Camera Calibration Based on 2D-plane - Academy Publisher

ISBN 978-952-5726-11-4
Proceedings of the Third International Symposium on Electronic Commerce and Security Workshops(ISECS ’10)
Guangzhou, P. R. China, 29-31,July 2010, pp. 365-368
<strong>Camera</strong> <strong>Calibration</strong> <strong>Based</strong> on 2D-plane
Guoquan Jiang 1 , Cuijun Zhao 2
1
School of Computer Science and Technology, Henan Polytechnic University, Jiaozuo, China
jiangguoquan@hpu.edu.cn
2
School of Resources and Environment Engineering, Henan Polytechnic University, Jiaozuo, China
zhaocuijun@hpu.edu.cn
Abstract—One goal of machine vision is to understand the
visible world by inferring 3D properties from 2D images.
<strong>Camera</strong> calibration is a process which models the
relationship between the 2D images and the 3D world. A
camera calibration method based on 2D-plane is used. In
the process of calibration, 25 images are shot from different
position. Use these images to obtain the intrinsic
parameters. Then a image is shot in a special position and
the extrinsic parameters are computed. Experimental
results show the method is accurate and robust.
Index Terms—camera calibration, intrinsic parameters,
extrinsic parameters
I. INTRODUCTION
<strong>Camera</strong> calibration is the first step towards
computational computer vision. <strong>Camera</strong> calibration is
divided into two phases. First, camera modeling deals
with the mathematical approximation of the physical and
optical behavior of the sensor by using a set of
parameters. The second phase of camera calibration deals
with the use of direct or iterative methods to estimate the
values of these parameters [1]. In general, camera
calibration can be classified two methods according to the
modeling of lens distortion. That is linear method and
non-linear camera calibration method [2, 3].
This calibration method uses a camera where its
position is fixed, move the plane in front of the camera
template (calibration reference), and take different
positions template image plane. For each position of the
images, extract the image on the four grid corner. The
corresponding corner relationship between the plane and
the image defined homography, n images from different
locations can be obtained n homography [4].
II. CAMERA CALIBRATION BASED ON 2D-PLANE
A. Mapping relationship between target plane and the
image plane
Three-dimensional points on target plane denotes
T
M = x, y,
z , two-dimensional point on the image
[ ]
plane denotes m
T
[ u,
v]
homogeneous coordinates is M [ x y z ]
m [ u v ]
= , The corresponding
= , , , 1 T
and
= , , 1 T
. <strong>Camera</strong> model is based on Pinhole
imaging model, the projective relationship between space
and image points is[5]
sm = A[ R T] M
(1)
S is the random non-vanishing scale factor, R、T is
the exterior parameter matrix, A is the Materials for
internal reference number matrix.
⎡ax
γ u0
⎤
A =
⎢
0 ay
v
⎥
⎢ 0 ⎥
(2)
⎢⎣
0 0 1 ⎥⎦
And, ( u0, v0)
is principal point picture element
coordinate, ax、 ay
respectively is the u、 v axis scale
factor, γ is the u、 v axis not vertical factor. Does not
lose the generality (guarantee revolving matrix
orthogonality), it may be supposed that plane template
located in world coordinate system's xy plane, namely.
Records the revolving matrix to list as, has
⎡x⎤
⎡u⎤
⎢ x
y
⎥
⎡ ⎤
s
⎢
v
⎥
A[ r1 r2 r3 t] ⎢ ⎥ A[ r1 r2
t
⎢
] y
⎥
⎢ ⎥
= =
⎢0⎥
⎢ ⎥
⎢⎣ 1⎥⎦
⎢ ⎥
⎢1⎥
1
⎣ ⎦
⎣ ⎦
(3)
T
Here M = [ x,
y]
, M = [ x, y, 1]
T
. Then between
point M and the corresponding image point has a matrix
transformation H:
And, H λK[ r1 r2
t]
factor. H = [ h1 h2 h3]
, then
[ h h h ] = λ A[ r r t]
sm
= HM
(4)
= is a 3×3 matrix, λ is constant
1 2 3 1 2
And, translation vector t is from world coordinate
system zero point to optical center vector; r 1
, r 2
is the
image plane two coordinate axes in the world coordinate
system's direction vector, obviously t will not be located
at the r 1
, r 2
plane, as a result of r 1
, r 2
orthogonal,
therefore, det([ r1, r2, t]) ≠ 0 , also det[ A]
≠ 0 ,
therefore det[ H ] ≠ 0 .
The computation of H causes between the actual image
coordinate m
i
and the image coordinate which according
to type (4) calculates the diverse smallest process. The
objective function is
2
min∑ m ˆ
i
− mi
(5)
i
© 2010 ACADEMY PUBLISHER
AP-PROC-CS-10CN008
365

B. Solve camera parameter matrix
The solution of the camera parameter matrix can be
seen Literature[6].
III. EXPERIMENT AND RESULTS
A. Experiment materials
KOKO camera; Image gathering card; Computer;
Tripod; 7×9 the black and white interaction's chess
discoid grid 2D plane target (as shown in Figure 1), each
grid size for 28×28 millimeter; Horizontal iron sheet
Figure 2. <strong>Calibration</strong> with the 25 images
Figure 1. 2D plane for camera calibration
B. Experiment procedure
1) Prints a calibration template to paste on the
horizontal iron sheet;
2) moves plane or camera to shot some template
images (more than or equal 20)from different angle;
3) detects characteristic point of the image;
4) obtains each image the unitary matrix H ;
5) computes camera's internal parameter by using
matrix H extracted in the premise of the distortion factor
being zero;
6) obtains a group of precision higher camera's
internal parameter by Further optimizing using the
counter-projection, simultaneously calculating each
distortion factor.
C. Experimental results
1) Lens' internal parameter
As shown in Figure 2, obtains internal parameter by
Carrying on the demarcation to the load 25 charts. Figure
5-8 shows the spatial distribution situation for the 25
images.
The intrinsic parameters after camera calibration is
Focal Length:
f = c
[3419.27498 3260.81444 ]
±[88.56950 99.83080]
Principle Point:
cc = [383.50000 287.50000]
± [0.00000 0.00000]
Skew:
alpha_c = [0.00000] ± [0.00000]
Figure 3. The spatial distribution for the 25 images
Distortation:
kc = [-0.41925 -46.21000 -0.00466 -0.02250 0.00000]
± [0.77140 78.34200 0.01162 0.00423 0.00000]
Pixel error:
err = [0.36731 0.79805]
2) Computation of the extrinsic parameters
We put the plane pattern on the ground and shoot a
image. Use this image to compute the extrinsic
parameters.
Extrinsic parameters:
Translation vector:
⎡-77.629688
⎤
T =
⎢
-98.708815
⎥
⎢ ⎥
⎢⎣
1755.086388⎥⎦
Rotation vector:
⎡ 0.019894 0.999776 -0.007244⎤
R =
⎢
0.433706 -0.015158 -0.900927
⎥
⎢ ⎥
⎢⎣
-0.900835 0.014781 -0.433911⎥⎦
Pixel error:
err = [0.32229 0.69439]
3) Experiment test
366

TABLE I.
THE COORDINATE OF MARKED POINTS AND THEIR CALIBRATION ERROR
Marked points
Image coordinate
Real world coordinate
Computational world
coordinate
u( pixels ) v( pixels ) x( cm ) ycm ( ) xˆ( cm ) ycm ˆ( )
x direction(
cm
)
error
y direction
( cm )
1 26 281 23.5 -9.5 21.74 -8.90 -1.76 0.60
2 47 354 32 -7.5 29.39 -7.42 -2.61 0.08
3 50 126 3 -10 3.00 -9.08 0.00 0.92
4 60 195 13.5 -8.5 11.85 -8.00 -1.65 0.50
5 81 496 46.5 -5 42.50 -5.17 -4.00 -0.17
6 104 76 -4.5 -7.5 -3.86 -6.73 0.64 0.77
7 183 258 20.5 -1.5 19.44 -1.85 -1.06 -0.35
8 208 163 9 -1 8.12 -1.00 -0.88 0.00
9 231 103 0 0 0.20 -0.03 0.20 -0.03
10 235 402 37 1.5 34.32 0.82 -2.68 -0.68
11 366 267 22 7.5 20.74 6.55 -1.26 -0.95
12 367 435 40 7.5 37.55 6.34 -2.45 -1.16
13 367 62 -6.5 7.5 -5.37 7.00 1.13 -0.50
14 370 206 14.5 8 13.72 6.85 -0.78 -1.15
15 373 122 3 8 3.06 7.18 0.06 -0.82
16 375 559 52 7.5 48.02 6.48 -3.98 -1.02
17 431 346 31 10.5 29.16 9.25 -1.84 -1.25
18 535 238 18 16 17.74 14.46 -0.26 -1.54
19 544 96 -1 17.5 -1.74 16.00 -0.74 -1.50
20 552 395 36 16 34.11 14.21 -1.89 -1.79
21 588 166 9.5 19.5 9.18 17.57 -0.32 -1.93
22 689 82 -3 26 -1.86 23.61 1.14 -2.39
23 721 430 40 23.5 37.58 20.98 -2.42 -2.52
24 723 295 25 25 24.34 22.51 -0.66 -2.49
25 725 515 48 23 44.92 20.34 -3.08 -2.66
26 730 164 9 27 9.18 24.51 0.18 -2.49
In order to verify the calibration algorithm’s accuracy,
we keep the camera’s position and orientation is
unchangeable. The world coordinate system
establishment is as the same as the Z. Zhang [6.]. 26
marked points are pasted on the laboratory ground. Figure
4 is the primary image shot by the camera. All the
marking points’ world coordinate can be obtained by
measuring, and the corresponding image coordinates can
be got by image processing. Fig. 5 show the centroid
coordinates of the marked point. We first use the image
coordinate of the 26 points to reconstruct their world
coordinate, and then compare them to the real world
coordinate. Their corresponding error is the experimental
error. The error can be evaluated by the following
expression:
m
2 2
( ∑ ( x ˆ ) ( ˆ
i
− xi + yi − yi)
i=
1
Qk
=
(6)
m
where , ( xi, y
i)
, ( i = 1,2, , m)
is the real world
coordinate of the marked point, ( xˆ, y ˆ)
, ( i = 1,2, , m)
is
the computed world coordinate by calibration method,
m is the number of marked points.
The experimental results can be seen from table 1.
The overall error of world coordinates is:
Figure 3. Marked points image
Figure 4. Extracted centroid of marked points
367

26
∑
( ( x − xˆ
) + ( y − yˆ
)
i=
1
Qk
=
26
= 2.0881( cm)
2 2
i i i i
CONCLUSIONS:
In this paper, a camera calibration method based on
2D-plane is adopted. It can obtain the intrinsic parameters
by move the 2D-plane orientations. The extrinsic
parameters can be computed by the specific orientation.
Experimental result shows that this camera calibration
method can quickly and accurately solve the internal and
external camera parameters.
ACKNOWLEDGMENT
This research is supported by the Doctoral Fund of
Henan Polytechnic University (B2010-27).
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