where the subscript
i
is the number of corresponding points.
2
A
in formula 20 presents the coefficient matrix of the additional error equations, which is linearized from formula
13:
1 2
3 4
5 6
7 8
9 2
2 1
5 4
8 7
3 1
6 4
9 7
3 2
6 5
9 8
2 2
2 2
2 2
2 2
2 r
r r
r r
r r
r r
r r
r r
r r
r r
r r
r r
r r
r r
r r
A
23
, B e x
is decomposed as:
1 2 3
1 1
1 1
1 1
1 2
2 2
2 2
2
1 7
,
k k
T
k k
k k
k k
i
bwxx bwxy
bwxz bwyx
bwyy bwyz
bwxx bwxy
bwxz bwyx
bwyy bwyz
bwxx bwxy
bwxz bwyx
bwyy bwyz
fr r
x bwxx
f e x B
e
2 8
3 9
4 7
5 8
6 9
, ,
, ,
i xi
i xi
i xi
i i
i i
i i
yi i
yi i
yi i
i i
i i
i
fr r x
fr r
x bwxy
bwxz Z
Z Z
fr r
y fr
r y
fr r
y bwyx
bwyy bwyz
Z Z
Z
24
1 1
1 1
2 2
2 2
2 2 2
2 2
2 1
3 4
1 1
1 4
1
,
1 2 2
3 ,
2 2
k k
T k
k k
k i
i i
i i
i i
i i
i i
bxx bxy
byx byy
bxx bxy
byx byy
bxx bxy
byx byy
bxx g x
g x g y
k x k y
bxy g y
g x k x y
b
f e x B
e
2 2
2 3
4 1
1 2
3 1
1 2 2
3 ,
2 2
i i
i i
i i
i i
i i i
yy g y
g x g y
k x k y
byx g x
g y k x y
25
3 6 6 6 6
3
,
T
f e x B
I e
26 here,
2
B
,
1
B
and
3
B
respectively denote the matrices of partial
derivatives of
2 e
, and
1
e
and
3
e
. According to formula 17,
, B e x
is :
2 1
2 6
2 6 2
3 6
6 2 6 3
3 k
k k
k k
k
B
B B
B
27 with the vector of misclosures:
2 6 1
,
k
ω B e
f e x
28 and cofactor matrices of
1
e ,
2
e
and
3
e
:
1 1
1 1
1 3 3
1 2
2 2
3 6 6 3
6 6
, ,
k k
k k
2
Q P
Q P
Q P
I
29 here,
is a sufficiently large constant which presents the weights of the six pseudo-observation equations. Considering
the correlation between the coordinates in the image coordinate system and the object world coordinate system, the more
general form of cofactor matrix is:
2 21
2 6
2 3
6 2 3
6 12
1 3
6 6 2
6 3 3
k k
k k
k k
k k
Q Q
Q Q
Q Q
30 where
21
Q
and
12
Q
denote the covariance matrix of
2 e
and
1
e
. So compared with the calculation process in Neitzel 2010, in
which the weighted matrix is diagonal, the observations here can be correlated.
The estimation for the unknown parameters from the solution of the linear equations system will be obtained as follows:
1 1
ˆ ˆ
ˆ
T T
B Q B
A λ
ω +
= 0 x
x A
31 and the first error vector is:
1 1
ˆ
T
e Q B
λ
32 This is an iterative calculation process. After stripping the
randomness of the solution
1
e
and
1
ˆx
, they are used in the next iteration step as their approximations.
4. EVALUATION
The evaluation method used in this paper is the multi-image intersection method. Intersection refers to the determination of a
point ’s position in object-space by intersecting the image rays
from two or more images. And it is the application of coilinearity equations which can be established as:
1 2
3 4
5 6
4 5
6 7
8 9
w w
w x
x w
w w
y w
w w
y y
w w
w z
r x r y
r z T
x y
r x r y
r z T
r x r y
r z T
y f
r x r y
r z T
33 After the calibration parameters are solved, with more than two
images, the 3D object world coordinates of the point can be calculated by the error equations:
w w
w x
x w
w w
w w
w x
y w
w w
x x
x v
x y
z x
x y
z y
y y
v x
y z
y x
y z
34 So the observations in this intersection solution are the image
coordinate measurements. Comparing the calculation results and the given coordinates of the control points, the correction
and accuracy of the calibration results will be evaluated.
5. CASE STUDY
In the following section, a numerical example based on actual experiments will be used to examine the camera model and the
parameter estimation strategy described in the previous sections. The setup used in our calibration experiments is shown in Fig. 1.
XXII ISPRS Congress, 25 August – 01 September 2012, Melbourne, Australia
144
Fig 1 calibration setup
In this calibration field, 58 mark points are mounted on the walls and steps. These points are measured by the total station,
whose angle measurement accuracy and ranging accuracy are 1″
and 0.6mm+2ppm, respectively. To promise the accuracy of every point within the millimeter level, the measuring distance
is less than 100 meters, and every point is measured 4 times. The weights for the 3D object coordinates are equal.
These 58 mark points are divided into two groups, including 38 control points and 20 check points.
The images were taken by the consumer-grade camera: Nikon D200, in which the effective part of the CCD sensor array is
3872
2592 pixels 23.6mm×15.8mm and the focal length is about 50 mm.
The corresponding image-point locations are estimated with sub-pixel accuracy.
In the experiment, eight camera stations are set up, and one image is taken on every station. The shooting distance is
between 15 and 20 meters. The sample is presented as Fig.2, and the 38 control points are remarked by red crosses.
Fig 2 the sample of images taken by the camera
After the initial values are calculated by formula 2 and 3 with LS adjustment, the improved two-step calibration method
is proceeded to optimize all the calibration parameters by formula 9 and 10. In order to solve this adjustment problem,
we compute this step by using the WLS and WTLS method, respectively. For the EIV model, we use the solution within the
iteratively linearized GH model. The weights for the six pseudo- observation equations are 10
10
. So the covariance matrix is:
3 2
2 2
3 6 2
3 6
3 3
10 6 6
10 10
k k
k k
k k
k k
I
Q I
I
35
Repeat the iteration until
1
ˆ ˆ
k k
x x
for a given
, in general,
10
10
. Here the superscript
k
denotes the iteration count.
The estimated calibration results are displayed in Table 1. The evaluation method is the multi-image intersection described
in section 4. The precision and accuracy of the solution will be evaluated by control points and check points respectively.
With the formula 34 and 35, the 3D object world coordinates of every point can be solved. Then the difference between the
calculation results and the given coordinates of the control points and check points will be computed, respectively. If we
use
2 -
x GCP
,
2 -
y GCP
,
2 -
z GCP
and
2 -CP
x
,
2 -
y CP
,
2 -
z CP
to represent the variance components of the ground control points
and check points;
2 0-GCP
and
2 0-CP
to delegate the variance components of the control points and check points, then the
evaluation results are shown in Table 2 and Table 3.
Tab.1 Calibration results Classical two-step method
Improved two-step method LS
WLS WTLS
x
p 0.01
0.00
y
p -0.00
-0.00
f
p 8624.53
8623.98 8623.11
x
S
1.000269 1.000269
1.000267
1 k
10
-10
p
-2
1.96 2.19
2.04
1 p
10
-8
p
-2
-0.6278 -0.6534
2 p
10
-8
p
-2
-0.3678 -0.8529
1 s
10
-8
p
-2
0.2213 0.2203
2 s
10
-8
p
-2
1.7280 1.5419
Tab.2 Precision of the calibration results calculated by control points Classical two-step method
Improved two-step method LS
WLS WTLS
2 -
x GCP
mm 1.1510
0.7087 0.4754
2 -
y GCP
mm 0.6382
0.4604 0.1524
2 -
z GCP
mm 0.1505
0.0592 0.0110
2 0-GCP
mm 1.9397
1.2283 0.6388
Tab.3 Accuracy of the calibration results calculated by check points Classical two-step method
Improved two-step method LS
WLS WTLS
2 -CP
x
mm 2.3900
1.3029 0.9014
2 -
y CP
mm 1.0167
0.6091 0.2805
2 -
z CP
mm 0.1854
0.1170 0.1099
2 0-CP
mm 3.5921
2.0290 1.2918
Comparing the results for the calibration parameters from Tables 1 and the evaluation results in Tables 2 and Table 3,
differences can be analyzed. 1 As can be seen from the calibration results shown in Table 3,
the offsets of the principle point and many kinds of parameters for camera distortion cannot be obtained by the classical two-
step calibration method. But for this lens, the decentering and thin prism distortions should not be neglected.
2 As shown in Table 1, no matter which calculation procedure is chosen, the calibration results solved by the improved two-
step method are similar. 3 From the evaluation results in Table 2, the variance
component of the control points solved by improved two-step calibration method is less than 1.5 millimeters, which is smaller
than the one calculated by the classical two-step method. And from Table 3, we can see that the accuracy of the calibration
results calculated by the improved two-step calibration method is higher than the classical one. However, if the camera
calibration model is the same, for example, as the improved two-step calibration method, with the EIV model, we can obtain
higher accurate calibration results than with the GM model.
4 Since the errors are obviously distributed in both the object world coordinate system and the image coordinate system, the
EIV model is preferable for solving this calibration problem. This can be detected also from the evaluation results in Table 2
and Table 3. The variance components calculated by the EIV model are much smaller than those calculated by the GM model.
XXII ISPRS Congress, 25 August – 01 September 2012, Melbourne, Australia
145
6. CONCLUDING REMARKS
