T
able
I. C
omputer
E
xecuting
T
ime
. T
he
number
of
iterations
indicated
in
the
first
column
reflect
the
quality
of
approximations
used
in
the
various
experi
-
ments with
the
collinearity
solution
.
Number of Iterations in
Collinearity Approach
Number of
Points Involved
Executing Time (secs)
Direct
Collinearity
2
43
6.16
12.56
4
6.16
22.14
5
6.16
27.21
No Convergence
6.16 No Solution
2
12
3.77
5.80
4
3.77
9.67
6
3.77
13.56
No Convergence
3.77 No Solution
T
able
II. S
tandard
E
rror
of
U
nit
W
eight
σ
0
. I
mage
coordinates
perturbed
by
a
random
error
of
normal
distribution with
the
standard
errors
indicated
in
the
column
“I
nput
”
and mean
zero
.
Number of
Points
σ
0
(µm)
Input
Output
Direct Method Conventional
43
3.000
2.781
2.995
5.000
4.635
4.992
10.000
9.271
9.985
20.000
18.545
19.969
12
3.000
2.435
2.692
10.000
8.117
8.972
T
able
III. A
ccuracy
of
O
bject
-S
pace
C
oordinates
O
btained
by
the
D
irect
and
the
C
ollinearity
A
pproaches
for
different
numbers
of
control
points
.
I
mage
coordinates
perturbed
by
a
random
error
of
normal
distribution with
standard
error
: 3.00 µ
m
and mean
zero
. Z-
coordinate
axis
along
camera
axis
. C
ollinearity
according
to
equation
12. (*M
ean
square
error
=
d
Sum of squares of residual errors
number of points
−
1
).
Method
Number
of
Control
Points
Mean Square Error* (µm)
Estimated
Standard
Error of Unit
Weight (µm)
X
Y
Z
Direct
5 NO SOLUTION POSSIBLE
Collinearity
5 NO SOLUTION POSSIBLE
Direct
6
205
179
408
1.770
Collinearity
6
200
174
407
2.005
Direct
10
165
135
334
2.293
Collinearity 10
172
137
353
2.261
Direct
20
157
94
342
2.792
Collinearity 20
161
95
356
2.744
Direct
30
138
100
301
2.888
Collinearity 30
141
101
310
2.892
Direct
43
135
94
297
2.782
Collinearity 43
135
92
303
2.995
T
able
IV. T
he
effect
of
nonperpendicularity
of
the
x
and
y
comparator
axes
.
I
mage
coordinates
perturbed
by
a
random
error
of
normal
distribution with
standard
error
: 3.000
um
,
and mean
zero
.
(*C
ollinearity
I:9
unknowns
, C
ollinearity
II, 11
unknowns
).
Angle between
x and y
Estimated Standard Error of Unit Weight (µm)
Collinearity I* Collinearity II** Direct Method
90°
2.970
2.995
2.781
91°
19.616
2.995
2.781
95°
97.616
2.995
2.781
99°
194.096
2.995
2.781
T
able
V. T
he
effect
of
systematic
differential
linear
distortion
along
the
x
and
y
comparator
axes
. I
mage
coordinates
perturbed
by
a
random
error
of
normal
distribution with
standard
error
: 3.000
um
and mean
zero
.
(*C
ollinearity
I: 9
unknowns
, ** C
ollinearity
II: 11
unknowns
.)
Scale Factor
Estimated Standard Error of Unit Weight (µm)
x
y
Collinearity I* Collinearity II** Direct Method
1.000 1.000
2.970
2.995
2.781
1.000 1.0001
3.234
2.995
2.781
1.0001 1.0001
2.970
2.995
2.781
1.000 1.0002
3.896
2.995
2.781
1.0002 1.0002
2.970
2.995
2.781
Extensive testing of the proposed method using real data is
currently underway.
7. Summary of Comparisons Between the Proposed and the
Conventional Approaches
The proposed approach is particularly suitable for non-metric
photography, where no fiducial marks are available, and can also
be applied with distinct advantages for data reduction in metric
photography. Following are some comments comparing the pro-
posed approach to the conventional collinearity approach.
a. The proposed method yields at least the same accuracy as
the conventional solution.
b. The proposed approach is a direct solution involving no
iterations and needs no initial approximations for the un-
knowns. Thus a solution is obtained even in cases where
the conventional collinearity approach fails due to the lack
of reasonable approximations for the unknown parameters
(inner and/or outer orientation elements). A case in point
here would be metric photography for which the outer ori-
entation is not known. It follows further that the proposed
solution is not subject to computational errors due to iter-
ation criteria nor to errors due to neglecting of second and
higher order terms in linearizing the observation equations.
c. The proposed solution is relatively easy to program since
it does not involve partial derivatives of the coefficients of
the observation equation.
d. The computer executing time and the computer memory
used are less in the proposed method than in the conven-
tional collinearity solution.
e. The number of unknowns in the proposed direct method is
the same as in the conventional approach, i.e. 11 (eleven).
Thus a minimum of 6 well distributed control points are
needed for a solution.
f. The proposed method is at a disadvantage in case of low
accuracy requirements where one can neglect the com-
parator calibration errors and lens and film distortions. In
this case the collinearity approach will have 9 unknowns
compared to the 11 unknowns of the proposed method.
106
February 2015
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
