The Projective Equation of a Circle and Its Application in Camera Calibration
Yinqiang Zheng and Yuncai Liu
Image Processing and Pattern Recognition Institute, Shanghai Jiao Tong University, P.R.China
Email: [email protected]
Abstract
In this article, we present the projective equation of a
circle in a perspective view, which naturally encodes such
important geometric entities as the projected circle center,
the vanishing point of the normal direction of the circle’s
supporting plane and the degenerate conic envelope
spanned by the image of circular points (ICPs). Based on
this projective equation, we propose an easy technique to
calibrate the focal length and the extrinsic parameters of
a camera merely by using one perspective view of two
arbitrary coplanar circles. Unlike existing optimization
algorithm, our method offers a closed form solution
through simple matrix manipulation. Experimental results
verify the correctness and efficiency of our proposed
technique.
1. Introduction
A variety of circle based calibration patterns, including
one circle with a bundle of lines passing through the
circle center, two concentric circles, two parallel circles
and N ≥ 3 coplanar circles, have been successfully used
in camera calibration [1,2].
In spite of the widespread application of circles, we do
not yet have a unified equation to represent the projection
of a circle in a perspective view. Inspired by the
projective equation of a sphere [3], which has
dramatically contributed to the application of spherical
features in camera calibration [4], we present the
projective equation of a circle in this work. We find that
such important geometric entities as the projected circle
center and the vanishing point of the normal direction of
the circle’s supporting plane are encoded in this
projective equation. We have already known that the ICPs
are quite essential in camera calibration [2,5]. It is
exciting to see that the degenerate conic envelope
spanned by the ICPs is included in this projective
equation as well. Existing circle based calibration
algorithms can benefit from this equation by either
simplifying their procedures or improving their
intelligibility.
In this paper, we use this projective equation to
determine the focal length and the extrinsic parameters of
a camera merely by using one perspective view of two
arbitrary coplanar circles. In many real scenarios, the pose
978-1-4244-2175-6/08/$25.00 ©2008 IEEE
of a camera is often changed and its focal length is
usually adjusted accordingly, while the remaining
intrinsic parameters, including the aspect ratio, the skew
factor and the principle point, keep almost unchanged [6].
Therefore, it is valuable to develop an easy technique to
fully determine the extrinsic parameters in addition to the
focal length of a camera, eliminating the inefficiency to
calibrate all the intrinsic parameters repeatedly by using
at least three images. Although Chen et al. [1] tackled this
problem, they merely proposed a method on the basis of
iterative optimization, whose convergence tightly depends
on proper initialization. Unfortunately, they did not show
how to initialize their algorithm. In contrast, our method
offers a closed form solution for this problem, whose
results can be directly used in real vision applications
with moderate noise levels or used to initialize Chen’s
algorithm for higher accuracy and speed. Extensive
experiments demonstrate the correctness and efficiency of
our proposed method.
2. The projective equation of a circle in one
perspective view
As shown in Fig.1, we assume the circle
Q lies on the
plane of z = 0 in the world coordinate system and the
circle center coincides with the origin. The rigid motion
between the camera and the world coordinate system is
denoted by R t . Therefore, the planar homography
[
]
matrix H between the image plane and the circle’s
supporting plane satisfies: H = K [ r1 r2 t ] , where
K is the intrinsic camera parameter matrix,
and rn , n = 1,2 , is the n-th column of R . Obviously, the
dual of the circle can be expressed in matrix form
*
−1
as: Q = Q
circle radius.
= diag{1,1,−1 / r 2 } , where r denotes the
*
According to [5], the dual circle image C satisfies:
kC * = HQ * H T
= K [r1
r2
t ]diag {1 1 − 1 / r 2 }[r1
= KK T − ( Kr3 )( Kr3 ) T −
r2
t] K T
1
( Kt )( Kt ) T ,
2
r
(1)
T
where k is an unknown scale factor and
r3 is the third
Zw
column of R .
Q1
Q2
O
X
t1 w(1)
t
O
2
2
l
C1
C2
Zw
π
Q
Y
L
Z
Ow
Xw
Yw
Y
Z
O
X
C
Fig.2 The projection of two coplanar circles
Y
X
O
3. Camera calibration
Fig.1. Basic projective model of a planar circle
Equation (1) is referred to as ‘the projective equation
of a circle’. Obviously, it naturally encodes the intrinsic
camera parameter matrix K and the circle pose
parameters (i.e. r3 and t ), promising its usefulness in both
intrinsic and extrinsic camera calibration. Furthermore,
according to [5], Kr3 denotes the vanishing point of the
normal
direction
of
the
supporting
plane
and
( Kt ) / r denotes the homogeneous coordinates of the
projected circle center, both of which are encoded in the
projective equation.
2.1. The degenerate conic envelope spanned by
the ICPs
The ICPs are very important in computer vision. In the
following, we shall show that the ICPs are implicitly
encoded in the projective equation.
Let
C *∞ denote the degenerate conic envelope spanned
*
by the two circular points, thus C ∞
= diag{1,1,0} . Then,
the degenerate conic envelop spanned by the ICPs,
C ∞'* here, can be computed by [5]:
C ∞'* ≡ HC ∞* H T
denoted by
= K [r1
= KK
T
r2 t ]diag{1,1,0}[r1 r2
T
3 3
t] K
T
T
(2)
'*
∞ is
incorporated in the projective
− Kr r K .
We can see that
C
T
equation as well. In addition, if the ICPs ( i, j ) are
denoted by the normalized homogeneous coordinate, i.e.
the third element equals to one, the degenerate conic
C∞'* can also be calculated by
C∞'* = (1 − r332 )(ij T + ji T ) / 2 ,
where r33 is the third element of r3 .
envelope
(3)
Now we utilize this projective equation to readily
determine the focal length and extrinsic parameters of a
zoom length camera merely using one image of two
arbitrary coplanar circles (see Fig.2). We assume the
aspect ratio, principle point and skew factor have already
been calibrated by using three or more images.
Specifically,
the
intrinsic
camera
parameter
matrix K = Adiag{ f , f ,1} , where A is known
and
f is the unknown camera focal length [5].
3.1. Simultaneously identifying the ICPs and
estimating the focal length f .
Generally, the two circle images C1 and C 2 have two
pairs of points in common, one pair of which is the ICPs
[2]. Since the ICPs lie on the image of absolute conic
(IAC),
whose
equation
is ω = K K = A diag{1 /
following equations hold
−T
−1
−T
f 2 ,1 / f 2 ,1} A−1 , the
iT ωi = 0 , and j T ωj = 0 ,
(4)
2
f .
Since the focal length f is positive, i.e. f > 0 , we
both of which are linear on
can identify the ICPs among the two point-pairs and
determine the focal length f simultaneously by solving
eq.(4).
3.2. Determining the rotation matrix R
As shown in Fig.2, we define a specific world
coordinate system ow − xw yw z w , whose origin o w
coincides with the circle center O1 of the left circle Q1 .
Its z - axis coincides with the normal direction of the
supporting plane and its x - axis is consistent with the
symmetric axis of the two circles O1O2 , denoted by L in
Fig.2. The extrinsic parameter is denoted by
where R
= [r1
r2
[R t ] ,
r3 ]. To eliminate uncertainty about
the direction of the world coordinate system, we assume
(1 0 0) ⋅ r1 ≥ 0 , and (0 0 1) ⋅ r3 ≥ 0 . (5)
In such coordinate configuration,
'*
where R1 and R2 denote the circle radii, and C∞ ,
representing the degenerate conic envelope spanned by
the ICPs, can be calculated from eq. (3).
p2
l
Kr3 represents the
homogeneous vanishing point of the normal direction of
the supporting plane, and Kr1 denotes the homogeneous
i
vanishing point of the symmetric axis L .
After the ICPs ( i, j ) are uniquely identified, we can
determine the vanishing line
l∞ by l∞ = i × j . Then, the
vanishing point Kr3 can be estimated by the pole-polar
constraints [5], i.e.
C1
Kr3 ∝ ω *l∞ , where ω * = KK T is
the dual image of the absolute conic (DIAC). Thus, r3 can
be estimated by
r3 = ± K −1ω *l ∞ / K −1ω *l∞ ,
Since the vanishing point
(6)
Kr1 is the intersection
l∞ and the image of the
symmetric axis l (see Fig.3), we should first identify l .
Let p1 and p2 be the projected circle centers
of C1 and C 2 , respectively. According to the pole-polar
between the vanishing line
constraint between the projected circle center and the
vanishing line with respect to the dual circle image [5],
we get:
p1 = Kt1 ∝ C1*l∞ , and p2 = Kt 2 ∝ C 2*l∞ ,
(7)
where t1 and t2 are the coordinates of the two circle
centers O1 and O2 in the camera coordinate system,
respectively. Thus, the image of the symmetric axis l
satisfies: l = p1 × p2 . Then, we get Kr1 ∝ l × l∞ , which
gives
r1 = ± K −1 (l × l∞ ) / K −1 (l × l∞ ) ,
(8)
By imposing the constraints from eq. (5) on eq. (6) and
(8), respectively, the sign of r3 and r1 can be uniquely
determined. Finally, the rotation matrix can be calculated
by R = [ r1 r3 × r1 r3 ] .
3.3. Determining the translation vector t
According to eq. (1), we get two similar equations:
⎧⎪k1C1* = C ∞'* − ( Kt1 / R1 )( Kt1 / R1 ) T
,
⎨
⎪⎩k 2 C 2* = C ∞'* − ( Kt 2 / R2 )( Kt 2 / R2 ) T
(9)
p1 j
l
q
C2
Fig.3 Two perspective ellipses on the image plane
k1 is the generalized eigen-value
'*
*
'*
*
of (C∞ , C1 ) so that the rank of C∞ − k1C1 equals to one.
Likewise, the parameter k2 can be determined.
Obviously,
Subsequently, eq. (9) can be rewritten as
⎧⎪(t1 / R 1 )(t1 / R 1 ) T = K −1 (C∞'* − k1C1* ) K −T
, (10)
⎨
⎪⎩(t 2 / R 2 )(t 2 / R 2 ) T = K −1 (C ∞'* − k 2 C 2* ) K −T
from which t1 / R1 and t 2 / R2 can be readily estimated
by matrix decomposition or a least square method. Note
that we can not fully determine the translation t unless
metric information about the two circles is known.
4. Experimental results
4.1. Synthetic data
The simulated camera has the following properties: the
image resolution is 1280*960, and the focal length equals
to 10 mm. We generate two separate circles by computer.
The world coordinate system, as defined in section 3.2,
relates to the camera coordinate system by a rigid body
motion, where the rotation axis is ψ 1
= 3 / 3(1,1,1)T ,
the rotation angle θ1 = π / 10 and the translation vector
t1=(20,10,1800)T (unit in millimeters). The two circles,
both with a radius of 200 mm, are 420 mm far away. In
this experiment, we test the performance of our algorithm
under varying noise levels. Gaussian noise with 0 mean
and σ standard deviation is added to the projected circles.
The estimated parameters are then compared with the
ground truth. We measure the relative error
of f , t1 and t 2 , and the absolute error of r1 , r2 and r3 ,
which is defined by their respective deviation angle (in
degrees) from their corresponding ground truth. We vary
the noise level from 0.2 pixels to 2 pixels. For each noise
level, we perform 100 independent trails and show the
average results. Fig.4 illustrates the results.
5
application of our proposed method to eliminate
perspective distortion.
4.5
t1
t2
f
4.5
r1
r2
r3
4
4
Absolute errors (degrees)
3.5
Relative errors (%)
3.5
3
2.5
2
1.5
3
2.5
2
1.5
1
1
0.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Noise levels δ (pixels)
1.6
1.8
2
0.5
0.2
0.4
(a)
0.6
0.8
1
1.2
1.4
Noise levels δ (pixels)
1.6
1.8
2
(b)
Fig.4. Performance vs. the noise levels. (a) Relative
errors of f , t1 and t 2 . (b) Absolute errors of r1 , r2
Fig. 6. Real application. (left) Original image. (right)
Image without perspective distortion.
and r3 .
5. Conclusions
Fig.5. Calibration pattern. (a) Two separate circles on
the checker pattern. (b) The extracted corner points
(red) and perspective ellipses (green).
In this article, we originally present the projective
equation of a circle and the closed form representation of
the degenerate conic envelope spanned by the ICPs.
Based on this projective equation, we propose a closed
form solution for camera calibration using one
perspective view of two arbitrary coplanar circles.
Compared with the iterative optimization technique, our
method is computationally efficient. In the future, we
shall extend the application of this projective equation.
4.2. Real images
6. References
The images are captured using a SCOR-14SOM
camera with 1280*960 image resolution. Two separate
circles, both with a radius of 45 mm, are attached on a
checker pattern and used as calibration pattern (see
Fig.5(a)). The distance between the two circle centers is
115mm. We first calibrate the camera by Zhang’s method
[7] and estimate the extrinsic parameters by computing
the homography matrix from the projected corner points
(see Fig.5 (b)), which are used as reference. Then, we
estimate the focal length and the extrinsic parameters
from our proposed method and those from Chen’s method
[1] when it is randomly initialized (Chen). We also
initialize Chen’s method by our estimated results to show
the improvement when it is properly initialized (Chen and
Ours). Table 1 shows the comparative results. We can see
that our results are acceptable in terms of accuracy. When
Chen’s method is initialized by our estimated results, its
performance can be further improved. Fig. 6 illustrates an
[1] Q. Chen, H. Wu and T. Wada, Camera Calibration with
Two Arbitrary Coplanar Circles, Proc. ECCV, pp. 521-532,
2004.
[2] P. Gurdjos, P. Sturm and Y.H. Wu, Euclidean Structure
from N ≥ 2 Parallel Circles: Theory and Algorithms,
Proc. ECCV, part I, pp. 238-252, 2006.
[3] M. Agrawal and L.S. Davis, Camera Calibration Using
Spheres: A Semi-Definite Programming Approach, Proc.
CVPR, pp. 782-789, 2003.
[4] X. Ying and H. Zha, “Geometric Interpretations of the
Relation between the Image of the Absolute Conic and
Sphere Images”, Trans. on PAMI, 28(12):2031-2036, 2006.
[5] R.Hartley and A. Zisserman, Multiple View Geometry in
Computer Vision, Cambridge Univ. Press, 2003.
[6] M.X. Li and J.M. Lavest, Some Aspects of Zoom Lens
Camera Calibration, Trans. on PAMI, 18(11):1105-1110,
1996.
[7] Z. Zhang, A Flexible New Technique for Camera
Calibration, Trans. on PAMI, 22(11): 1330-1334, 2000.
(a)
(b)
Table 1 Comparison of the estimated results from different methods
24.691
Err.
(%)
-
-27.801
-29.484
Our method
23.945
3.001
-27.617
-29.329
Chen
23.456
5.027
-25.314
-30.129
1637.247
4.608
-0.204
0.471
0.858
4.315
Chen and Ours
24.245
1.806
-27.931
-29.804
1706.421
0.577
-0.156
0.477
0.865
3.460
Approach
f
Zhang(ground truth)
1716.319
Err.
(%)
-
-0.157
0.423
0.892
Err.
(degree)
-
1661.124
3.215
-0.142
0.339
0.930
5.354
t1
r3