Badly calibrated camera in ego-motion estimation -- propagation of uncertainty
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October 8, 1997
Reference
Czech Technical University, Faculty of Electrical Engineering Department of Control Engineering, Center for Machine Perception 121 35 Prague 2, Karlovo namest 13, Czech Republic fax +420 2 24357385, phone +420 2 24357465, http://cmp.felk.cvut.cz Get this publication from ftp://cmp.felk.cvut.cz/pub/cmp/articles/svoboda/svcaip97.ps.gz .
?
This research was supported by Region Rh^ne Alpes with the program TEMPRA o { East Europe between l'INPG Grenoble and the Czech Technical University, the Czech Ministry of Education grant No. VS 96049, grants 102/97/0480, 102/97/0855 and 102/95/1378 of the Grant Agency of the Czech Republic and European Union grant Copernicus CP941068, and the grant of the Czech Technical University No 3097472.
Badly Calibrated Camera in Ego-motion Estimation, Propagation of Uncertainty
Tomas Svoboda Peter Sturm
svoboda@cmp.felk.cvut.cz
Tomas Svoboda and Peter Sturm. A badly calibrated camera in ego-motion estimation, propagation of uncertainty. In Gerald Sommer, Kostas Daniilidis, and Josef Pauli, editors, 7-th International Conference Computer Analysis of Images and Patterns, pages 183{190. Springer Verlag in the LNCS series, September 1997 and V are orthonormal matrices and D is the diagonal matrix containing singular values of E . The rotation matrix R and the matrix S can be computed as S = V ZV T ; R = UY V T or UY T V T ; (4) where Z and Y are known constant matrices 2]. The Euler angles characterizing the rotation can be recovered from R using a relationship from 5]. Localized correspondences are expressed in pixel coordinates, q. However to estimate motion, normalized image coordinates u is needed, see equation (2). Assume the classical pinhole camera model. Introducing a calibration matrix K we can transform pixel coordinates q into normalized image coordinates u:
1
Abstract. This paper deals with the ego-motion estimation (motion of the camera) from two views. To estimate an ego-motion we have to nd correspondences and we need a calibrated camera. In this paper we solve the problem how to propagate known camera calibration errors into the uncertainty of the motion parameters. We present a linear estimate of the uncertainty in ego-motion based on the uncertainty in the calibration parameters. We show that the linear estimate is very stable.
. u0 T SRu = u0T E u = 0;
op tic al ax is
l sys
tem
f
(2)
where E is the essential matrix. The equations above was proposed by Longuett Higgins in 6]. The essential matrix E can be reversely decomposed into the motion parameters using method e.g. by Hartley 2]. If E is estimated correctly, it has rank 2 and its nonzero singular values are equal 6]. Using the singular value decomposition we can factorize E as
1 Introduction
Let us assume to have two images of a static scene captured by the same camera from two di erent viewpoints. Having at least eight corresponding points and knowing the calibration parameters of the camera, the camera motion can be estimated up to a similarity using a linear algorithm. The similarity reconstruction of the scene can also be done. This problem has been solved over years 6, 11]. To calibrate the camera, some of many methods developed for the o -line camera calibration as Tsai's method 10], can be used. In this paper we deal with two views and an innacurately calibrated camera. We use a linear method intended for a calibrated camera to estimate the egomotion. Because of noise in the estimation process, an error analysis is needed. There are two sources of errors: (a) noise in correspondences and (b) noise in the calibration parameters. Weng 11] studied the in uence of the noise on the motion parameters. Florou and Mohr 1] used the statistic approach to study reconstruction errors with respect to the calibration parameters. In our paper, we present a linear algorithm to estimate a credibility of the motion parameters, based on the uncertainty in the calibration parameters.
2 Fundamentals
2.1 Ego-Motion from Point Correspondences
Let the motion between two positions of the camera be desribed by a rotation matrix R and a translation vector t, and let u and u0 be normalized image coordinates of corresponding points, see Figure 1, then the coplanarity constraint can be written as: u0T (t Ru) = 0: (1) Introducing the antisymmetric matrix S containing elements of the translation vector t, we can rewrite the coplanarity constraint above as
y_w
z_w
M x_w W
M
X
u q_u θ
q_v
c R
y x z C F
pixe
u
u’
cam
era
syst
em
C
R,t
C’
Fig. 1. Two views, the geometry and the Fig. 2. The pinhole camera, retinal (R)
coplanarity constraint. C denotes the center of projection and focal (F) plane and three coordinate systems.
Badly Calibrated Camera in Ego-motion Estimation { Propagation of Uncertainty ?
Tomas Svoboda1 and Peter Sturm2
Center for Machine Perception, Czech Technical University, Karlovo nam. 13, CZ 121-35 Praha 2, Czech Republic, svoboda@cmp.felk.cvut.cz 2 GRAVIR{IMAG, project MOVI, INRIA Rh^ne{Alpes, 655, avenue de l'Europe, o 38330 Monbonnot, Grenoble, France, peter.sturm@inrialpes.fr
Reference
Czech Technical University, Faculty of Electrical Engineering Department of Control Engineering, Center for Machine Perception 121 35 Prague 2, Karlovo namest 13, Czech Republic fax +420 2 24357385, phone +420 2 24357465, http://cmp.felk.cvut.cz Get this publication from ftp://cmp.felk.cvut.cz/pub/cmp/articles/svoboda/svcaip97.ps.gz .
?
This research was supported by Region Rh^ne Alpes with the program TEMPRA o { East Europe between l'INPG Grenoble and the Czech Technical University, the Czech Ministry of Education grant No. VS 96049, grants 102/97/0480, 102/97/0855 and 102/95/1378 of the Grant Agency of the Czech Republic and European Union grant Copernicus CP941068, and the grant of the Czech Technical University No 3097472.
Badly Calibrated Camera in Ego-motion Estimation, Propagation of Uncertainty
Tomas Svoboda Peter Sturm
svoboda@cmp.felk.cvut.cz
Tomas Svoboda and Peter Sturm. A badly calibrated camera in ego-motion estimation, propagation of uncertainty. In Gerald Sommer, Kostas Daniilidis, and Josef Pauli, editors, 7-th International Conference Computer Analysis of Images and Patterns, pages 183{190. Springer Verlag in the LNCS series, September 1997 and V are orthonormal matrices and D is the diagonal matrix containing singular values of E . The rotation matrix R and the matrix S can be computed as S = V ZV T ; R = UY V T or UY T V T ; (4) where Z and Y are known constant matrices 2]. The Euler angles characterizing the rotation can be recovered from R using a relationship from 5]. Localized correspondences are expressed in pixel coordinates, q. However to estimate motion, normalized image coordinates u is needed, see equation (2). Assume the classical pinhole camera model. Introducing a calibration matrix K we can transform pixel coordinates q into normalized image coordinates u:
1
Abstract. This paper deals with the ego-motion estimation (motion of the camera) from two views. To estimate an ego-motion we have to nd correspondences and we need a calibrated camera. In this paper we solve the problem how to propagate known camera calibration errors into the uncertainty of the motion parameters. We present a linear estimate of the uncertainty in ego-motion based on the uncertainty in the calibration parameters. We show that the linear estimate is very stable.
. u0 T SRu = u0T E u = 0;
op tic al ax is
l sys
tem
f
(2)
where E is the essential matrix. The equations above was proposed by Longuett Higgins in 6]. The essential matrix E can be reversely decomposed into the motion parameters using method e.g. by Hartley 2]. If E is estimated correctly, it has rank 2 and its nonzero singular values are equal 6]. Using the singular value decomposition we can factorize E as
1 Introduction
Let us assume to have two images of a static scene captured by the same camera from two di erent viewpoints. Having at least eight corresponding points and knowing the calibration parameters of the camera, the camera motion can be estimated up to a similarity using a linear algorithm. The similarity reconstruction of the scene can also be done. This problem has been solved over years 6, 11]. To calibrate the camera, some of many methods developed for the o -line camera calibration as Tsai's method 10], can be used. In this paper we deal with two views and an innacurately calibrated camera. We use a linear method intended for a calibrated camera to estimate the egomotion. Because of noise in the estimation process, an error analysis is needed. There are two sources of errors: (a) noise in correspondences and (b) noise in the calibration parameters. Weng 11] studied the in uence of the noise on the motion parameters. Florou and Mohr 1] used the statistic approach to study reconstruction errors with respect to the calibration parameters. In our paper, we present a linear algorithm to estimate a credibility of the motion parameters, based on the uncertainty in the calibration parameters.
2 Fundamentals
2.1 Ego-Motion from Point Correspondences
Let the motion between two positions of the camera be desribed by a rotation matrix R and a translation vector t, and let u and u0 be normalized image coordinates of corresponding points, see Figure 1, then the coplanarity constraint can be written as: u0T (t Ru) = 0: (1) Introducing the antisymmetric matrix S containing elements of the translation vector t, we can rewrite the coplanarity constraint above as
y_w
z_w
M x_w W
M
X
u q_u θ
q_v
c R
y x z C F
pixe
u
u’
cam
era
syst
em
C
R,t
C’
Fig. 1. Two views, the geometry and the Fig. 2. The pinhole camera, retinal (R)
coplanarity constraint. C denotes the center of projection and focal (F) plane and three coordinate systems.
Badly Calibrated Camera in Ego-motion Estimation { Propagation of Uncertainty ?
Tomas Svoboda1 and Peter Sturm2
Center for Machine Perception, Czech Technical University, Karlovo nam. 13, CZ 121-35 Praha 2, Czech Republic, svoboda@cmp.felk.cvut.cz 2 GRAVIR{IMAG, project MOVI, INRIA Rh^ne{Alpes, 655, avenue de l'Europe, o 38330 Monbonnot, Grenoble, France, peter.sturm@inrialpes.fr