The Automatic Selection of Radial Distortion Models

Abstract

Several kinds of distortions exist in imaging systems which in specific circumstances may affect an image's geometry without debilitating quality or diminishing the information existing in the image. The most important type is the radial distortion which represents high distortion accurately. Many lens distortion models exist combined with several variations where different techniques are used to calibrate each distortion model. This paper presents an algorithm to select automatically the best lens distortion model for four lenses of different focal length using different statistical information criterion without sacrificing a significantly lower error. The used method requires a simple chessboard pattern, which observed from different position and calibrated using Zhang method, to compute the complexity of the lens distortion model automatically. The result shows the 6 th order radial distortion model is the best model with the minimum error about-0.273 for lens of focal length 30.64 mm using MDL criteria, while at 4 th order the minimum error about-0.177, and at 2 nd order about-0.112.

Full text

International Journal of Computer Applications (0975 – 8887)
Volume 179 – No.40, May 2018
9
The Automatic Selection of Radial Distortion Models
Rihab K. Hamad
Optoelectronics Department/
College of Engineering
University of Technology/ Iraq
Baidaa Hamed
Physics Science Department/
College of Science
Mustansiriyah University/ Iraq
H. A. Hassonny
Control and system
Engineering Department/
College of Engineering
University of Technology/ Iraq
ABSTRACT
Several kinds of distortions exist in imaging systems which in
specific circumstances may affect an image’s geometry
without debilitating quality or diminishing the information
existing in the image. The most important type is the radial
distortion which represents high distortion accurately. Many
lens distortion models exist combined with several variations
where different techniques are used to calibrate each
distortion model. This paper presents an algorithm to select
automatically the best lens distortion model for four lenses of
different focal length using different statistical information
criterion without sacrificing a significantly lower error. The
used method requires a simple chessboard pattern, which
observed from different position and calibrated using Zhang
method, to compute the complexity of the lens distortion
model automatically.
The result shows the 6th order radial distortion model is the
best model with the minimum error about - 0.273 for lens of
focal length 30.64 mm using MDL criteria, while at 4th order
the minimum error about -0.177, and at 2nd order about -
0.112.
Keywords
Camera Calibration, Radial Distortion, Lenses
1. INTRODUCTION
Camera lens distortion is significant in a medium to wide
angle lenses. At present many distortion models and executing
are available for evaluating camera lens distortion choosing
the right model and executing could provide accurate result.
The simpler methods which are used by the computer vision
community have been developed due to the complexity of the
evaluation trials and advances in the use of computers for
analytical analysis [1, 2 and 3]. Numerous PC vision
calculations critically depended upon the supposition of a
linear pinhole camera, especially structure from motion
algorithms. Some of the proposed distortion calibration
methods consider an automatic distortion model selection
method [4, 5 and 6]. There are several kinds of lens distortion
but the most effective kind is the radial distortion. Which
bends straight lines into circular arcs, and it’s important in
higher quality cameras that introduce error into 3 dimension
reconstruction processes [7, 8]. Radial distortion may appear
as a barrel distortion, which arising usually at short focal
lengths, or pincushion distortion, which arising usually at
longer focal lengths.
2. RADIAL DISTORTION
It’s the main kind of lens distortion which formed by defects
in lens shape and displays itself only as radial positional error.
While the other types of distortion are usually created by
indecorous lens and camera gathering, produce both radial
and tangential errors in point positions. Radial distortion is a
linear motion for the image point radially from or to the image
center, due to the fact that objects at several angular spaces
from the lens axis suffer many amplifications and its lack in a
straight line transmission. It is important to indicate that radial
distortion is exceedingly connected with focal length, even if
it’s not modeled within intrinsic parameters of the camera [9,
10].
The following polynomial equation symbolizes the radial
distortion [11]:
r f(r)= r (1+ k1r2+ k2r4+ ..) (1)
Where k1, k2 are the distortion coefficients and r2 = x2 + y2.
The most commonly radial distortion models that have been
used are still in the polynomial form of (1) until recently.
The previous polynomial radial distortion models are used
with n-orders act as standards for evaluating the performance
of the radial distortion models automatically which will be
presented in results section.
3. DISTORTION MODEL SELECTION
When several competing models for a given system can
represent the distortion, the task of distortion model selection
would be good for choosing the finest model. Using the most
fitting and instructions model will give both better precision
and decreased computational model complexity. The model
with more degrees of freedom in most cases will fit the data
closer than other less complex models [5, 7]. Sum of common
and actual model selection criteria is commonly used in
different applications. These criterions shows a good perform
in computer vision applications.
In the following criteria the parameters of the equations as
follow: N refers to the number of samples, k is the parameters
number in the model, and SSE Refers to the sum-square-error
(SSE) computed as: SSE=
2
 Where 2
= -` is
the difference between the measured and estimated image
points.
3.1 Akaike Information Criterion (AIC):
AIC is the first criteria presented in statistics literature for
model selection. It has the following form:
AIC = N ln (SSE N) + 2 k (2)
The model that can depict any future information adequately
with a similar distribution is correct and chosen by AIC.
In other words, AIC selects the model that reduces the error of
another perception. And it can compare very different models
[8].
3.2 Bayesian Information Criterion (BIC):
BIC is presented by Schwarz in 1978; the principle of this
criterion is as following:

International Journal of Computer Applications (0975 – 8887)
Volume 179 – No.40, May 2018
10
Choose the model that maximizes the conditional probability
data. This data is constrained by some priori information
data. The BIC can take different structures depending on the
supposed priori data [8]. It has the following form:
BIC = N ln (SSE N) + k logN (3)
3.3 Consistent AIC (CAIC):
Bozdogan presented CAIC in 1987, it is an endeavor to
conquer the tendency of the AIC to overestimate the
complexity of the model [9]. It has the following form:
CAIC = N ln (SSE N) + k (logN + 1) (4)
3.4 Minimum Description Length (MDL):
MDL is presented by rissanen in 1978, the simplest
model that depicts the information adequately will be chosen
in MDL [10]. It has the following form:
MDL = N ln (SSE N) + 2 k logN (5)
the explained model selection criterions are shown in Table 1
Table 1. Model Selection Criterions
Model selection criterions
Formula
AIC
N ln (SSE N) + 2 k
BIC
N ln (SSE N) + k logN
CAIC
N ln (SSE N) + k (logN + 1)
MDL
N ln (SSE N) + 2 k logN
4. RESULTS
The presented algorithm shown in figure (1) consists of two
process calibration and model selection process. The
calibration was done for the three radial models (2nd, 4th and
6th order) and four lenses of different focal length 70 mm,
100mm, 150 mm, and190mm using Zhang method. A simple
chessboard pattern was observed from different position with
resolution of 1830×1330, and the model plane contained of
8×11=88 points corner. For each lens five different images
were used and calibrated. The equations (2, 3, 4 and 5) of the
criteria were applied to select the best model and show the
performance of the different radial distortion models. Figure
(2) shows one image for each lens.
Fig 1. The Presented Algorithm
Fig 2. Images captured with (a) 190mm, (b) 150mm, (c)
100mm, and (d) 70mm lens.
Tables (2, 3, and 4) show the calibration results for 2nd, 4th and
6th order radial distortion model after optimization for the
used lenses.
Table 2. Calibration result after optimization using 2nd
order distortion model for Several Lenses
different lenses
of focal length
(mm)
Radial distortion
coefficient
Pixel error
k1
u
v
30.64
0.25901
2.04594
1.91832
27.77
0.94332
3.46324
4.13295
26.31
0.52793
1.94436
2.96566
21.81
0.36694
4.43100
4.46265
Corner
detection
Camera parameters
estimation
Fc, cc, alpha_c, kc, err
Calibration
Error analysis
Camera parameters
estimation
Fc, cc, alpha_c, kc,
err
Apply Statistic
Information Criteria
AIC, BIC, CAIC,
MDL
Select the model
with minimum score
Print Error
Model Selection Process
Calibration Process

International Journal of Computer Applications (0975 – 8887)
Volume 179 – No.40, May 2018
11
Table 3: Calibration results after optimization using 4th
order distortion model for Several Lenses
different
lenses of
focal
length
(mm)
Radial distortion
coefficient
Pixel error
k1
k2
u
v
30.64
0.07210
-
1.49916
1.84158
2.07128
27.77
0.94949
-
1.86983
3.18991
3.97695
26.31
0.56427
8.14431
2.37465
2.23881
21.81
0.37225
0.31389
4.24701
4.44458
The complexity of 2nd order, 4th order and 6th order radial
distortion models are tested analytically using different model
selection criterions, to choose automatically the best distortion
model.
Table 4. Calibration result after optimization using 6th
order distortion model for Several Lenses
differen
t
lenses
of focal
length
(mm)
Radial distortion
coefficient
Pixel error
k1
k2
k3
u
v
30.64
0.2656
7
-
0.12787
0.0442
8
1.9329
7
1.9052
7
27.77
0.1702
1
-
0.09652
0.0297
0
2.0263
8
2.1845
5
26.31
0.6462
8
19.2709
9
-
0.0064
1.9503
8
2.4210
6
21.81
0.1376
0
0.45825
0.0422
1
2.4322
1
2.3575
2
Table 5. Complexity of four lenses using 2nd order radial
distortion model
Lenses of
focal length
(mm)
AIC
BIC
CAIC
MDL
30.64
4.839
0.936
3.936
-0.112
27.77
8.091
4.187
7.187
3.139
26.31
5.909
2.006
5.005
0.957
21.81
8.879
4.975
7.975
3.927
Table 6. Complexity of four lenses using 4th order radial
distortion model
Lenses of focal
length (mm)
AIC
BIC
CAIC
MDL
30.64
4.775
0.871
3.871
-0.177
27.77
7.799
3.895
6.895
2.847
26.31
5.597
1.694
4.694
0.646
21.81
8.764
4.860
7.860
3.812
Table 7. Complexity of four lenses using 6th order radial
distortion model
Lenses of focal
length (mm)
AIC
BIC
CAIC
MDL
30.64
4.677
0.774
3.774
-0.273
27.77
5.141
1.238
4.238
0.190
26.31
5.327
1.424
4.424
0.376
21.81
5.785
1.882
4.882
0.834
MDL is utilized over the other entire criteria in comparing the
radial distortion models and picking the best one, which it’s a
good estimate of the least complex model that will give a
model which model the distortion competently. This is
compatible with Broaddus opinion in [10], where MDL choose
the complexity equal to or less than that of the other criteria
without sacrificing a significantly lower error.
Fig 3. the complexity using MDL criterion of the 2nd and
6th order radial distortion model for four different lenses
(1) 30.64mm, (2) 27.77mm, (3) 26.31mm and (4) 21.81mm
5. CONCLUSION
A complete automatic selection for the camera lens distortion
was presented for the use with wide angle camera. Some of
information criteria have successful results in choosing the
distortion model complexity in automatic way.
Results show that the 6th order radial distortions model it’s the
best model to use which contain minimum error than 4th and 2nd
orders. The model complexity decreases gradually depending
on the focal length of the lens, as clarified in figure (3).
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