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American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS)
ISSN (Print) 2313-4410, ISSN (Online) 2313-4402
© Global Society of Scientific Research and Researchers
http://asrjetsjournal.org/
Application of Robust Estimation Methods for Detecting
and Removing Gross Errors from Close-Range
Photogrammetric Data
Khaled M. Zakya*, Ashraf A. Ghonemb
aAssociate Professor of Surveying, Shoubra Faculty of Engineering, Benha University, 108 Shoubra Street,
Cairo16929, Egypt
bLecturer of Surveying, Shoubra Faculty of Engineering, Benha University, 108 Shoubra Street, Cairo16929,
Egypt
aEmail: khaled.hassan@feng.bu.edu.eg
bEmail: dr.ghonem@hotmail.com
Abstract
Systematic, random and gross errors are considered the main problems facing those working in the photo-
triangulation processes. The influence of systematic errors on photo-measurements may include lens distortion,
film deformation, refraction and other distortions. Usually, these types of errors can be solved by the calibration
process. Meanwhile, the traditional least-squares method was used to adjust photogrammetric data in order to
solve the problems of random errors. In case observations contain gross errors, the reliability of least-squares
estimates is strongly affected. In this paper, two independent mathematical models (photo-variant self-
calibration and robust estimation) are combined for solving and processing the problems of systematic and gross
errors in one step. Also, this paper investigated the effectiveness of robust estimation models on solving gross
errors in close-range photogrammetric data sets that require photo bundle adjustment solution. The results of
investigation indicate that all robust methods have the advantage of detecting and removing gross errors over the
least squares method especially in cases of observation contains large-sized errors. Moreover, the Modified M-
estimator (IGGIII) method has the best performance and accuracy. Furthermore, gross error was also revealed in
the residuals.
Keywords: Gross Errors; Least-square; Robust Estimation Methods; Close-Range Photogrammetry; Photo
Bundle adjustment; Photo-Variant Self-Calibration.
-----------------------------------------------------------------------
* Corresponding author.
American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 111-120
112
1. Introduction
Robust estimation methods were used in different surveying applications such as triangulation and leveling
networks for the first time in 1964 [7,10]. Subsequently, several weight functions were developed for the robust
estimators. Although most of these weight functions have no theoretical basis, weight functions were empirical
[1,2,3,4]. The weight functions could be selected for robust estimates if only their values are less and residuals
are larger. This is due to the measures of robustness for robust estimates being not unique [5]. Based on the least
squares estimate, this research studies the robustness, defines a measurement for it and deduces the
corresponding robust estimate. Robust estimation methods are able to simultaneous parameter estimation and
outlier elimination during the estimation process [6]. If observation equations contain additional parameters to
model the effect of systematic errors, then the use of iteratively reweighted least squares with an appropriately
chosen estimation gives us a tool for simultaneous treatment of all errors [6,14]. The main objective of this
research is to study the effect of gross errors on close-range photogrammetric data and the ability to detect and
remove this type of errors using a combination model from photo-variant self-calibration and robust estimation.
Robust estimation methods were used in different surveying applications such as triangulation and leveling
networks for the first time in 1964 [7,10]. Subsequently, several weight functions were developed for the robust
estimators. Although most of these weight functions have no theoretical basis, weight functions were empirical
[1,2,3,4]. The weight functions could be selected for robust estimates if only their values are less and residuals
are larger. This is due to the measures of robustness for robust estimates being not unique [5]. Based on the least
squares estimate, this research studies the robustness, defines a measurement for it and deduces the
corresponding robust estimate. Robust estimation methods are able to simultaneous parameter estimation and
outlier elimination during the estimation process [6]. If observation equations contain additional parameters to
model the effect of systematic errors, then the use of iteratively reweighted least squares with an appropriately
chosen estimation gives us a tool for simultaneous treatment of all errors [6,14]. The main objective of this
research is to study the effect of gross errors on close-range photogrammetric data and the ability to detect and
remove this type of errors using a combination model from photo-variant self-calibration and robust estimation.
Robust estimation methods were used in different surveying applications such as triangulation and leveling
networks for the first time in 1964 [7,10]. Subsequently, several weight functions were developed for the robust
estimators. Although most of these weight functions have no theoretical basis, weight functions were empirical
[1,2,3,4]. The weight functions could be selected for robust estimates if only their values are less and residuals
are larger. This is due to the measures of robustness for robust estimates being not unique [5]. Based on the least
squares estimate, this research studies the robustness, defines a measurement for it and deduces the
corresponding robust estimate. Robust estimation methods are able to simultaneous parameter estimation and
outlier elimination during the estimation process [6]. If observation equations contain additional parameters to
model the effect of systematic errors, then the use of iteratively reweighted least squares with an appropriately
chosen estimation gives us a tool for simultaneous treatment of all errors [6,14]. The main objective of this
research is to study the effect of gross errors on close-range photogrammetric data and the ability to detect and
remove this type of errors using a combination model from photo-variant self-calibration and robust estimation.
American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 111-120
113
2. Solving Gross Errors
Close-range applications do not produce large data sets as in aerial triangulation. The failed process acquires
data affected by gross errors. Data acquired with a non-metric camera for close-range applications should use a
photo-variant bundle solution. Almost, the measurements are acquired with gross errors. Robust estimation
methods are then suitable for use in adjusting these measurements. A robustfied bundle adjustment procedure
has been developed along this direction [8]. This method showed the exact values of gross errors in the
residuals. On the other hand, classical least-squares method distributed these gross errors to other un-affected
measurements [9]. This research discusses the combination of photo-variant self-calibration and different robust
estimation methods for processing close-range measurements.
3. Image-Variant Parameters of Interior Orientation
The photogrammetric data is influenced by systematic errors which may be lens distortion, film deformation,
refraction, etc. The values of these types of errors can be modeled and determined from the camera calibration
processes in close-range photogrammetry. Metric cameras have stable interior geometry over a period of time.
The parameters solved by calibration are always carried as constant from photograph to another. An advanced
data processing model allows for the distortions to vary from a photograph to another. This process is known as
photo-variant solution [16]. The use of non-metric camera for close-range photogrammetry has been improved
by many researchers [12,17,18]. Nowadays, digital consumer cameras of high-resolution are vastly available
and used in close-range photogrammetry. The mechanical construction of this type of cameras oftentimes does
not achieve the demands of close-range photogrammetry, so they have to be modeled sufficiently. A camera
modeling is called image-variant parameters of interior orientation should be applied. Significant improvements
of object accuracy have been achieved with respect to standard calibration techniques based on self-calibrating
bundle adjustment [18].
3.1. Camera Parameters Model (Image-variant parameters)
Camera parameters of interior orientation are applied for all images of photogrammetric projects. Parameters of
distortion are normally defined by the photo's principal point. So, the equation of the standard observation given
by the following equations:
x – xo += ()()()
()()()
y – yo+= ()()()
()()()
where Δxp , Δyp are offsets from the principle point to the centre of image frame, is the nominal camera focal
length, (Xo , Yo , Zo) are the ground coordinates of the projection centre, (Xi , Yi , Zi) are the ground coordinates
of point i and m11 to m33 are elements of rotation matrix.
American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 111-120
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4. Robust Estimation Methods
Usually, the influence of systematic errors on photo- measurements can be solved by calibration reports. The
conventional least squares method leads to unbiased estimates with minimum variance if the model is entirely
correct [17]. But in case observations contain gross errors or single-point displacement, the outcome is no longer
correct. These type of errors or displacements contaminate the estimates of parameters and are distributed over
the whole residual vector. Hence, it is extremely difficult and sometimes impossible to locate blunders by
screening the residuals. This disadvantage can be avoided, if a robust estimation method is applied. Robust
estimates are not influenced by blunders as long as the majority of points conform to the modeled trend [13].
The mathematical models of the robust estimation methods used in this research are summarized in table (1).
Table 1: Models of robust estimation methods and their weight functions
Robust
method
Weight function
Critical values
Reference
R-Estimators
()
where
is the rank of
weighted residuals and
is the score function
1.5 : 2
[1]
S-Estimator
=
in which
is the posteriori
scale factor given by the robust estimator
1.5 : 2
[21]
Danish
exp (/) ||>
1 ||
1.5 : 2
[3]
Andrews
M-estimators
sin(v/)/(/) ||
0 ||>
1.5 : 2
[2]
Modified M-
estimator
(IGGIII)
1 ||
|| <||
0 ||>
c
o
= 2.0 : 3.0
c1 = 4.5 : 8.5
[20]
Huber
1 ||
/|| ||>
1.5 : 2
[11]
Where so is the a priori standard error of the unit weight given by:
/0.6745,
American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 111-120
115
vi is the residual of observation, med is the median, pi is the weight of observation, ci is the constant critical
value given by: = so . . ,/ in which is the cofactor matrix of the residuals, p is the weight
matrix of the observations, f is the degree of freedom, α is the significance level, and t represents t-table. So, the
critical value can be calculated as follows: =/
in which n is the number of observations.
5. Numerical Example
Cannon digital IXUS 990 IS, digital compact camera equipped with 5x zoom lens and has been calibrated at the
widest view of its zoom lenses. It was also calibrated with a lens equivalent to 35 mm film format. A building
was photographed with one stereo-pair by the target camera for the calibration purpose. The distance between
the target camera and the photographed building was 15 m. the distance between the two projection centers was
3 m. The control and check points have been marked on the acquired image. The coordinates of the control and
check points have been measured from the stereo-model. The differences between their coordinates derived
from ground surveying technique and those defined by the photogrammetric process have been computed. A
photogrammetric software is called BUNDLEH Lite [15,19] has been used for the process of non-metric camera
calibration. Four values of gross errors: zero, 5 mm, 10 mm and 20 mm were added to one coordinate of an
image point. The different robust methods in table (1) were used to process the data in turn while the photo-
variant self-calibration mode was activated in the adjustment.
6. Results and Analysis
Results of the test are tabulated in tables (2), (3), (4) and (5) with the Root Mean Square Errors (RMSE) of
check points when zero, 5 mm, 10 mm and 20 mm blunders were introduced to a point coordinates obtained
from traditional least-squares and different robust estimation methods respectively.
Table 2: Root Mean Square Error (RMSE) of check points in case of zero gross error using different robust
methods
Method of adjustment
RMSE
X (mm)
Y (mm)
Z (mm)
Least Squares
0.28
0.25
0.79
R-Estimators
0.25
0.27
0.92
S-Estimator
0.22
0.23
0.75
Danish
0.12
0.14
0.83
Andrews
M-estimators
0.20
0.24
0.69
Modified M-estimator
(IGGIII)
0.16
0.15
0.50
Huber
0.36
0.41
0.94
American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 111-120
116
Table 3: Root Mean Square Error (RMSE) of check points in case of 5 mm gross error using different robust
methods
Method of adjustment
RMSE
X (mm)
Y (mm)
Z (mm)
Least Squares
12.15
10.16
99.79
R-Estimators
0.35
0.38
1.03
S-Estimator
0.33
0.34
1.26
Danish
0.23
0.25
1.01
Andrews
M-estimators
0.31
0.35
1.34
Modified M-estimator
(IGGIII)
0.27
0.26
1.03
Huber
0.48
0.52
1.56
Table 4: Root Mean Square Error (RMSE) of check points in case of 10 mm gross error using different robust
methods
Method of adjustment
RMSE
X (mm)
Y (mm)
Z (mm)
Least Squares
25.15
30.16
134.45
R-Estimators
0.33
0.41
1.12
S-Estimator
0.36
0.36
1.34
Danish
0.28
0.29
1.11
Andrews
M-estimators
0.37
0.38
1.35
Modified M-estimator
(IGGIII)
0.29
0.29
1.12
Huber
0.52
0.55
1.58
Table 5: Root Mean Square Error (RMSE) of check points in case of 20 mm gross error using different robust
methods
Method of adjustment
RMSE
X (mm)
Y (mm)
Z (mm)
Least Squares
32.15
42.16
134.45
R-Estimators
0.35
0.43
1.45
S-Estimator
0.38
0.38
1.55
Danish
0.31
0.31
1.23
Andrews
M-estimators
0.38
0.40
1.56
Modified M-estimator
(IGGIII)
0.30
0.32
1.34
Huber
0.54
0.56
1.59
Statistical tests have been performed for detecting and decreasing the effect of gross errors and then the
American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 111-120
117
adjustment should be performed again. Least-squares method was carried out with blunder-free measurements
as a reference adjustment. Six robust estimation methods were used to adjust the same measurements after
adding blunders to a point coordinates.
It is easy to notice that the effect of zero or small-sized gross errors on the adjusted coordinates was minimal
and can be neglected as shown in table (2). On the other hand, large-sized gross errors breakdown the adjusted
coordinates completely as shown in tables (3), (4) and (5). Furthermore, gross errors were also included in the
residuals as shown in tables (6) and (7). The results were not sufficiently accurate in case of using least-squares
method for adjusting close-range photogrammetric data that contains blunder values of 10 mm and 20 mm,
while there was an improvement in accuracy when robust estimators were used.
The Modified M-estimator (IGGIII), which is one of the robust estimation methods, gives better accuracy than
other robust methods (see tables (3), (4) and (5)). From table (3), it can be noticed that the gross error was
forbade from share in the solution, then the accuracy of the solution obtained earlier from the least squares
method can be improved, provided sound geometry is still maintained. Table (8) shows that gross error was
detected in the residual. The Modified M-estimator (IGGIII) method performance was the best for X, Y and Z
coordinates without iterations.
Huber's estimation method has shown the least accurate results in both plane and height coordinates accuracy
with few iterations. In case of large-sized blunders (10 mm and 20 mm), the Modified M-estimator (IGGIII)
method has the best performance and accuracy without iterations.
Table 6: Robust Estimator for detecting gross errors of selected points on images in case of use 5 mm error
Point-id
V
X
(mm)
Weight
V
Y
(mm)
Weight
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0.0004
0.0000
-0.0009
0.0006
0.0005
0.0007
0.0003
0.0003
0.0003
0.0029
0.0003
0.0004
0.0002
-0.0005
0.0002
0.685
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
0.000
1.000
1.000
1.000
1.000
1.000
0.0008
0.0001
0.0002
0.0003
0.0002
0.0001
0.0002
0.0001
0.0001
0.0009
0.0003
0.0001
0.0002
0.0002
0.0002
0.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
0.000
1.000
1.000
1.000
1.000
1.000
American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 111-120
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Table 7: Robust Estimator for detecting gross errors of selected points on the images in case of use 10mm error
Point-id
V
X
(mm)
Weight
V
Y
(mm)
Weight
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0.0004
0.0000
-0.0002
0.0002
0.0002
0.0001
0.0002
0.0001
0.0001
0.0089
0.0003
0.0001
0.0002
-0.0005
0.0002
0.685
1.000
0.897
0.754
1.875
0.987
0.943
0.989
0.978
0.000
0.764
0.990
0.986
0.987
0.885
0.0007
-0.0001
-0.0007
0.0001
0.0002
0.0001
0.0005
0.0004
0.0003
0.0000
0.0009
0.0007
0.0006
0.0008
0.0005
0.423
0.965
0.453
0.990
0.954
0.989
0.897
0.912
0.876
1.000
0.345
0.453
0.654
0.534
0.867
Table 8: Robust Estimator for detecting gross errors of selected points on the images in case of use 20mm error
Point-id
V
X
(mm)
Weight
V
Y
(mm)
Weight
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0.0005
0.0000
-0.0002
0.0001
0.0001
0.0002
0.0002
0.0002
0.0002
15.432
0.0002
0.0002
0.0001
-0.0001
0.0002
0.355
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
0.000
1.000
1.000
1.000
1.000
1.000
0.0007
0.0000
-0.0005
0.0004
0.0001
0.0004
0.0003
0.0001
0.0003
0.0006
0.0005
0.0006
0.0004
0.0008
0.0002
0.312
1.000
0.342
0.423
1.000
0.412
0.543
1.000
0.654
0.353
0.342
0.234
0.542
0.234
0.864
American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 111-120
119
7. Conclusion
The main objective of this paper was to apply different methods of robust estimation with the classical least-
square method for detecting and solving the gross errors in close range-photogrammetric data. For this purpose,
one stereo-pair terrestrial photos were taken by non-metric digital camera. A calibration process was also carried
out using BUNDLEH Lite software. Four values of gross errors: zero, 5 mm, 10 mm and 20 mm were added to
one coordinate of an image point. The different robust methods in table (1) were used to process the data in turn
while the photo-variant self-calibration mode was activated in the adjustment. The results indicate that the effect
of zero or small-sized gross errors on the adjusted coordinates was minimal and can be neglected. Large-sized
gross errors breakdown the adjusted coordinates completely. Gross errors were also included in the residuals.
The results were not sufficiently accurate in case of using least squares method for adjusting close-range data
that contains 10 mm and 20 mm, while there was an improvement in accuracy when robust estimators were
used. The Modified M-estimator (IGGIII), which is one of the robust estimation methods, gives better accuracy
than other robust methods.
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