transformation and the
Minimum Number of GCPs Required