submitted for
GA
. Different objective functions as described
in Table 1 are used as fitness functions with
GA
parameters
shown in Figure 5a and the value of
m
is varied from 10
percent to 70 percent in steps of 10. The obtained results are
tabulated as shown in Figure 5. The result expected is that the
GA
would not select or include the points from
Q
as
GCP
s and
the
RMSE
at good checkpoints in each case would be similar
to the value when
m = 0
. Figure 5b indicates
RMSE
at all the
good checkpoints, i.e., for points in
R
. Figure 5c indicates the
percentage of area covered under the selected
GCP
s for various
experiments. The x-axis indicates
m
,
i.e., size of
Q
(percent-
age of points chosen for adding errors) for both the plots.
Figure 5d shows the corresponding
GA
objective function’s
convergences for one of the cases, i.e., when
m = 40%,
the
plot shows generations on x-axis and the best chromosome’s
objective value on y-axis. It is also to be noted that
RMSE
and
CE90
are minimization objective (decreasing plot) while others
are maximization (increasing plot).
Following are the conclusions of the exercises:
• When there were no errors in the data, all the objective
functions yielded similar results. However, when errors
were introduced, as expected, each of the objective
functions showed small changes in the results, but
overall the results were converging to a similar value to
that of the “no errors(
m
= 0)” case. This implies the
GA
had successfully eliminated the erroneous checkpoints
and chosen good checkpoints as
GCP
s.
• Considering “Area” alone as the objective function, the
solution produced higher errors than others and hence,
was found not suitable as the objective function espe-
cially when the datasets are known to have errors.
• A/E objective function resulted in consistent behavior
but resulted in relatively higher errors than some of the
other functions. However, the errors were less than a
pixel resolution of reference image.
•
RMSE
,
CE90
, and A/E50 objective functions showed sim-
ilar behavior and showed consistent results even when
the percentage of errors increased up to 50 percent
(
m
= 50). The error was approximately 10 m, which is
within single pixel accuracy. The results also matched
the solution of the ‘no errors case. ‘
• The solution obtained by using ‘A/E50’ objective
resulted in similar accuracies as that of
CE90
, and
RMSE
,
but covered a larger area under the selected
GCP
s.
• If the errors cross more than 50 percent, the results are
inconsistent and inconclusive.
• It is also obvious that A/E50 objective function conver-
gence is faster than other objective functions.
Figure 5. Objective function derivation and comparison with other techniques.
T
able
1. O
bjective
F
unctions
for
C
ase
S
tudy
Case study Objective
Goal
1
Minimize RMSE (in meters)
RMSE error at all the points should be minimum
2
Minimize CE90 (in meters)
Minimum Error containing 90% of population
3
Maximize Area covered under GCPs
Well distributed points only
4
Maximize A/E
Considering both maximum area and minimal error
5
Maximize A/E50
Considering both maximum area and minimal error
382
May 2016
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
