PE&RS July 2015 - page 569

20 pixels in the boundaries, because additional operations are
required to test whether the ground points are imaged within
the valid range). For each image, 1,000,000 random points
were first simulated and then backprojected to the object space
with the help of given terrain elevations, and these backpro-
jection results (i.e., ground point coordinates) were treated as
true values for testing the different scan-line search algorithms
used in the object-to-image projection. Finally, the image
points obtained from the object-to-image projection were again
backprojected to the object space. If the discrepancy between
the planimetric coordinates of a ground point and its true
value was smaller than a pre-defined threshold (1 percent of
the ground sampling distance used in the experiments), it was
considered that the ground point was correctly projected.
Six scan-line search schemes were evaluated in the com-
parative experiments.
1. Iterative search based on the ΔxF criterion (IF).
This algorithm was introduced by Zhao and Li (2006),
and their step size equation was given by:
x x
x
step
F
F ∆ ∆=
δ (14)
where δ is the CCD pixel size. In the experiment, the
iteration was terminated when |x x
x
step
F
F ∆ ∆=
δ | was less than
0.008 pixels (the converged case) or the number of
iterations exceeded 30 (the non-converged case).
It should be mentioned that we made a small modi-
fication to Zhao and Li’s algorithm by adding an affine
transformation at the beginning, which can provide a
rough initial value for the iterative search and, accord-
ingly, can reduce the number of iterations required, to
some extent.
2. Revised IF scheme ( RIF).
This is a simple modification of the IF scheme, using
Equation 5 to calculate the step size.
3. Iterative search and sequential search based on the ΔxF
criterion (IFSF).
This is the proposed algorithm. An affine transforma-
tion and a sub-pixel interpolation were used at the
beginning and the end of the search, respectively.
4. Iterative searches based on the D and ΔxF criteria (IDIF).
This algorithm was designed by Wang et al. (2009), and
we made a small modification. In the original version
of this algorithm, the step size and convergence condi-
tion used in the iterative search step based on the D cri-
terion were given by Equations 6 and 15, respectively.
D i · Di+1 < 0 (15)
where D i is the corresponding point-plane distance of
the current scan line i, and i +1 refers to the next scan
line. According to the test results, there is a very large
chance that it meets d > D i > D i+1 > 0 in the iterative pro-
cess, and in this scenario, the step size calculated from
Equation 6 will be 0, and Equation 15 will never be satis-
fied. Therefore, we added a new convergence condition:
xD
step = 0. (16)
As for the iterative search step based on the ΔxF
criterion, the convergence threshold and maximum
iteration number threshold used in this scheme were
the same as those used in the IF scheme.
5. Revised IDIF scheme (RIDIF).
This scheme is the same as the last one except that
Equation 5 was used to calculate the step size in the
iterative search step based on the ΔxF criterion.
Iterative search based on the D criterion and sequential
search based on the ΔxF criterion (IDSF).
This is another version of the proposed algorithm, and the
only difference with the IFSF scheme is that the D criterion
was used in the iterative search step.
Results and Analysis
Table 2 lists the number of ground points that were wrongly
projected to the image space (only the results of flight strips
A and B are shown). Flight strip A has a quite complex line
imaging geometry, which is illustrated by the many errors oc-
curring in some of the scan-line search schemes. As for flight
strip B, the line imaging geometry appears close to the ideal
condition because there are no errors in the results of all the
tested scan-line search schemes. By comparing the results of
the IF and IDIF schemes with their revised versions (RIF and
RIDIF schemes), we found that the convergence of the iterative
search based on the ΔxF criterion is highly related to the step
size equation used.
As schematically shown in Figure 7, when strong atmo-
spheric turbulence is encountered, scan planes may substan-
tially jerk backward and forward with respect to their ideal
positions (Gehrke and Uebbing, 2011; Wohlfeil, 2012), caus-
ing notable texture compression and stretching. In the case
of Figure 8, a consecutive number of scan planes quickly jerk
forward, which causes the step size calculated from Equation
14 to be twice the scan-line coordinate error (i.e., the discrep-
ancy between the current and the correct scan-line coordi-
nates) when the iteration number exceeds five. Thus, the step
sizes of two adjacent iterations have exactly the same absolute
values but different signs (Figure 8a), and the iteration oscil-
lates at both sides of the correct scan-line position (Figure 8b).
The experimental results in Table 2 show that the pro-
posed algorithm (IFSF and IDSF schemes) can always yield
correct results, which can be explained by the fact that the
algorithm employs more stringent convergence conditions
and a sequential search in the fine search stage. The first
Table 1. Two Flight S trips Used in the Experiments
Flight strip Image View angle (deg) Number of scan lines
A
Backward −14 49,272
Nadir 2 98,552
Forward 27 49,272
B
Backward −14 377,072
Nadir 2 377,072
Forward 27 377,072
Table 2. Number of Errors in 1,000,000 Test Points
Flight strip Image IF RIF IFSF IDIF RIDIF IDSF
A
Backward 11,957 174 0 4,557 67 0
Nadir 310 123 0 61 23 0
Forward 24,492 914 0 12,602 471 0
B
Backward 0 0 0 0 0 0
Nadir 0 0 0 0 0 0
Forward 0 0 0 0 0 0
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