Analysis of the Effects of Three Line Scanner’s Focal Length Bias

Journal of the Korean Society of Surveying, Geodesy, Photogrammetry and Cartography.
2014.
Jan,
32(1):
1-8

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

- Received : November 09, 2013
- Accepted : February 21, 2014
- Published : January 30, 2014

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The positions, attitudes, and internal orientation parameters of three line scanners are critical factors in order to acquire the accurate location of objects on the ground. Based on the assumption that positions and attitudes of the sensors are derived either from direct geo-referencing which of using Global Positioning Systems (GPS) and Inertial Navigation Systems (INS), or from indirect geo-referencing which of using Ground Control Points (GCPs), this paper describes on biased effects of Internal Orientation Parameter (IOP) on the ground. The research concentrated on geometrical explanations of effects from different focal length biases on the ground. The Synthetic data was collected by reasonable flight trajectories and attitudes of three line scanners. The result of experiments demonstrated that the focal length bias in case of indirect geo-referencing does not have critical influences on the quality of reconstructed ground space. Also, the relationships between IO parameters and EO parameters were found by the correlation analysis. In fact, the focal length bias in case of the direct geo-referencing caused significant errors on coordinates of reconstructed objects. The RMSE values along the vertical direction and the amount of focal length bias turned out to be almost perfect linear relationship.
Three line scanner
An example of images, taken by a three line scanner, is shown in
Figure 2.
A major advantage of this system is that it can provide images with three different looking angles with almost no time gap between images. Also, unlike with the traditional frame system, the three line geometry is characterized by nearly parallel projection along the flight direction and perspective projection across the flight direction (
Gruen and Zhang, 2003
).
Forward, nadir, backward images which were taken by a three line scanner, Courtesy of Gruen and Zhang (2003)
x_{p}
,
y_{p}
,
f
) and lens distortions. For developing the formula and the level of importance of interior orientation parameters, the focal length among the IO parameters is selected and analyzed by geometrical impact of its bias in a section. The relationship between the IO parameters and EO parameters will be mentioned in later Section 4.
Figure 3
indicates three-line scanner with and without a focal length bias, while
Figure 4
and
5
are referring shapes in upper direction of a three line scanner with and without bias, respectively.
Side view of three line scanner with and without focal length bias
Top view of three line scanner without focal length bias
Top view of three line scanner with focal length bias
In
Figure 3
,
H, B, f
, and
a
present flying height, base distance, focal length, and perspective center, respectively If there is the bias in amount of Δ
f
in the focal length, an original position of CCD array (i.e., b) changes to the position of
c
, and the position of
r
changes to
s
in the object space.
First of all, we examined the case without any bias through the
Figure 4.
As seen in the figure, distance between
i
and
j
points was expressed as Equation 1.
l
indicates the length of CCD array. According to
Figure 3
, Equation (1) can be modified to Equation (2) by using
tan θ = B / 2H
and
sin θ
=
As seen in
Figure 5
, if there is focal length bias in the three line scanner, the distance between points
p
and
q
can be calculated wit Equation (3):
According to
Figure 3
, Equation (3) can be modified to Equation (4) by using tan θ
^{'}
=
B /2H^{'}
and
From Equations (2) and (4), the distance (
) between point
i
and
j
, and the distance (
) between point
p
and
q
are determined to be same by
.
On the other hand,
and
can be derived from
Figure 3
by using triangle similarity. Thus, the value
H′-H
is same as
From the above variations, we were able to find that the focal length bias does not have effects along and across scan lines in three line scanner, however, there were some effects in accuracies of the vertical direction.
Experimental conditions for generating the synthetic data
Table 2
shows information about high and low altitude sensors and amounts of IOP biases. The simulation of high altitude space-borne was based on IKONOS, while the low altitude scanner case was based on ADS40 sensor information. A flying height of three line of a high altitude sensor is 680km above ground. The focal length is 10m in both forward and backward directions, which is 8.98794 m for the nadir case. At this point, noted that the focal lengths in nadir direction and in off-nadir cases (i.e., forward and backward directions) are different, but they have the same perspective center. Please, refer to
Figure 1
for clear understanding.
Specifications and the amounts of bias in both low and high altitude sensors
The Ground Sampling Distance (GSD) is 0.68m. The amount of principal point bias is around 50um. Also, the range of focal length bias is from 50um to 50,000um.
In the case of low altitude, the flying height is 5,500m above the ground. Focal length is 75mm in both forward and backward directions, and 67.40955mm for the nadir direction. The Ground Sampling Distance (GSD) is 0.73m. The amount of principal point bias is around 0.375um. Also, the range of focal length biases is from 0.375um to 375um.
x
_{p}
in the high altitude case(the experiments from H_A1 to H_A4), while there is about 3.8m error in the scan line (Y-direction) by adding the bias into
y
_{p}
As the bias values in the focal length increase, RMSE along the Z-direction increases proportionally, which has been proven mathematically in Section 3. In fact, this phenomena also occurs similar in low altitude cases (the experiments from L_A1 to L_A4).
Figure 6
and
7
illustrate the visual understanding of effects from the focal length bias. As seen in figures, relationship between the focal length bias and the RMSE along Z direction is almost perfect linear for both high and low altitudes three line scanners. The Pearson’s correlation coefficients between the two variables in both high and low altitudes were 1 and more than 0.999, respectively.
Effects of focal length bias in high altitude (R^{2} = 1)
Effects of focal length bias in low altitude (R^{2} > 0.999)
High altitude three line scanner (direct geo-referencing)
Low altitude three line scanner (direct geo-referencing)
High altitude three line scanner (indirect geo-referencing)
Low altitude three line scanner (indirect geo-referencing)
We also carried out the correlation analysis to figure out which IO parameters and EO parameters are coupled under different strength levels.
Table 7
indicates a correlation matrix of IO and EO parameters. According to the
table 7
,
x
_{p}
is somewhat correlated with φ(rotation along Y-axis). The table also refers that
y
_{p}
is strongly correlated with ω (rotation along X-axis) and κ(rotation along Z-axis) as well as it is somewhat correlated with Y
_{0}
(translation along Y-direction). For
f
, it is strongly correlated with X
_{0}
(translation along X-direction) and Z
_{0}
(translation along Z-direction). Also, it is somewhat correlated with Y
_{0}
and φ. In fact, some of the EO parameters are strongly correlated with each other, such as X
_{0}
& Z
_{0}
and ω & κ.
Correlation matrix of the IO and EO parameters (absolute values)

1. Introduction

The main purpose of photogrammetric field is to reconstruct object space accurately (
Fraser ., 2002
;
Lee and Bethel, 2004
). To achieve the purpose, processes to determine the interior and exterior orientation parameters of cameras (or sensors) are significant. The interior orientation determines internal characteristics of a sensor and parameters, such as focal length, principal point, lens distortion, while the exterior orientation’s parameters include a position and an attitude of the sensor. Sensors, mounted on the crafts (i.e., space-born or air-borne crafts), are calibrated in the laboratory before launching crafts. Interior Orientation Parameters (IOPs) to explain physical phenomena of the sensor can be determined through camera calibration (or sensor calibration) (
McGlone ., 2004
;
Tsai, 1986
;
Brown, 1971
;
Lichti, ., 2010
;
Lichti and Kim, 2011
;
Rau ., 2011
).
Geometrical images of the sensor might change time to time due to various physical effects, which indicate thermal influences from the sun in the orbit and vibrations after or during launching the craft. Therefore, the on-orbit sensor calibration is also important after launching. The On-orbit sensor calibration requires a test field of containing different types of variations in heights, and choosing accurate ground control points (
Jacobsen ., 2005
). In fact, although the on-orbit calibration is carried out, it is not easy to accurately determine the sensor’s internal orientation parameters, especially in high-resolution satellite sensors (e.g., IKONOS or Quickbird) with narrow Angular Field Of View (AFOV). This is due to the high correlations between the interior and exterior orientation parameters of the sensor (
Jacobsen, 2005
;
Baltsavias and Zhang, 2005
). Many researchers have presented the studies about camera calibration, however, the researches about accuracy effects of the internal orientation parameters on the ground are rarely found, especially in three line scanners. For this reason, the study focused on investigation of the IOP bias (especially, focal length bias) effects on the quality of reconstructed object space. This paper will start with the description of three line scanner, which is widely used in photogrammetric field, in Section 2. At next, the paper will geometrically prove the bias including into the focal length, which can have effects on accurate locations of interesting points on the ground. The experimental results and analysis will present on Section 4. Also, during experiments, there were two cases of geo-referencing approaches (i.e., direct and indirect ones). Lastly, the concluding remarks and future work plans are addressed in the last section.
2. Three-Line Scanner System

The three line scanner system, primarily developed by Starlabo Corporation in Tokyo, captures digital image with three different looking angles (
Murai, Matsumoto, 2000
,
Murai, 2001
;
Chen ., 2001
). At the same time, the system produced seamless three different digital images. Theoretically, there should be different sets of values for six exterior orientation parameters in each scan line. Therefore, a good trajectory model is necessary for this system.
Figure 1
shows a conceptual structure of the sensor, which of onedimensional scan lines on forward, nadir, and backward. Three scanners with different looking angles share the same perspective center. The MOMS, developed by Germany, and the ADS40, developed by Leica and DLR, can be also classified into three line scanners.
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3. Geometric Understanding of Focal Length Bias Effects on the Ground

Recall that the interior orientation parameters include the location of principal point (
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4. Experimental Results and Analysis

This study evaluated accuracies of direct and indirect geo-referencing for a three line scanner, by changing the amounts of the focal length biases. In the study of direct geo-referencing, external orientation parameters were set as true values for investigating the bias effects only.
Table 1
shows the specific experimental conditions of sensor trajectory, image measurement error, control information, accuracy of Global Positioning Systems (GPS), accuracy of Inertial Navigation Systems (INS), accuracy of Ground Control Points (GCP), and height variation of GCP. 100 Ground Control Points (GCPs) were created by changing the altitude of ±500m. GPS accuracies of high altitude are 4cm for all X, Y, Z directions, and also accuracies of low altitude are 15cm for all directions. INS accuracies of high altitude were 7 seconds, but 15 seconds for low altitude case. When indirect geo-referencing were performed, 10 GCPs, among the 100 GCPs, were used for sensor modeling, and others were used for Check Points (CP). Since direct geo-referencing did not require any of GCPs, all 100 GCPs were used as check points.
Experimental conditions for generating the synthetic data

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Specifications and the amounts of bias in both low and high altitude sensors

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- 4.1. Focal Length Bias Effects In the Case Of Direct Geo-referencing

Tables 3
and
4
display results for the direct georeferencing with high and low altitudes, respectively. In direct orientation, the achieved accuracy is significantly affected by the bias in internal orientation parameters, as seen in the experiments H_A1 to H_A4, L_A1 to L_A4. There is an error between 3.8 and 4.0m approximately across the san lines (X-direction) by adding the bias into
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High altitude three line scanner (direct geo-referencing)

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Low altitude three line scanner (direct geo-referencing)

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- 4.2. Focal Length Bias Effects In the Case Of Indirect Geo-referencing

On the other hand, the effects of IOP biases are insignificant on the reconstructed object space for indirect geo-referencing according to
Tables 5
and
6
from H_A5 to H_A8 and L_A5 to L_A8. This result was expected because EOP changes to compensate for the IOP biases. In other words, derived attitudes and positions of the scanner absorbed effects from IOP biases. Most of the RMSE values in the experiments from H_A5 to H_A8 are close to zeroes, since we deal with a scanner of a narrow Angular Field of View (AFOV) and a high flying height. In this case, internal orientation parameters correlate with external orientation parameters. All relevant experiments, except for L_A8 (i.e., L_A5 to L_A7) of low-altitude scanner, also had the RMSE values close to zeroes. A reason for the result of L_A8 is that attitudes and positions of the scanner do not absorb all effects from IOP biases comparing to the high-altitude scanner, when the scanner does not have narrow AFOV.
Habib . (2007)
has addressed the relationship of AFOV and correlation between IOP and EOP. For more detailed information, please refer to
Habib . (2007)
.
High altitude three line scanner (indirect geo-referencing)

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Low altitude three line scanner (indirect geo-referencing)

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Correlation matrix of the IO and EO parameters (absolute values)

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5. Concluding Remarks and Future Work

In this study, the effects of focal length bias on the coordinates of the reconstructed objects were mainly analyzed for three line scanners in space-borne (i.e., high altitude) and airborne (i.e., low altitude). This research applied two different geo-referencing approaches, which are direct and indirect ones. The accuracy of the reconstructed object space for analyzing the bias effects was evaluated by adding artificial biases to the principal point and focal length. Based on the real trajectory of sensors, a topography of the object space was simulated, and trajectories of sensors were also determined. Firstly, one of the focal length bias effects was proven through mathematical derivations. As more experiments were carried out, those results also showed the similar trends to one that is proven through mathematical proofs. In fact, IOP biases of the direct georeferencing generated serious errors on determination of three-dimensional coordinates; as the bias value in the focal length increases, error along the vertical direction increases proportionally. The Pearson’s correlation coefficients, in which between the focal length biases and the RMSE values along the vertical direction, showed 1 and more than 0.999 for the high and low altitude cases, respectively.
On the other hand, the effects of IOP biases are insignificant on the reconstructed object space, when indirect geo-referencing is carried out. In this case, the derived attitudes and positions of the scanner absorbed the effects from the biases. Similar phenomena occur for the low altitude experiments. However, when the amount of the bias is very large, the attitudes and positions of the scanner do not absorb all the effects from the bias. Additionally, the correlation analysis is carried out, and the correlation between the IO and the EO parameters are proven as well.
Simulated data was used for this study due to difficulties in data acquisition of the IOP bias, however, future research will be better to concentrate on analysis of the bias effects with real data. Moreover, the terrain roughness effects in case of low altitude scanner will be studied in the future research.
Acknowledgements

This study was supported the intramural research funds from SANGMYUNG University, 2012.

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Citing 'Analysis of the Effects of Three Line Scanner’s Focal Length Bias
'

@article{ GCRHBD_2014_v32n1_1}
,title={Analysis of the Effects of Three Line Scanner’s Focal Length Bias}
,volume={1}
, url={http://dx.doi.org/10.7848/ksgpc.2014.32.1.1}, DOI={10.7848/ksgpc.2014.32.1.1}
, number= {1}
, journal={Journal of the Korean Society of Surveying, Geodesy, Photogrammetry and Cartography}
, publisher={Korean Society of Surveying, Geodesy, Photogrammetry and Cartography}
, author={김, 창재}
, year={2014}
, month={Jan}