The problem of calibrating a stereo rig is extremely important for practical applications. Existing work is based on the use of a calibration pattern whose 3D model is a priori known. We show theoretically and with experiments on real images, how it is possible to completely calibrate a stereo rig, that is to determine each camera’s intrinsic parameters and the relative displacement between the two or three cameras, using only point matches obtained during unknown motions, without any a priori knowledge of the scenes.
The first part of the chapter is devoted to the computation of the intrinsic parameters of the cameras by a method based upon the estimation of the so-called fundamental matrix associated with camera displacement. Three different displacements are sufficient to solve the Kruppa equations which yield these parameters.
The second part of the chapter is devoted to the computation of the extrinsic parameters. We first explain how to recover the unknown motions previously used, once we have an estimate of the intrinsic parameters and the fundamental matrices. The computation is quite robust to the inaccuracy of the determination of the camera parameters. We then present the equations which allow us, from two displacements of the stereo rig, for which the camera motions are computed independently, to compute the relative displacement between the cameras. This technique allows us to compute the relative displacement between two or three cameras and complete the full calibration of the rig.
- Camera Motion
- Fundamental Matrix
- Intrinsic Parameter
- Camera Parameter
- Principal Point
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J. L. Mundy, A. Zisserman (eds.) Geometric invariance in computer vision. MIT Press, 1992.
J.G. Semple, G.T. Kneebone. Algebraic projective geometry. Oxford Science Publication, 1952.
O.D. Faugeras, G. Toscani. The calibration problem for stereo. In Proceedings of CVPR’86, pp. 15–20, 1986.
R.Y. Tsai. An Efficient and Accurate Camera Calibration Technique for 3D Machine Vision. In Proceedings CVPR ‘86, Miami Beach, Florida, 364–374. IEEE, June 1986.
H. Asada, M. Brady. The curvature primal sketch. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8, 2–14, 1986.
G. Medioni. Y. Yasumuto. Corner detection and curve representation using cubic b-spline. In Proc. International Conference on Robotics and Automation,pp. 764–769, San Francisco, 1986. IEEE.
F. Veillon, R. Horaud, T. Skordas. Finding geometric and relational structures in an image. In O. Faugeras (ed.) Computer Vision–ECCV’90. (Lecture Notes in Computer Science, Vol. 427) pp. 374–384, Springer, Berlin Heidelberg, 1990.
M.A. Shah, R. Jain. Detecting time-varying corners. Computer Vision, Graphics, and Image Processing 28, 345–355, 1984.
C. Harris, M. Stephens. A combined corner and edge detector. In Proc. Alvey Vision Conference, 189–192, Manchester, 1988.
R. Deriche, G. Giraudon. A Computational Approach for Corner and Vertex Detection. The International Journal of Computer Vision 10 (2), 101–124, 1993.
A. Guiducci. Corner characterization by differential geometry techniques. Pattern Recognition Letters 8, 311–318, 1988.
K. Rohr. Modelling and identification of characteristic intensity variations. Image and Vision Computing 10 (2), 66–76, 1992.
R. Deriche and T. Blaszka. Recovering and characterizing image features using an efficient model based approach. In Proc. International Conference on Computer Vision and Pattern Recognition, pp. 530–535, New York, 1993. IEEE.
R.E. Kelly, P.R.H. McConnell, S.J. Mildenberger. The gestalt photomapper. Photogrammetric Engineering and Remote Sensing 43, 1407–1417, 1977.
W. Forstner, A. Pertl. Photogrammetric standard methods and digital image matching techniques for high precision surface measurements. In Gelsema, E.S., Kanal, L.N., (eds.), Pattern Recognition in Practice II, pp. 57–72. Elsevier Science Publishers, 1986.
D.B. Gennery. Modelling the Environment of an Exploring Vehicle by means of Stereo Vision. PhD thesis, Stanford University, June 1980.
R. Deriche, O.D. Faugeras. Tracking line segments. Image and Vision Computing, 8(4), 261–270, 1990. A shorter version appeared in the Proceedings of the 1st ECCV.
O.D. Faugeras. What can be seen in three dimensions with an uncalibrated stereo rig. In G. Sandini (ed.) Computer Vision–ECCV’92. (Lecture Notes in Computer Science, Vol. 588) pp. 563–578, Springer, Berlin Heidelberg, 1992.
R. Hartley, R. Gupta, T. Chang. Stereo from uncalibrated cameras. In Proc. of the Conference on Computer Vision and Pattern Recognition, pp. 761–764, Urbana, 1992.
L. Robert. Reconstruction de courbes et de surfaces par vision stéréoscopique. Applications a la robotique mobile. PhD thesis, Ecole Polytechnique, 1993.
A. Shashua. Projective structure from two uncalibrated images: structure from motion and recognition. Technical Report A.I. Memo No. 1363, MIT, Sept 1992.
H.C. Longuet-Higgins. A Computer Algorithm for Reconstructing a Scene from Two Projections. Nature 293, 133–135, 1981.
Q.-T. Luong, R. Deriche, O.D. Faugeras, T. Papadopoulo. On determining the Fundamental matrix: analysis of different methods and experimental results. Technical Report 1894, INRIA, 1993.
E. Kruppa. Zur Ermittlung eines Objektes aus zwei Perspektiven mit innerer Orientierung. Sitz.-Ber. Akad. Wiss., Wien, math. naturw. Kl., Abt. IIa. 122, 1939–1948, 1913.
O.D. Faugeras, S.J. Maybank. Motion from point matches: multiplicity of solutions. The International Journal of Computer Vision, 4(3), 225–246, 1990. also INRIA Tech. Report 1157.
S.J. Maybank, O.D. Faugeras. A Theory of Self-Calibration of a Moving Camera. The International Journal of Computer Vision, 8 (2), 123–151, 1992.
O.D. Faugeras, Q.-T. Luong, and S.J. Maybank. Camera self-calibration: theory and experiments. In G. Sandini (ed.) Computer Vision–ECCV’92. (Lecture Notes in Computer Science, Vol. 588) pp. 563–578, Springer, Berlin Heidelberg, 1992.
A. Morgan. Solving polynomial systems using continuation for engineering and science problems. Prentice-Hall, 1987.
Q.-T. Luong, O.D. Faugeras. Self-calibration of a camera using multiples images. In Proc. International Conference on Pattern Recognition, pp. 9–12, Den Haag, 1992. IEEE.
Q.-T. Luong. Matrice fondamentale et calibration visuelle sur l’environnement: vers une plus grande autonomie des systèmes robotiques. PhD thesis, Université de Paris-Sud., Dec. 1992.
O.D. Faugeras, F. Lustman, G. Toscani. Motion and Structure from point and line matches. In Proc. International Conference on Computer Vision, pp. 25–34, London, June 1987. IEEE
M.E. Spetsakis, J. Aloimonos. Optimal computing of structure from motion using point correspondences in two frames. In Proc. International Conference on Computer Vision, pp. 449–453, Tarpen Springs, FL, 1988. IEEE.
J. Weng, N. Ahuja, T.S. Huang. Optimal motion and structure estimation. In Proc. International Conference on Computer Vision and Pattern Recognition,pp. 144–152, San Diego, 1989. IEEE.
B.K.P. Horn. Relative orientation. The International Journal of Computer Vision, 4 (1), 59–78, 1990.
R.Y. Tsai, T.S. Huang. Uniqueness and estimation of three-dimensional motion parameters of rigid objects wirth curved surfaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 13–27, 1984.
R.I. Hartley. Estimation of relative camera positions for uncalibrated cameras. In G. Sandini (ed.) Computer Vision–ECCV’92. (Lecture Notes in Computer Science, Vol. 588) pp. 563–578, Springer, Berlin Heidelberg, 1992.
R. Kumar and A. Hanson. Sensibility of the pose refinement problem to accurate estimation of camera parameters. In Proceedings of the International Conference on Computer Vision, pp. 365–369, Osaka, Japan, 1990.
J.C.K. Chou and M. Kamel. Quaternions approach to solve the kinematic equation of rotation, AaAx = AxAa, of a sensor-mounted robotic manipulator. In Proc. International Conference on Robotics and Automation,pp. 656–662, Philadelphia, 1988. IEEE.
Y.S. Shiu, S. Ahmad. Calibration of wrist-mounted robotic sensors by solving homogeneous transform equations of the form AX = XB. IEEE Transactions on robotics and automation, 5 (1), 16–29, 1989.
R.Y. Tsai, R.K. Lenz. Real time versatile robotics hamd/eye calibration using 3D machine vision. In Proc. International Conference on Robotics and Automation,pp. 554–561, Philadelphia, 1988. IEEE.
H.H. Chen. A screw motion approach to uniqueness analysis of head-eye geometry. In Proc. of the Conference on Computer Vision and Pattern Recognition,pp. 145–151, Maui, Hawaii, June 1991. IEEE.
E. Pervin, J.A. Webb. Quaternions in computer vision and robotics. In Proc. International Conference on Computer Vision and Pattern Recognition,pp. 382–383, Arlington, VA, 1983. IEEE.
L. Robert, O.D. Faugeras. Curve-Based Stereo: Figural Continuity And Curvature. In IEEE Proc. International Conference on Computer Vision and Pattern Recognition,pp. 57–62, Maui, Hawaii, June 1991. IEEE.
P.R. Beaudet. Rotational invariant image operators. In Proc. International Conference on Pattern Recognition,pp. 579–583, Kyoto, 1978. IEEE.
L. Dreschler, H.H. Nagel. On the selection of critical points and local curvature extrema of region boundaries for interframe matching. In Proc. International Conference on Pattern Recognition,pp. 542–544, Munich, 1982. IEEE.
L. Kitchen, A. Rosenfeld. Gray-level corner detection. Pattern Recognition Letters 95–102, 1982.
H.H. Nagel. Constraints for the estimation of displacement vector fields from image sequences. In Proc. International Joint Conference on Artificial Intelligence,pp. 156–160, Karlsruhe, 1983. Morgan Kaufman.
O.A. Zuniga, R.M. Haralick. Corner detection using the facet model. In Proc. International Conference on Computer Vision and Pattern Recognition,pp. 30–37, Arlington, VA, 1983. IEEE.
Q.-T. Luong, O.D. Faugeras. The fundamental matrix: theory, algorithms, and stability analysis. The International Journal of Computer Vision, 17 (1), 43–76, 1996.
Q.-T. Luong, O.D. Faugeras. Self-calibration of a moving camera from point correspondences and fundamental matrices. The International Journal of Computer Vision, 22 (3), 261–289, 1997.
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Luong, QT., Faugeras, O.D. (2001). Self-Calibration of a Stereo Rig from Unknown Camera Motions and Point Correspondences. In: Gruen, A., Huang, T.S. (eds) Calibration and Orientation of Cameras in Computer Vision. Springer Series in Information Sciences, vol 34. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04567-1_8
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