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张正友标定原理原版 a flexible new technique for camera calibration
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关于摄像机定位中很重要的一个步骤:摄像机标定的一种常用的方法。由微软张正友提出
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A Flexible New Technique for Camera
Calibration
Zhengyou Zhang
December 2, 1998
(updated on December 14, 1998)
(updated on March 25, 1999)
(updated on Aug. 10, 2002; a typo in Appendix B)
(updated on Aug. 13, 2008; a typo in Section 3.3)
(last updated on Dec. 5, 2009; a typo in Section 2.4)
Technical Report
MSR-TR-98-71
Citation: Z. Zhang, “A flexible new technique for camera calibration”,
IEEE Transactions on Pattern Analysis and Machine Intelligence,
22(11):1330–1334, 2000.
Microsoft Research
Microsoft Corporation
One Microsoft Way
Redmond, WA 98052
zhang@microsoft.com
http://research.microsoft.com/˜zhang
A Flexible New Technique for Camera Calibration
Zhengyou Zhang
Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399, USA
zhang@microsoft.com http://research.microsoft.com/˜zhang
Contents
1 Motivations 2
2 Basic Equations 3
2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Homography between the model plane and its image . . . . . . . . . . . . . . . . . 4
2.3 Constraints on the intrinsic parameters . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.4 Geometric Interpretation
†
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Solving Camera Calibration 5
3.1 Closed-form solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2 Maximum likelihood estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.3 Dealing with radial distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Degenerate Configurations 8
5 Experimental Results 9
5.1 Computer Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5.2 Real Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.3 Sensitivity with Respect to Model Imprecision
‡
. . . . . . . . . . . . . . . . . . . . 14
5.3.1 Random noise in the model points . . . . . . . . . . . . . . . . . . . . . . . 14
5.3.2 Systematic non-planarity of the model pattern . . . . . . . . . . . . . . . . . 15
6 Conclusion 17
A Estimation of the Homography Between the Model Plane and its Image 17
B Extraction of the Intrinsic Parameters from Matrix B 18
C Approximating a 3 × 3 matrix by a Rotation Matrix 18
D Camera Calibration Under Known Pure Translation
§
19
†
added on December 14, 1998
‡
added on December 28, 1998; added results on systematic non-planarity on March 25, 1998
§
added on December 14, 1998, corrected (based on the comments from Andrew Zisserman) on January 7, 1999
1
A Flexible New Technique for Camera Calibration
Abstract
We propose a flexible new technique to easily calibrate a camera. It is well suited for use
without specialized knowledge of 3D geometry or computer vision. The technique only requires
the camera to observe a planar pattern shown at a few (at least two) different orientations. Either
the camera or the planar pattern can be freely moved. The motion need not be known. Radial lens
distortion is modeled. The proposed procedure consists of a closed-form solution, followed by a
nonlinear refinement based on the maximum likelihood criterion. Both computer simulation and
real data have been used to test the proposed technique, and very good results have been obtained.
Compared with classical techniques which use expensive equipment such as two or three orthog-
onal planes, the proposed technique is easy to use and flexible. It advances 3D computer vision
one step from laboratory environments to real world use.
Index Terms— Camera calibration, calibration from planes, 2D pattern, absolute conic, projective
mapping, lens distortion, closed-form solution, maximum likelihood estimation, flexible setup.
1 Motivations
Camera calibration is a necessary step in 3D computer vision in order to extract metric information
from 2D images. Much work h as been done, starting in the photogrammetry community (see [2,
4] to cite a few), and more recently in computer vision ([9, 8, 23, 7, 26, 24, 17, 6] to cite a few).
We can classify those techniques roughly into two categories: photogrammetric calibration and self-
calibration.
Photogrammetric calibration. Camera calibration is performed by observing a calibration object
whose geometry in 3-D space is known with very good precision. Calibration can be done very
efficiently [5]. The calibration object usually consists of two or three planes orthogonal to each
other. Sometimes, a plane undergoing a precisely known translation is also used [23]. These
approaches require an expensive calibration apparatus, and an elaborate setup.
Self-calibration. Techniques in this category do not use any calibration object. Just by moving a
camera in a static scene, the rigidity of the scene provides in general two constraints [17, 15]
on the cameras’ internal parameters from one camera displacement by using image informa-
tion alone. Therefore, if images are taken by the same camera with fixed internal parameters,
correspondences between three images are sufficient to recover both the internal and external
parameters which allow us to reconstruct 3-D structure up to a similarity [16, 13]. While this ap-
proach is very flexible, it is not yet mature [1]. Because there are many parameters to estimate,
we cannot always obtain reliable results.
Other techniques exist: vanishing points for orthogonal directions [3, 14], and calibration from pure
rotation [11, 21].
Our current research is focused on a desktop vision system (DVS) since the potential for using
DVSs is large. Cameras are becoming cheap and ubiquitous. A DVS aims at the general public,
who are not experts in computer vision. A typical computer user will perform vision tasks only from
time to time, so will not be willing to invest money for expensive equipment. Therefore, flexibility,
robustness and low cost are important. The camera calibration technique described in this paper was
developed with these considerations in mind.
2
The proposed technique only requires the camera to observe a planar pattern shown at a few (at
least two) different orientations. The pattern can be printed on a laser printer and attached to a “rea-
sonable” planar surface (e.g., a hard book cover). Either the camera or the planar pattern can be moved
by hand. The motion need not be known. The proposed approach lies between the photogrammet-
ric calibration and self-calibration, because we use 2D metric information rather than 3D or purely
implicit one. Both computer simulation and real data have been used to test the proposed technique,
and very good results have been obtained. Compared with classical techniques, the proposed tech-
nique is considerably more flexible. Compared with self-calibration, it gains considerable degree of
robustness. We believe the new technique advances 3D computer vision one step from laboratory
environments to the real world.
Note that Bill Triggs [22] recently developed a self-calibration technique from at least 5 views of
a planar scene. His technique is more flexible than ours, but has difficulty to initialize. Liebowitz and
Zisserman [14] described a technique of metric rectification for perspective images of planes using
metric information such as a known angle, two equal though unknown angles, and a known length
ratio. They also mentioned that calibration of the internal camera parameters is possible provided at
least three such rectified planes, although no experimental results were shown.
The paper is organized as follows. Section 2 describes the basic constraints from observing a
single plane. Section 3 describes the calibration procedure. We start with a closed-form solution,
followed by nonlinear optimization. Radial lens distortion is also modeled. Section 4 studies con-
figurations in which the proposed calibration technique fails. It is very easy to avoid such situations
in practice. Section 5 provides the experimental results. Both computer simulation and real data are
used to validate the proposed technique. In the Appendix, we provides a number of details, including
the techniques for estimating the homography between the model plane and its image.
2 Basic Equations
We examine the constraints on the camera’s intrinsic parameters provided by observing a single plane.
We start with the notation used in this paper.
2.1 Notation
A 2D point is denoted by m = [u, v]
T
. A 3D point is denoted by M = [X, Y, Z]
T
. We use
x to denote
the augmented vector by adding 1 as the last element:
m = [u, v, 1]
T
and
M = [X, Y, Z, 1]
T
. A camera
is modeled by the usual pinhole: the relationship between a 3D point M and its image projection m is
given by
s
m = A
R t
M , (1)
where s is an arbitrary scale factor, (R, t), called the extrinsic parameters, is the rotation and trans-
lation which relates the world coordinate system to the camera coordinate system, and A, called the
camera intrinsic matrix, is given by
A =
α γ u
0
0 β v
0
0 0 1
with (u
0
, v
0
) the coordinates of the principal point, α and β the scale factors in image u and v axes,
and γ the parameter describing the skewness of the two image axes.
We use the abbreviation A
−T
for (A
−1
)
T
or (A
T
)
−1
.
3
2.2 Homography between the model plane and its image
Without loss of generality, we assume the model plane is on Z = 0 of the world coordinate system.
Let’s denote the i
th
column of the rotation matrix R by r
i
. From (1), we have
s
u
v
1
= A
r
1
r
2
r
3
t
X
Y
0
1
= A
r
1
r
2
t
X
Y
1
.
By abuse of notation, we still use M to denote a point on the model plane, but M = [X, Y ]
T
since Z is
always equal to 0. In turn,
M = [X, Y, 1]
T
. Therefore, a model point M and its image m is related by a
homography H:
s
m = H
M with H = A
r
1
r
2
t
. (2)
As is clear, the 3 × 3 matrix H is defined up to a scale factor.
2.3 Constraints on the intrinsic parameters
Given an image of the model plane, an homography can be estimated (see Appendix A). Let’s denote
it by H =
h
1
h
2
h
3
. From (2), we have
h
1
h
2
h
3
= λA
r
1
r
2
t
,
where λ is an arbitrary scalar. Using the knowledge that r
1
and r
2
are orthonormal, we have
h
T
1
A
−T
A
−1
h
2
= 0 (3)
h
T
1
A
−T
A
−1
h
1
= h
T
2
A
−T
A
−1
h
2
. (4)
These are the two basic constraints on the intrinsic parameters, given one homography. Because a
homography has 8 degrees of freedom and there are 6 extrinsic parameters (3 for rotation and 3 for
translation), we can only obtain 2 constraints on the intrinsic parameters. Note that A
−T
A
−1
actually
describes the image of the absolute conic [16]. In the next subsection, we will give an geometric
interpretation.
2.4 Geometric Interpretation
We are now relating (3) and (4) to the absolute conic.
It is not difficult to verify that the model plane, under our convention, is described in the camera
coordinate system by the following equation:
r
3
r
T
3
t
T
x
y
z
w
= 0 ,
where w = 0 for points at infinity and w = 1 otherwise. This plane intersects the plane at infinity at
a line, and we can easily see that
r
1
0
and
r
2
0
are two particular points on that line. Any point on it
4
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资源评论
- xhjhoney2013-09-05是张正有的英文原版,没错。
- daihao_zhuzhu2014-04-25这个资源直接共享就好了,还要分,不好不好
- 龚1正2018-11-21英文论文,这个资源直接共享就好了,还要分,不好不好
- liaofengyi2013-10-30与yanmo的一样,重复了
- baidu_279036892015-06-16东西是好的 能共享就更好了
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