Auto-calibration from the orthogonality constraints - Pattern ...
Auto-Calibration from the Orthogonality Constraints
.
Yongduek Seot and Anders Heyden3 t EE Dept. Pohang University of Science and Technology, Pohang, 790-784, Republic of Korea, Centre for Mathematical Sciences, Lund University, Box 118, SE-221 00 Lund, Sweden*
Abstract
intrinsic parameters, called self-calibration, which leads to the so called Kruppa equations, see [8], [3].
This paper describes an iterative algorithm for making Euclidean reconstruction of a scenefi-om an image sequence captured by a camera with zero skew. The output consists of both the Euclidean reconstruction and the intrinsic parameters of the camera at the different imaging instants, i.e. it also provides a camera calibration. The problem is solved in two different steps. Firstly, the projective structure is obtained from a factorization method followed by a bundle adjustment method. Secondly, the Euclidean reconstruction is obtained from an iterative method that estimates the location of the absolute conic and the intrinsic parameters iteratively, using linear operations in each iteration. In this method a new constraint, called the orthogonality constraint, is used to constrain the absolute conic. Results are shown on experiments on both synthetic and real data.
Several attempt has been made to develop autocalibration techniques under less severe restrictions on the intrinsic parameters of the camera. One approach using the so called 'modulus constraints' in [ll], where the selfcalibration method presented in [ 121is extended to allowing changing focal length. However, the practical implications of this result is limited since when the focal length varies, by zooming, the principal point varies also. Later on, in [51, an existence proof (along with a computation method) for the possibility to make Euclidean reconstruction under the assumption of known skew and aspect ratio. Simultaneously, it was proven in [6] and [lo] that it is sufficient to know the skew. In fact, it was even shown in the former paper the more general result that it is sufficient to know any one of the intrinsic parameters. Observe that all other intrinsic parameters are unknown and allowed to vary between the different imaging instances. Finally, it has been shown in [7] that it is possible to make Euclidean reconstruction when only one of the intrinsic parameters is assumed to be constant. In this paper a novel algorithm for making flexible calibration from the assumption of zero skew will be presented. The algorithm is based on an initial projective reconstruction, in the form of a sequence of camera matrices. The Euclidean reconstruction is obtained from an iterative scheme, where the intrinsic parameters and the absolute conic are estimated sequentially, using only linear operations in each step. In this step a new constraint, called the orthogonality constraint, is used to constrain the location of the absolute conic. The purpose of the algorithm is to obtain good initial estimates for a subsequent bundle adjustement method or to make fast approximate auto-calibration.
1. Introduction One of the main goals of computer vision is to extract three-dimensional structure from a number of twodimensional perspective images, the so called the structure and motion problem. Either pre-calibrated cameIas are used, making it possible to reconstruct the object and the motion up to an unknown similarity transformation, see[ 1J or un-calibrated cameras are used, restricting the result to a reconstruction up to an unknown projective transformation, see [13,9,2]. During the last years there has been an intensive research on the possibility to obtain reconstructions up to an unknown similarity transformation (often called Euclidean reconstructions) without using fully calibrated cameras. In this case, it is necessary to have some additional information about either the intrinsic parameters, the extrinsic parameters or the object in order to obtain the desired Euclidean reconstruction. One way is to use a camera with constant
2. Background on Auto-Calibration In this section a brief background on auto-calibration will be given along with some notations. The camera is
*Sponsored by the Swedish Research Council for Engineering Sciences
(TFR), project 95-64-222 and JIG 99-241
0-7695-0750-6/00 $10.00 0 2000 IEEE
67
3. The Orhogonality Constraints
modeled by the well known camera equation r-
r
~
Let {Pi} denote the sequence of projective camera matrices obtained through a projective reconstruction method. Assume that we have an estimate of the location of the principal point and have chosen coordiantes in the images such that this estimate is located at the origin. Let (Sxi,Syi)}i denote the remaining error in the principal point. Introduce
A x = K [ R ]- R t ] X = P X . Here X = [ X Y 2 1 1' denotes object coordinates in extended form and x = [ x y 1 denotes extended image coordinates. The scale factor A, called the depth, accounts for perspective effects and ( R ,t ) represent a rigid transformation of the object, i.e. R denotes a 3 x 3 rotation matrix and t a 3 x 1 translation vector. Finally, the parameters in the calibration matrix, K , represent intrinsic properties of the image formation system: f represents focal length, y represents the aspect ratio, s represents the skew, i.e. non-rectangular light sensitive arrays can be modeled, and (20, yo) is called the principal point and is interpreted as the orthogonal projection of the focal point onto the image plane. The parameters in R and t are called extrinsic parameters and the parameters in K are called the intrinsic parameters. Finally, a camera that can be modeled as in (l),with s = 0 is called a zero-skew camera. In the case of a sequence of m images the camera equation is written as
'3
Ax = Ki[ Ri
I
- Riti
i = 1 , . .. , m
] X = Pix,
.
where q k , k = 1 , 2 , 3 denotes the k:th row, represented by a four-dimensional column vector, of the principal-pointfree part of the projective camera matrix and the index i is removed in p k and g k for simpler noi;ation. Now, rewriting the dual absolute conic projection equation (5) for the camera Pi together with the zero-skew condition, we have
(%w+ (W
[
N
Pi"
N
KRi[ I
1
- ti
] ,
(2)
6xi
Proposition 3.1. Let 41. q2 and q:3 denote the rows of a camera matrix, Q , as in ( 1 ) with zero skew ( s = 0 ) and principal point located at the origin in the image (XO = yo = 0). Then the following (orthogonality constraints) holds = qlRq3 = q262qS = 0
(3)
K=
N
Define R = HH',
(8)
[:: : :] 0
f 0
in the Kruppa constraints in (3,which gives
-
K R ~ R ' K ~= K K .~
.
Proof. Introduce the intrinsic parameter matrix
where Ri denote orthogonal matrices, denote equality up to scale and Ki denote the appropriate intrinsic parameters. Let H denote the 4 x 3 matrix consisting of the first three columns of H , giving Pifi KRi and multiplying this equation with its transpose gives
P,HH~P?
6Zi
(7) where [ljmR$z] denotes the 3 x 3 rnatrix whose (m,n):th component is IjmCl$;. We have the following proposition.
Assume that we already have obtained a projective reconstruction, i.e. a sequence of camera matrices, Pi, and a structure X obeying (2). Since both Pi and X only are defined up to an unknown projective transformation, also PiH and H-lX represents valid projective reconstructions of the camera matrix sequence and the structure, where H denotes an arbitrary 4 x 4 non-singular matrix. Assume that Pi and X are the obtained projective reconstruction, related to the desired Euclidean reconstruction Pi" and XE by a projective transformation, H ,
PiH
6&,
&i&i
(4)
i.e. the dual of the absolute conic, and
and the assertion follows.
w = KK', i.e. the dual to the image of the absolute conic.
0
Then (4) can be written as W i N
PiRP? ,
Now, we would like to apply this proposition to (7). Introduce the notation ( p l ,p2)Q for q1 Rgz, where Q denotes the 10 parameters in the symmetric matrix fi written as a vector. Define three 3 x 10 matrices Mi, M z i and Mvi
(5)
which says that the absolute conic is projected to the image of the absolute conic.
68
In step 2, the matrix H is given by H = UD6 after computing the SVD of R = U D V T and setting the smallest singular value in D to zero, giving D. Needless to say, we need a stop criterion. One possible choice might be to check the magnitude of the deviations { ( & ~ k ) , d y ~ k ) ) } However, ~l. in this case the rank 3 conis not expected. Therefore, our dition of the solution of R(')). That policy is to check the singular value is, the solution corresponding to the smallest singular value through the whole iteration will be selected to be the best solution in practice.
according to
02)~
4. Experiments Experiments have been carried out on both simulated and real data to show the applicability of the presented iterative techniques. Simulated Data: Experiments were performed with 27 points in 10 images. The points were positioned regularly with coordinates between -500 and +500 units. The camera positions were chosen at random approximately 1000 units away. The orientation were chosen at random. The intrinsic parameters were chosen as follows: f = 1000 N(O,50), s = 0, 7 = 1.2, (ZO,YO) = (0,O) ( N ( 0 ,lo), N ( 0 , lo)), where N ( 0 , o ) denotes a Gaussian random variable with mean zero and standard deviation CT.After generating synthetic data points, projective reconstruction was obtained by projective factorization [4] followed by projective bundle adjustment, and then the linear iteration method was applied. The left hand side graph in Figure 1 shows the evolutionary loci of (dzi,6, i) for four different cameras. Since the principal points were assumed to be the image center at the beginning of the iteration, they are far from the (0,O) location in the figure which means zero error in (bzi,dyi). However, after a few iterations, the distance between the true and the estimated principal points decreased as one can see in the plot. The right hand side graph in Figure 1 shows the minimum singular values obtained when we computed the 10dimensional vector Q ( k )using the matrix M and H using
implying the following relation
MZQ
+
= dz2MziQ dY,MYi"
.
(13)
The following observations can be made from these equations. 1. When (dzi7dyi)is small or equal to zero, (10) will yield an estimate of the dual absolute conic R linearly from the orthogonality constraints, since then MiQ M 0. This estimate could be obtained by constructing the matrix M = [ M T ,...,MZlT for all available camera matrices and finding the vector that minimizes IIMQll through SVD of
+
M. 2. When the dual absolute conic 2 is given, the principal point of each projective camera matrix Pi or the deviation (dzi7S y i ) from the true principal point can be computed linearly using (13) for all available image, which again gives an over-determined system. 3. From the above two observations, we may establish an iterative linear algorithm to estimate both the principal points and the dual absolute conic, by computing Q and {(dzi,dyi)}Elrepeatedly. We propose the following iterative algorithm:
+
a(").
1. Construct the 3m x 10 matrix M and compute from the orthogonality constraints.
Observe that the skew is not exactly zero due to the iterative algorithm and that the accuracy of the aspect ratio is very good. Real Data: Figure 4 shows one of 14 images used in the real data experiment. The corner points were detected as the intersections of the lines of the black rectangles. The projective reconstruction was obtained in the same way as before, and we used the image center as the start principal point of the iteration. Notice that the minimum singular value for H changes abruptly in Figure 4. At the minimum peak of the graph, the solution R was chosen as the best one.
2. Compute the projective-to-Euclidean transformation matrix H through SVD of a(')), and record the smallest singular value (This step is needed to assure that rankR = 3.)
02)~.
3. Compute the deviations { (d,~"), dyik))}gl using the orthogonality constraints and the R obtained in the previous step and make a coordinate transformation in the image such that the new estimate of the principal point becomes at the origin.
69
Figure 1.0 = 0. Left: The evolution of ( & i , 6 y i )with respect to iteration (oi = 0). Right: The singular values vs. iteration number.
Figure 3. Left: The evolution of the two smallest singular values through the iteration. The other two graphs, shown for comparison, are the next smallest singular values of M and Q, respectively. Right: Angles measured after Euclidean reconstruction, compared with the result of the Euclidean bundle adjustment. Their true values are 90".
References
Figure 4 shows the measured angles after Euclidean reconstruction, compared with the result of the Euclidean bundle adjustment.
H I K. B. Atkinson. Close Range Photogrammetry and Machine Vision. Whittles Publishing, 1996. t21 0. D. Faugeras. What can be seen in three dimensions with an uncalibrated stereo rig? In G. Sandini, editor, Proc. 2nd European Conf. on Computer Vision,Santa Margherita Ligure, Italy, pages 563-578. Springer-Verlag, 1992. [31 0. D. Faugeras, Q.-T. Luong, and S . J. Maybank. Camera self-calibration: Theory and experiments. In G. Sandini, editor, ECCV'92, volume 588 of Lecture notes in Computer Science, pages 321-334. Springer-'Verlag, 1992. 141 A. Heyden. Projective structure and motion from image sequences using subspace methods. In Proc. 10th Scandinavian Conference on Image Analysis, pages 963-968, 1997. 151 A. Heyden and K. Astrom. Euclidean reconstruction from image sequences with varying and iinknown focal length and principal point. In Proc. Conf. Conzputer Vision and Pattern Recognition, pages 438-443, 1997. 161 A. Heyden and K. Astrom. Minimal conditions on intrinsic parameters for Euclidean reconstruction. In Proc. 22nd Asian Conf. on Computer Vision,Hong Kong, China, 1998.
5. Conclusions The presented iterative methods have been shown to give good results on both simulated and real data, without needing any initial data apart from the projective reconstruction. The only constraint used is zero skew, but the method can be generalized to more general situations, e.g. zero skew and constant aspect ratio. Further work will be directed towards examining the properties of the algorithm by making more
experiments and a theoretical analysis if possible. Another line of investigation is to generalize the algorithm to other constraints on the intrinsic parameters.
Acknowledgements Yongduek Seo thanks professor Ki-Sang Hong for the financial support during his visition to MIG group of Lund University. 70
Figure 2. The first image used in the real experiment (image size is640 x 480).
A. Heyden and K. Wstrom. Flexible calibration: Minimal cases for auto-calibration. In Proc. 7th Int. Con& on Computer Vision, Kerhyra, Greece, 1999. Q.-T. Luong. Matrice Fondamentale et Calibration Visuelle sur I' Environnement-Vers une plus grande uutonomie des systkmes robotiques. PhD thesis, Universitt de Paris-Sud,
Centre d'Orsay, 1992. R. Mohr and E. Arbogast. It can be done without camera calibration. Pattern Recognition Letters, 12(1):39-43, 1991. M. Pollefeys, R. Koch, and L. Van Gool. Self-calibration
and metric reconstruction in spite of varying and unknown internal camera parameters. In Proc. 6th Int. Conf. on Compute?-Vision, Mumbai, India, 1998. M. Pollefeys, L. Van Gool, and A. Oosterlinck. Euclidean reconstruction from image sequences with variable focal length. In B. Buxton and R. Cipolla, editors, ECCV'96, volume 1064 of Lecture notes in Computer Science, pages 3 1-44. Springer-Verlag, 1996. M. Pollefeys, L. Van Gool, and M. Oosterlinck. The modulus constraint: A new constraint for self-calibration. In Proc. International Conference on Pattern Recognition, Vienna, Austria, pages 349-353, 1996. G. Span: An algebraic-analytic method for affine shapes of point configurations. In Proc. 7th Scandinavian Conference on Image Analysis, pages 274-281, 1991. Also in Theoiy & Applications of Image Analysis I , pages 87-98,.
71
.
Yongduek Seot and Anders Heyden3 t EE Dept. Pohang University of Science and Technology, Pohang, 790-784, Republic of Korea, Centre for Mathematical Sciences, Lund University, Box 118, SE-221 00 Lund, Sweden*
Abstract
intrinsic parameters, called self-calibration, which leads to the so called Kruppa equations, see [8], [3].
This paper describes an iterative algorithm for making Euclidean reconstruction of a scenefi-om an image sequence captured by a camera with zero skew. The output consists of both the Euclidean reconstruction and the intrinsic parameters of the camera at the different imaging instants, i.e. it also provides a camera calibration. The problem is solved in two different steps. Firstly, the projective structure is obtained from a factorization method followed by a bundle adjustment method. Secondly, the Euclidean reconstruction is obtained from an iterative method that estimates the location of the absolute conic and the intrinsic parameters iteratively, using linear operations in each iteration. In this method a new constraint, called the orthogonality constraint, is used to constrain the absolute conic. Results are shown on experiments on both synthetic and real data.
Several attempt has been made to develop autocalibration techniques under less severe restrictions on the intrinsic parameters of the camera. One approach using the so called 'modulus constraints' in [ll], where the selfcalibration method presented in [ 121is extended to allowing changing focal length. However, the practical implications of this result is limited since when the focal length varies, by zooming, the principal point varies also. Later on, in [51, an existence proof (along with a computation method) for the possibility to make Euclidean reconstruction under the assumption of known skew and aspect ratio. Simultaneously, it was proven in [6] and [lo] that it is sufficient to know the skew. In fact, it was even shown in the former paper the more general result that it is sufficient to know any one of the intrinsic parameters. Observe that all other intrinsic parameters are unknown and allowed to vary between the different imaging instances. Finally, it has been shown in [7] that it is possible to make Euclidean reconstruction when only one of the intrinsic parameters is assumed to be constant. In this paper a novel algorithm for making flexible calibration from the assumption of zero skew will be presented. The algorithm is based on an initial projective reconstruction, in the form of a sequence of camera matrices. The Euclidean reconstruction is obtained from an iterative scheme, where the intrinsic parameters and the absolute conic are estimated sequentially, using only linear operations in each step. In this step a new constraint, called the orthogonality constraint, is used to constrain the location of the absolute conic. The purpose of the algorithm is to obtain good initial estimates for a subsequent bundle adjustement method or to make fast approximate auto-calibration.
1. Introduction One of the main goals of computer vision is to extract three-dimensional structure from a number of twodimensional perspective images, the so called the structure and motion problem. Either pre-calibrated cameIas are used, making it possible to reconstruct the object and the motion up to an unknown similarity transformation, see[ 1J or un-calibrated cameras are used, restricting the result to a reconstruction up to an unknown projective transformation, see [13,9,2]. During the last years there has been an intensive research on the possibility to obtain reconstructions up to an unknown similarity transformation (often called Euclidean reconstructions) without using fully calibrated cameras. In this case, it is necessary to have some additional information about either the intrinsic parameters, the extrinsic parameters or the object in order to obtain the desired Euclidean reconstruction. One way is to use a camera with constant
2. Background on Auto-Calibration In this section a brief background on auto-calibration will be given along with some notations. The camera is
*Sponsored by the Swedish Research Council for Engineering Sciences
(TFR), project 95-64-222 and JIG 99-241
0-7695-0750-6/00 $10.00 0 2000 IEEE
67
3. The Orhogonality Constraints
modeled by the well known camera equation r-
r
~
Let {Pi} denote the sequence of projective camera matrices obtained through a projective reconstruction method. Assume that we have an estimate of the location of the principal point and have chosen coordiantes in the images such that this estimate is located at the origin. Let (Sxi,Syi)}i denote the remaining error in the principal point. Introduce
A x = K [ R ]- R t ] X = P X . Here X = [ X Y 2 1 1' denotes object coordinates in extended form and x = [ x y 1 denotes extended image coordinates. The scale factor A, called the depth, accounts for perspective effects and ( R ,t ) represent a rigid transformation of the object, i.e. R denotes a 3 x 3 rotation matrix and t a 3 x 1 translation vector. Finally, the parameters in the calibration matrix, K , represent intrinsic properties of the image formation system: f represents focal length, y represents the aspect ratio, s represents the skew, i.e. non-rectangular light sensitive arrays can be modeled, and (20, yo) is called the principal point and is interpreted as the orthogonal projection of the focal point onto the image plane. The parameters in R and t are called extrinsic parameters and the parameters in K are called the intrinsic parameters. Finally, a camera that can be modeled as in (l),with s = 0 is called a zero-skew camera. In the case of a sequence of m images the camera equation is written as
'3
Ax = Ki[ Ri
I
- Riti
i = 1 , . .. , m
] X = Pix,
.
where q k , k = 1 , 2 , 3 denotes the k:th row, represented by a four-dimensional column vector, of the principal-pointfree part of the projective camera matrix and the index i is removed in p k and g k for simpler noi;ation. Now, rewriting the dual absolute conic projection equation (5) for the camera Pi together with the zero-skew condition, we have
(%w+ (W
[
N
Pi"
N
KRi[ I
1
- ti
] ,
(2)
6xi
Proposition 3.1. Let 41. q2 and q:3 denote the rows of a camera matrix, Q , as in ( 1 ) with zero skew ( s = 0 ) and principal point located at the origin in the image (XO = yo = 0). Then the following (orthogonality constraints) holds = qlRq3 = q262qS = 0
(3)
K=
N
Define R = HH',
(8)
[:: : :] 0
f 0
in the Kruppa constraints in (3,which gives
-
K R ~ R ' K ~= K K .~
.
Proof. Introduce the intrinsic parameter matrix
where Ri denote orthogonal matrices, denote equality up to scale and Ki denote the appropriate intrinsic parameters. Let H denote the 4 x 3 matrix consisting of the first three columns of H , giving Pifi KRi and multiplying this equation with its transpose gives
P,HH~P?
6Zi
(7) where [ljmR$z] denotes the 3 x 3 rnatrix whose (m,n):th component is IjmCl$;. We have the following proposition.
Assume that we already have obtained a projective reconstruction, i.e. a sequence of camera matrices, Pi, and a structure X obeying (2). Since both Pi and X only are defined up to an unknown projective transformation, also PiH and H-lX represents valid projective reconstructions of the camera matrix sequence and the structure, where H denotes an arbitrary 4 x 4 non-singular matrix. Assume that Pi and X are the obtained projective reconstruction, related to the desired Euclidean reconstruction Pi" and XE by a projective transformation, H ,
PiH
6&,
&i&i
(4)
i.e. the dual of the absolute conic, and
and the assertion follows.
w = KK', i.e. the dual to the image of the absolute conic.
0
Then (4) can be written as W i N
PiRP? ,
Now, we would like to apply this proposition to (7). Introduce the notation ( p l ,p2)Q for q1 Rgz, where Q denotes the 10 parameters in the symmetric matrix fi written as a vector. Define three 3 x 10 matrices Mi, M z i and Mvi
(5)
which says that the absolute conic is projected to the image of the absolute conic.
68
In step 2, the matrix H is given by H = UD6 after computing the SVD of R = U D V T and setting the smallest singular value in D to zero, giving D. Needless to say, we need a stop criterion. One possible choice might be to check the magnitude of the deviations { ( & ~ k ) , d y ~ k ) ) } However, ~l. in this case the rank 3 conis not expected. Therefore, our dition of the solution of R(')). That policy is to check the singular value is, the solution corresponding to the smallest singular value through the whole iteration will be selected to be the best solution in practice.
according to
02)~
4. Experiments Experiments have been carried out on both simulated and real data to show the applicability of the presented iterative techniques. Simulated Data: Experiments were performed with 27 points in 10 images. The points were positioned regularly with coordinates between -500 and +500 units. The camera positions were chosen at random approximately 1000 units away. The orientation were chosen at random. The intrinsic parameters were chosen as follows: f = 1000 N(O,50), s = 0, 7 = 1.2, (ZO,YO) = (0,O) ( N ( 0 ,lo), N ( 0 , lo)), where N ( 0 , o ) denotes a Gaussian random variable with mean zero and standard deviation CT.After generating synthetic data points, projective reconstruction was obtained by projective factorization [4] followed by projective bundle adjustment, and then the linear iteration method was applied. The left hand side graph in Figure 1 shows the evolutionary loci of (dzi,6, i) for four different cameras. Since the principal points were assumed to be the image center at the beginning of the iteration, they are far from the (0,O) location in the figure which means zero error in (bzi,dyi). However, after a few iterations, the distance between the true and the estimated principal points decreased as one can see in the plot. The right hand side graph in Figure 1 shows the minimum singular values obtained when we computed the 10dimensional vector Q ( k )using the matrix M and H using
implying the following relation
MZQ
+
= dz2MziQ dY,MYi"
.
(13)
The following observations can be made from these equations. 1. When (dzi7dyi)is small or equal to zero, (10) will yield an estimate of the dual absolute conic R linearly from the orthogonality constraints, since then MiQ M 0. This estimate could be obtained by constructing the matrix M = [ M T ,...,MZlT for all available camera matrices and finding the vector that minimizes IIMQll through SVD of
+
M. 2. When the dual absolute conic 2 is given, the principal point of each projective camera matrix Pi or the deviation (dzi7S y i ) from the true principal point can be computed linearly using (13) for all available image, which again gives an over-determined system. 3. From the above two observations, we may establish an iterative linear algorithm to estimate both the principal points and the dual absolute conic, by computing Q and {(dzi,dyi)}Elrepeatedly. We propose the following iterative algorithm:
+
a(").
1. Construct the 3m x 10 matrix M and compute from the orthogonality constraints.
Observe that the skew is not exactly zero due to the iterative algorithm and that the accuracy of the aspect ratio is very good. Real Data: Figure 4 shows one of 14 images used in the real data experiment. The corner points were detected as the intersections of the lines of the black rectangles. The projective reconstruction was obtained in the same way as before, and we used the image center as the start principal point of the iteration. Notice that the minimum singular value for H changes abruptly in Figure 4. At the minimum peak of the graph, the solution R was chosen as the best one.
2. Compute the projective-to-Euclidean transformation matrix H through SVD of a(')), and record the smallest singular value (This step is needed to assure that rankR = 3.)
02)~.
3. Compute the deviations { (d,~"), dyik))}gl using the orthogonality constraints and the R obtained in the previous step and make a coordinate transformation in the image such that the new estimate of the principal point becomes at the origin.
69
Figure 1.0 = 0. Left: The evolution of ( & i , 6 y i )with respect to iteration (oi = 0). Right: The singular values vs. iteration number.
Figure 3. Left: The evolution of the two smallest singular values through the iteration. The other two graphs, shown for comparison, are the next smallest singular values of M and Q, respectively. Right: Angles measured after Euclidean reconstruction, compared with the result of the Euclidean bundle adjustment. Their true values are 90".
References
Figure 4 shows the measured angles after Euclidean reconstruction, compared with the result of the Euclidean bundle adjustment.
H I K. B. Atkinson. Close Range Photogrammetry and Machine Vision. Whittles Publishing, 1996. t21 0. D. Faugeras. What can be seen in three dimensions with an uncalibrated stereo rig? In G. Sandini, editor, Proc. 2nd European Conf. on Computer Vision,Santa Margherita Ligure, Italy, pages 563-578. Springer-Verlag, 1992. [31 0. D. Faugeras, Q.-T. Luong, and S . J. Maybank. Camera self-calibration: Theory and experiments. In G. Sandini, editor, ECCV'92, volume 588 of Lecture notes in Computer Science, pages 321-334. Springer-'Verlag, 1992. 141 A. Heyden. Projective structure and motion from image sequences using subspace methods. In Proc. 10th Scandinavian Conference on Image Analysis, pages 963-968, 1997. 151 A. Heyden and K. Astrom. Euclidean reconstruction from image sequences with varying and iinknown focal length and principal point. In Proc. Conf. Conzputer Vision and Pattern Recognition, pages 438-443, 1997. 161 A. Heyden and K. Astrom. Minimal conditions on intrinsic parameters for Euclidean reconstruction. In Proc. 22nd Asian Conf. on Computer Vision,Hong Kong, China, 1998.
5. Conclusions The presented iterative methods have been shown to give good results on both simulated and real data, without needing any initial data apart from the projective reconstruction. The only constraint used is zero skew, but the method can be generalized to more general situations, e.g. zero skew and constant aspect ratio. Further work will be directed towards examining the properties of the algorithm by making more
experiments and a theoretical analysis if possible. Another line of investigation is to generalize the algorithm to other constraints on the intrinsic parameters.
Acknowledgements Yongduek Seo thanks professor Ki-Sang Hong for the financial support during his visition to MIG group of Lund University. 70
Figure 2. The first image used in the real experiment (image size is640 x 480).
A. Heyden and K. Wstrom. Flexible calibration: Minimal cases for auto-calibration. In Proc. 7th Int. Con& on Computer Vision, Kerhyra, Greece, 1999. Q.-T. Luong. Matrice Fondamentale et Calibration Visuelle sur I' Environnement-Vers une plus grande uutonomie des systkmes robotiques. PhD thesis, Universitt de Paris-Sud,
Centre d'Orsay, 1992. R. Mohr and E. Arbogast. It can be done without camera calibration. Pattern Recognition Letters, 12(1):39-43, 1991. M. Pollefeys, R. Koch, and L. Van Gool. Self-calibration
and metric reconstruction in spite of varying and unknown internal camera parameters. In Proc. 6th Int. Conf. on Compute?-Vision, Mumbai, India, 1998. M. Pollefeys, L. Van Gool, and A. Oosterlinck. Euclidean reconstruction from image sequences with variable focal length. In B. Buxton and R. Cipolla, editors, ECCV'96, volume 1064 of Lecture notes in Computer Science, pages 3 1-44. Springer-Verlag, 1996. M. Pollefeys, L. Van Gool, and M. Oosterlinck. The modulus constraint: A new constraint for self-calibration. In Proc. International Conference on Pattern Recognition, Vienna, Austria, pages 349-353, 1996. G. Span: An algebraic-analytic method for affine shapes of point configurations. In Proc. 7th Scandinavian Conference on Image Analysis, pages 274-281, 1991. Also in Theoiy & Applications of Image Analysis I , pages 87-98,.
71
