PE&RS May 2016 - page 326

assumption is ineffective. Hence, the radial distortion cor-
rection must be considered. Existing methods for distortion
correction can be divided into two categories. One is tradi-
tional correction algorithms, which the interior orientation
parameters and radial distortion coefficients are estimated
together (Tsai, 1987; Heikkila, 2000; Fraser, 2103; Habib and
Morgan, 2013). These methods have relatively high accuracy,
but they require a known pattern or image sequences, and the
processes of correction are also complex. Rather than using a
known template or images, plumb-line methods estimate lens
distortion directly based on distorted straight line segments
in images (Brown, 1971; van den Heuvel, 1999; Devernay and
Faugeras, 2001; Btauer-Burchardt and Voss, 2002; González-
Aguilera and Gómez-Lahoz, 2011). The plumb-line methods
are extremely useful for a single architectural image. How-
ever, some of these methods require the separating process
of different straight line segments and the average of results
(Brown, 1971; Devernay and Faugeras, 2001; González-Agu-
ilera and Gómez-Lahoz, 2011). Additionally, some algorithms
assume the distortion center as the image center (Btauer-
Burchardt and Voss, 2001; Alvarez
et al
., 2009), which is not
always possible. Some existing methods estimate the distor-
tion center and lens distortion simultaneously using an itera-
tive process. However, it has been demonstrated that these
methods may cause the unstable solutions (Brown, 1971;
Swaminathan and Nayar, 2000).
Actually, there is a lot of geometric information in man-
made scenes, such as straight lines and circles/ellipses. In this
paper, a new self-calibration approach with radial distortion
from a single image is proposed. The approach makes full use
of line segments and an ellipse in the image when the scene
contains straight lines and a circle/ellipse. The first contribu-
tion is to estimate the interior orientation parameters based on
multiple pairs of orthogonal vanishing points from line seg-
ments and an ellipse in the image. The second contribution
is to iteratively optimize the interior orientation parameters
and the radial distortion coefficient. In the process of radial
distortion correction, the distortion center is assumed to be
known. After the principal point is obtained in the process
of the interior orientation parameters calculation, the distor-
tion center is set as the current calculated principal point.
Then the above procedures are iteratively
continued until the principal point reaches
a stable solution.
Overview of the Approach
The overview of the proposed approach
is illustrated in Figure 1. There are three
main steps: radial distortion correction,
the interior orientation parameters calcu-
lation, and loop optimization. The data
for the method are line segments and an
ellipse extracted from the image. In the
first step, the radial distortion coefficients
along with vanishing points are estimated
using a least squares adjustment. The
adjustment model is established from
line segmented in the image. Combined
with these corrected vanishing points and
the ellipse, multiple pairs of orthogonal
vanishing points are then calculated from
the pole-polar relationship. The principal
distance and principal point are optimized
with these pairs of orthogonal vanishing
points. Last, the current computed princi-
pal point is used as the distortion center.
Then the above two steps are iteratively
continued until a stable solution of the
principal point is achieved.
The radial distortion correction and the interior orientation
parameters calculation are integrated. The interior orienta-
tion parameters calculation receives initial radial distortion
coefficient and vanishing points, and feeds back the refined
principal point and principal distance. Therefore, it helps
compensate for their disadvantages and improves the accu-
racy of the approach.
The Framework of the Approach
Preliminaries
Under a perspective transformation, a point (
X,Y, Z
) in object
space is projected to an image point (
x,y
) in image space by
the collinearity equation.
x x x c
r X X r Y Y r Z Z
r X X r Y Y
S
S
S
S
S
− + = −
(
)
+ −
(
)
+ −
(
)
(
)
+ −
(
)
0
11
12
13
31
32
+ −
(
)
− + = −
(
)
+ −
(
)
+ −
(
)
(
r Z Z
y y y c
r X X r Y Y r Z Z
r X X
S
S
S
S
S
33
0
21
22
23
31
)
+ −
(
)
+ −
(
)
r Y Y r Z Z
S
S
32
33
(1)
where, the principal distance
c
and the principal point (
x
0
,y
0
)
are the interior orientation parameters. The rotation matrix
R
r r r
r r r
r r r
=
11 12 13
21 22 23
31 32 33
is determined by three orientation angles
(
ω
,
φ
,
κ
). The perspective center (
X
S
,Y
S
, Z
S
) and the three
orientation angles (
ω
,
φ
,
κ
) which represent camera position
and orientation are the exterior orientation parameters. (
Δ
x
,
Δ
y
) are the terms which model the effects of symmetrical and
asymmetrical lens distortion in the image coordinate system.
In general, the first radial distortion coefficient is enough
to introduce error into geometric measurement for most mod-
ern standard camera lens. The distortion center is assumed to
be identical to the principal point. Thus, the distortion model
is expressed as
Figure 1. Overview of the approach.
326
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