Page 203 - Digital Analysis of Remotely Sensed Imagery

168 Cha pte r F i v e

pinpointed. If more GCPs than are necessary are retained in the trans-
formation equations, the coefficients of the multiple equations are then
solved using the principle of least-squares adjustment. According to
this principle, the coefficients are so estimated that the sum of the
squared residuals at each of the GCP is minimal. Therefore, residuals
among all the retained GCPs are dependent on one another.
No matter how many GCPs are selected, they must always have a
balanced spatial distribution to achieve quality rectification. All selected
GCPs must be dispersed widely throughout the entire image. If the
area of interest makes up only a portion of the whole scene, then the
GCPs should be distributed well beyond its border. In this way all por-
tions of the rectified image have the same geometric reliability. Any
areas that are not adequately represented by ground control are virtu-
ally extrapolated from the covered area. The final output image gener-
ated from extrapolation is much less reliable geometrically than that
produced from interpolation. Any areas that lack GCPs have a lower
geometric reliability than that indicated by the overall accuracy.

5.5.3 Accuracy of Image Transform
Application of Eqs. (5.13) and (5.14) results in one set of coordinates
called estimated (E, N) for all retained GCPs. They also have another set
ˆ ˆ
of observed coordinates ( ,EN ) that are obtained either from a topo-
graphic map or using a GPS unit. Owing to the aforementioned reasons
(e.g., coordinate inaccuracy, inaccuracy in identifying the GCPs and in
reading their coordinates, and the use of more GCPs than is necessary),
these two sets of coordinates are unlikely to be identical. Their differ-
ence, termed rectification residual, is rarely equal to zero. Instead, it varies
from GCP to GCP. At a given point, the residual in easting may differ
from that in northing. Statistical analysis of residuals at all GCPs yields
an accuracy indicator called root-mean-square-error (RMSE). It is
derived using the following equations in both easting and northing:

n
n
RMSE = 1 ∑ δ 2 = 1 ∑ ( E − E ) 2
ˆ
E n Ei n i i (5.18)
i=1 i=1
and
1 n 1 n
ˆ
RMSE = ∑ δ 2 = ∑ ( N − N ) 2 (5.19)
N n Ni n i i
i=1 i=1
where n = the total number of GCPs finally retained in a
rectification
E and N = easting and northing coordinates of the ith GCP
i i
calculated from the established functions f and f
2
1
ˆ ˆ
, EN = reference coordinates in easting and northing that

are obtained from topographic maps or using a GPS