In this case, a5 in equations (1) becomes zero and the rela-
tionship between comparator coordinates and image coordi-
nates can be expressed as:
x– = a
1
+ a
2
x + a
3
y
y– = a
4
+ a
6
y
(10)
Combining equations (10) and (3) one gets
a
11
(X – X
0
) + a
12
(Y – Y
0
) + a
13
(Z – Z
0
)
a
31
(X – X
0
) + a
32
(Y – Y
0
) + a
33
(Z – Z
0
)
a
1
+ a
2
x + a
3
y – c · = 0
(11)
a
21
(X – X
0
) + a
22
(Y – Y
0
) + a
23
(Z – Z
0
)
a
31
(X – X
0
) + a
32
(Y – Y
0
) + a
33
(Z – Z
0
)
a
4
+ a
6
y – c · = 0
Equation (11) has 12 unknowns, but they are not linearly
independent. These unknowns can be reduced to 11 linearly
independent unknowns by eliminating a
2
and a
6
and intro-
ducing two unknowns C
x
and C
y
to replace C;
( C
c
a
C
c
a
x
y
=
=
2
6
;
, C
x
and C
y
reflect possible differential linear
distortions along x and y comparator axes). Equation 11 can
thus be rewritten as:
a–
1
+ a–
2
x – C
x
= 0
a
11
(X – X
0
) + a
12
(Y – Y
0
) + a
13
(Z – Z
0
)
a
31
(X – X
0
) + a
32
(Y – Y
0
) + a
33
(Z – Z
0
)
(12)
a–
3
+ y – C
y
= 0
a
11
(X – X
0
) + a
12
(Y – Y
0
) + a
13
(Z – Z
0
)
a
31
(X – X
0
) + a
32
(Y – Y
0
) + a
33
(Z – Z
0
)
Equations (12) represent the basic equations in the conven-
tional (collinearity) approach. As explained above, these
equations take into consideration the non-prependicularity
between comparator axes, and differential linear distortions
along x and y axes.
4. Observation Equations
Expanding equations (12) by Taylor’s series and neglecting
second and higher order items, one gets:
V
x
+ a
1y
V
y
+ b
1x
Δω
+ b
2x
Δϕ
+ b
3x
Δκ
+ b
4x
Δ
X
0
+ b
5x
Δ
Y
0
+ b
6x
Δ
Z
0
+
b
7x
Δ
C
x
+ b
8x
Δ
C
y
+ b
9x
Δ
a–
1
+ b
10x
Δ
a–
2
+ b
11x
Δ
a–
3
+ F
x
° = 0
and
(13)
V
y
+ b
1y
Δω
+ b
2y
Δϕ
+ b
3y
Δκ
+ b
4y
Δ
Y
0
+ b
5y
Δ
Y
0
+ b
6y
Δ
Z
0
+ b
7y
Δ
C
y
+ b
8y
Δ
C
y
+ b
9y
Δ
a–
1
+ b
10y
Δ
a–
2
+ b
11y
Δ
a–
3
+ F
y
° = 0
Where
V
x
, V
y
are errors in x and y
a
1y
= a
2
, is partial derivative of F
x
w.r.t. y
b
1x
, b
1y
are the partial derivatives of F
x
and F
y
(see footnote
below) w.r.t.
ω
b
2x
, b
2y
are the partial derivatives of F
x
and F
y
w.r.t.
ϕ
b
3x
, b
3y
are the partial derivatives of F
x
and F
y
w.r.t.
κ
b
4x
, b
4y
are the partial derivatives of F
x
and F
y
w.r.t. X
0
b
5x
, b
5y
are the partial derivatives of F
x
and F
y
w.r.t. Y
0
b
6x
, b
6y
are the partial derivatives of F
x
and F
y
w.r.t. Z
0
b
7x
, b
7y
are the partial derivatives of F
x
and F
y
w.r.t. C
x
b
8x
, b
8y
are the partial derivatives of F
x
and F
y
w.r.t. C
y
b
9x
, b
9y
are the partial derivatives of F
x
and F
y
w.r.t. a–
1
b
10x
, b
10y
are the partial derivatives of F
x
and F
y
w.r.t. a–
2
b
11x
, b
11y
are the partial derivatives of F
x
and F
y
w.r.t. a–
3
F
x
o
and F
y
o
are functions of approximate values of the un-
known parameters.
Equations (13) represent the observation equations in the con-
ventional collinearity approach. The observation equations in
the proposed direct approach may be obtained by expanding
equations (9) and including all the zero terms (e.g. 0
l
5
and 0
l
2
)
for ease of reference:
w
1
v
x
+ X
l
1
+ Y
l
2
+ Z
l
3
+
l
4
+ 0
l
5
+ 0
l
6
+ 0
l
7
+ 0
l
8
+ xX
l
9
+ xY
l
10
+ xZ
l
11
+ x = 0,
and
(14)
w
2
v
y
+ 0
l
1
+ 0
l
2
+ 0
l
3
+ 0
l
4
+ X
l
5
+ Y
l
6
+ Z
l
7
+
l
8
+ yX
l
9
+ yY
l
10
+ yZ
l
11
+ y = 0.
In equations (14) the factor w1 and w2 may be considered as
weight factors, and their value can be easily determined in
the solution.
A comparison between equations (13) and equations (14)
indicate the simplicity of the proposed solution.
5. Analysis of Errors
Both the conventional and proposed approaches are influ-
enced by the following errors:
a. Uncertainties in comparator measurements and errors
in object space coordinates of control points.
b. Errors in mathematical modeling of film and lens dis-
tortions (random errors as well as unrepresented – or
residual – systematic errors).
In addition, the conventional iterative approach is subject to
computational errors due to:
a. Iteration criteria
b. Neglecting of second and higher terms in the lineariza-
tion of the observation equations (13).
Obviously, the proposed direct solution is not subject to
these computational errors.
6. Fictitious Data Tests
A number of fictitious data tests were conducted to assess
the capabilities of the proposed solution (equations 9) and
compare them to the capabilities of the conventional ap-
proach (equations 12.) As a datum for comparison of the
two approaches, data from the collinearity approach with 9
unknowns (only parameters of inner and outer orientation are
included, errors due to comparator adjustment, lens distor-
tion, and film deformation are not considered) were used. The
test covered the following aspects: handling of differential
linear distortions along the x and y comparator coordinate
axes, correction for non-prependicularity of the comparator
axes, accuracy of determination of the unknowns (standard
error of unit weight), and computer executing time.
Tables I through V summarize the results of these prelimi-
nary tests.
a
11
(X – X
0
) + a
12
(Y – Y
0
) + a
13
(Z – Z
0
)
a
31
(X – X
0
) + a
32
(Y – Y
0
) + a
33
(Z – Z
0
)
F
x
= a–
1
+ x + a–
2
y – C
x
· = 0
a
11
(X – X
0
) + a
12
(Y – Y
0
) + a
13
(Z – Z
0
)
a
31
(X – X
0
) + a
32
(Y – Y
0
) + a
33
(Z – Z
0
)
F
y
= a–
3
+ y – C
y
· = 0
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
February 2015
105
