Sensors 2019, 19, 3166 5 of 21
Define
α
x
= f /dX
, which represent the ratio of focal length f to the physical dimension of a pixel
in the u-axis direction. Define
α
y
= f /dY
, which represent the ratio of focal length f to the physical
dimension of a pixel in the v-axis direction. Then, the above formula can be equivalent to:
Z
c
u
v
1
=
α
x
0 u
0
0
0 α
y
v
0
0
0 0 1 0
"
R t
0
T
1
#
X
w
Y
w
Z
w
1
= M
1
M
2
X
w
Y
w
Z
w
1
(2)
M
1
is a 3
×
4 matrix determined by parameters such as
α
x
,
α
y
, u
0
and v
0
. These four parameters
are related to the internal structure of the camera called the internal parameters of the camera. The
rotation matrix R and the translation vector t are determined by the orientation relationship between
the camera and the world coordinates. Therefore, M
2
is called the external parameters of the camera.
The internal and external parameters of the camera were obtained through multiple experiments and
calculations, and the process of camera calibration was completed.
2.2. Image Distortion Removal
In recent years, vehicle-mounted cameras often used wide-angle lens to obtain an enlarged
field of vision, and the image distortion changes appeared remarkable. Image distortion is generally
divided into radial and tangential distortions. Compared with the actual road scene, the common
edge distortion of lanes, vehicles and backgrounds is called radial distortion. If the camera lens is not
parallel to the image plane of the visual sensor, the collected image will be tilted, thus making the lane,
vehicle and background look closer or farther than in the actual three-dimensional world. This type
of image distortion is called tangential distortion, which is more disadvantageous to reflect the real
driving environment.
The mathematical model of radial distortion is defined as follows:
(
u
0
= u(1 + k
1
r
2
+ k
2
r
4
+ k
3
r
6
)
v
0
= v(1 + k
1
r
2
+ k
2
r
4
+ k
3
r
6
)
(3)
where, r
2
= x
2
+ y
2
, (u
0
,v
0
) is the coordinate of point d in the Image Pixel Coordinate.
The mathematical model of tangential distortion is presented as follows:
u
0
= u +
h
2p
1
v + p
2
(r
2
+ 2u
2
)
i
v
0
= v +
h
p
1
(r
2
+ 2v
2
) + 2p
2
u
i
(4)
Equations (3) and (4) demonstrate that image distortion is related to five parameters—k
1
, k
2
,
k
3
, p
1
and p
2
—which are collectively called distortion coefficients. By calibrating the camera and
acquiring the distortion coefficient, the distorted image can be corrected. In this study, the checkerboard
calibration plate was used to remove image distortion. The checkerboard was composed of black and
white grids with regular shapes to easily find the coordinates of corner points and observe the image
correction conveniently. After determining the coordinates of the corners, a transformation matrix was
created to map the distorted points to the undistorted points, and image distortion was removed by
using coordinate transformation. Figure 3 shows the correction of the distorted image, in which the
left and right halves are the original and undistorted images, respectively.
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