Abstract
Two images of a single scene/object are related by the epipolar geometry, which can be described by a 3×3 singular matrix called the essential matrix if images' internal parameters are known, or the fundamental matrix otherwise. It captures all geometric information contained in two images, and its determination is very important in many applications such as scene modeling and vehicle navigation. This paper gives an introduction to the epipolar geometry, and provides a complete review of the current techniques for estimating the fundamental matrix and its uncertainty. A well-founded measure is proposed to compare these techniques. Projective reconstruction is also reviewed. The software which we have developed for this review is available on the Internet.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Aggarwal, J. and Nandhakumar, N. 1988. On the computation of motion from sequences of images-A review. In Proc. IEEE, Vol. 76,No. 8, pp. 917-935.
Aloimonos, J. 1990. Perspective approximations. Image and Vision Computing, 8(3):179-192.
Anderson, T. 1958. An Introduction to Multivariate Statistical Analysis. John Wiley & Sons, Inc.
Ayache, N. 1991.Artificial Vision for Mobile Robots. MIT Press.
Ayer, S., Schroeter, P., and Bigün, J. 1994. Segmentation of moving objects by robust motion parameterestimation over multiple frames. In Proc. of the 3rd European Conf. on Computer Vision, J.-O. Eklundh (Ed.), Vols. 800-801 of Lecture Notes in Computer Science, Springer-Verlag: Stockholm, Sweden, Vol. II, pp. 316- 327.
Beardsley, P., Zisserman, A., and Murray, D. 1994. Navigation using affine structure from motion. In Proc. of the 3rd European Conf. on Computer Vision, J.-O. Eklundh (Ed.), Vol. 2 of Lecture Notes in Computer Science, Springer-Verlag: Stockholm, Sweden, pp. 85- 96.
Boufama, B. and Mohr, R. 1995. Epipole and fundamental matrix estimation using the virtual parallax property. In Proc. of the 5th Int. Conf. on Computer Vision, IEEE Computer Society Press: Boston, MA, pp. 1030-1036.
Carlsson, S. 1994. Multiple image invariance using the double algebra. In Applications of Invariance in Computer Vision, J.L. Mundy, A. Zissermann, and D. Forsyth (Eds.), Vol. 825 of Lecture Notes in Computer Science, Springer-Verlag, pp. 145-164.
Csurka, G. 1996. Modélisation projective des objets tridimensionnels en vision par ordinateur. Ph.D. Thesis, University of Nice, Sophia-Antipolis, France.
Csurka, G., Zeller, C., Zhang, Z., and Faugeras, O. 1996. Characterizing the uncertainty of the fundamental matrix. Computer Vision and Image Understanding, 68(1):18-36, 1997. Updated version of INRIA Research Report 2560, 1995.
Deriche, R., Zhang, Z., Luong, Q.-T., and Faugeras, O. 1994. Robust recovery of the epipolar geometry for an uncalibrated stereo rig. In Proc. of the 3rd European Conf. on Computer Vision, J.-O. Eklundh (Ed.), Vols. 800-801 of Lecture Notes in Computer Science, Springer Verlag: Stockholm, Sweden, Vol. 1, pp. 567-576.
Enciso, R. 1995. Auto-calibration des capteurs visuels actifs. Reconstruction 3D active. Ph.D. Thesis, University Paris XI Orsay.
Faugeras, O. 1992. What can be seen in three dimensions with an uncalibrated stereo rig. In Proc. of the 2nd European Conf. on Computer Vision, G. Sandini (Ed.), Vol. 588 of Lecture Notes in Computer Science, Springer-Verlag: Santa Margherita Ligure, Italy, pp. 563-578.
Faugeras, O. 1993. Three-Dimensional Computer Vision: A Geometric Viewpoint. The MIT Press.
Faugeras, O. 1995. Stratification of 3-D vision: Projective, affine, and metric representations. Journal of the Optical Society of America A, 12(3):465-484.
Faugeras, O. and Lustman, F. 1988. Motion and structure from motion in a piecewise planar environment. International Journal of Pattern Recognition and Artificial Intelligence, 2(3):485- 508.
Faugeras, O., Luong, T., and Maybank, S. 1992. Camera selfcalibration: Theory and experiments. In Proc. 2nd ECCV, G. Sandini (Ed.), Vol. 588 of Lecture Notes in Computer Science, Springer-Verlag: Santa Margherita Ligure, Italy, pp. 321-334.
Faugeras, O. and Robert, L. 1994. What can two images tell us about a third one?. In Proc. of the 3rd European Conf. on Computer Vision, J.-O. Eklundh (Ed.), Vols. 800-801 of Lecture Notes in Computer Science, Springer-Verlag: Stockholm, Sweden. Also INRIA Technical report 2018.
Faugeras, O. and Mourrain, B. 1995. On the geometry and algebra of the point and line correspondences between nimages. In Proc. of the 5th Int. Conf. on Computer Vision, IEEE Computer Society Press: Boston, MA, pp. 951-956.
Fischler, M. and Bolles, R. 1981. Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography. Communications of the ACM, 24:381- 385.
Golub, G. and van Loan, C. 1989. Matrix Computations. The John Hopkins University Press.
Haralick, R. 1986. Computer vision theory: The lack thereof. Computer Vision, Graphics, and Image Processing, 36:372-386.
Hartley, R. 1993. Euclidean reconstruction from uncalibrated views. In Applications of Invariance in Computer Vision, J. Mundy and A. Zisserman (Eds.), Vol. 825 of Lecture Notes in Computer Science, Springer-Verlag: Berlin, pp. 237-256.
Hartley, R. 1994. Projective reconstruction and invariants from multiple images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(10):1036-1040.
Hartley, R. 1995. In defence of the 8-point algorithm. In Proc. of the 5th Int. Conf. on Computer Vision, IEEE Computer Society Press: Boston, MA, pp. 1064-1070.
Hartley, R., Gupta, R., and Chang, T. 1992. Stereo from uncalibrated cameras. In Proc. of the IEEE Conf. on Computer Vision and Pattern Recognition, Urbana Champaign, IL, pp. 761-764.
Hartley, R. and Sturm, P. 1994. Triangulation. In Proc. of the ARPA Image Understanding Workshop, Defense Advanced Research Projects Agency, Morgan Kaufmann Publishers, Inc., pp. 957- 966.
Heeger, D.J. and Jepson, A.D. 1992. Subspace methods for recovering rigid motion I: Algorithm and implementation. The International Journal of Computer Vision, 7(2):95-117.
Hesse, O. 1863. Die cubische gleichung, von welcher die Lösung des problems der homographie von M. Chasles Abhängt. J. Reine Angew. Math., 62:188-192.
Huang, T. and Netravali, A. 1994. Motion and structure from feature correspondences: A review. In Proc. IEEE, 82(2):252-268.
Huber, P. 1981. Robust Statistics. John Wiley & Sons: New York.
Laveau, S. 1996. Géométrie d'un système de Ncaméras. Théorie. Estimation. Applications. Ph.D. Thesis, École Polytechnique.
Longuet-Higgins, H. 1981. A computer algorithm for reconstructing a scene from two projections. Nature, 293:133-135.
Luong, Q.-T. 1992. Matrice Fondamentale et CalibrationVisuelle sur l'Environnement-Vers une plus grande autonomie des systèmes robotiques. Ph.D. Thesis, Université de Paris-Sud, Centre d'Orsay.
Luong, Q.-T. and Viéville, T. 1994. Canonic representations for the geometries of multiple projective views. In Proc. of the 3rd European Conf. on Computer Vision, J.-O. Eklundh (Ed.), Vols. 800- 801 of Lecture Notes in Computer Science, Springer-Verlag: Stockholm, Sweden, Vol. 1, pp. 589-599.
Luong, Q.-T. and Faugeras, O.D. 1996. The fundamental matrix: Theory, algorithms and stability analysis. The International Journal of Computer Vision, 1(17):43-76.
Maybank, S. 1992. Theory of Reconstruction from Image Motion. Springer-Verlag.
Maybank, S.J. and Faugeras, O.D. 1992. A theory of self-calibration of a moving camera. The International Journal of Computer Vision, 8(2):123-152.
Mohr, R., Boufama, B., and Brand, P. 1993a. Accurate projective reconstruction. In Applications of Invariance in Computer Vision, J. Mundy and A. Zisserman (Eds.), Vol. 825 of Lecture Notes in Computer Science, Springer-Verlag: Berlin, pp. 257-276.
Mohr, R., Veillon, F., and Quan, L. 1993b. Relative 3d reconstruction using multiple uncalibrated images. In Proc. of the IEEE Conf. on Computer Vision and Pattern Recognition, pp. 543-548.
More, J. 1977. The levenberg-marquardt algorithm, implementation and theory. In Numerical Analysis, G.A. Watson (Ed.), Lecture Notes in Mathematics 630, Springer-Verlag.
Mundy, J.L. and Zisserman, A. (Eds.) 1992.Geometric Invariance in Computer Vision. MIT Press.
Odobez, J.-M. and Bouthemy, P. 1994. Robust multiresolution estimation of parametric motion models applied to complex scenes. Publication Interne 788, IRISA-INRIA Rennes, France.
Olsen, S. 1992. Epipolar line estimation. In Proc. of the 2nd European Conf. on Computer Vision, Santa Margherita Ligure, Italy, pp. 307-311.
Ponce, J. and Genc, Y. 1996. Epipolar geometry and linear subspace methods: A new approach to weak calibration. In Proc. of the IEEE Conf. on Computer Vision and Pattern Recognition, San Francisco, CA, pp. 776-781.
Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T. 1988. Numerical Recipes in C. Cambridge University Press.
Quan, L. 1993. Affine stereo calibration for relative affine shape reconstruction. In Proc. of theFourth BritishMachineVision Conf., Surrey, England, pp. 659-668.
Quan, L. 1995. Invariants of six points and projective reconstruction from three uncalibrated images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 17(1).
Rey W.J. 1983. Introduction to Robust and Quasi-Robust Statistical Methods. Springer: Berlin, Heidelberg.
Robert, L. and Faugeras, O. 1993. Relative 3d positioning and 3d convex hull computation from a weakly calibrated stereo pair. In Proc. of the 4th Int. Conf. on Computer Vision, IEEE Computer Society Press: Berlin, Germany, pp. 540-544. Also INRIA Technical Report 2349.
Rothwell, C., Csurka, G., and Faugeras, O. 1995. A comparison of projective reconstruction methods for pairs of views. In Proc. of the 5th Int. Conf. on Computer Vision, IEEE Computer Society Press: Boston, MA, pp. 932-937.
Rousseeuw, P. and Leroy, A. 1987. Robust Regression and Outlier Detection. John Wiley & Sons: New York.
Shapiro, L. 1993. Affine analysis of image sequences. Ph.D. Thesis, University of Oxford, Department of Engineering Science, Oxford, UK.
Shapiro, L., Zisserman, A., and Brady, M. 1994. Motion from point matches using affine epipolar geometry. In Proc. of the 3rd European Conf. on Computer Vision, J.-O. Eklundh (Ed.), Vol. II of Lecture Notes in Computer Science, Springer-Verlag: Stockholm, Sweden, pp. 73-84.
Shapiro, L. and Brady, M. 1995. Rejecting outliers and estimating errors in an orthogonal-regression framework. Phil. Trans. Royal Soc. of Lon. A, 350:407-439.
Shashua, A. 1994a. Projective structure from uncalibrated images: structure from motion and recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(8):778-790.
Shashua, A. 1994b. Trilinearity in visual recognition by alignment. In Proc. of the 3rd European Conf. on Computer Vision, J.-O. Eklundh (Ed.), Vols. 800-801 of Lecture Notes in Computer Science, Springer-Verlag: Stockholm, Sweden, pp. 479-484.
Spetsakis, M. and Aloimonos, J. 1989. A unified theory of structure from motion. Technical Report CAR-TR-482, Computer Vision Laboratory, University of Maryland.
Sturm, R. 1869. Das problem der projektivitt und seine anwendung auf die flächen zweiten grades. Math. Ann., 1:533-574.
Torr, P. 1995. Motion segmentation and outlier detection. Ph.D. Thesis, Department of Engineering Science, University of Oxford.
Torr, P. and Murray, D. 1993. Outlier detection and motion segmentation. In Sensor Fusion VI, SPIE Vol. 2059, P. Schenker (Ed.), Boston, pp. 432-443.
Torr, P., Beardsley, P., and Murray, D. 1994. Robust vision. British Machine Vision Conf., University of York, UK, pp. 145-154.
Torr, P., Zisserman, A., and Maybank, S. 1995. Robust detection of degenerate configurations for the fundamental matrix. In Proc. of the 5th Int. Conf. on Computer Vision, IEEE Computer Society Press: Boston, MA, pp. 1037-1042.
Torr, P., Zisserman, A., and Maybank, S. 1996. Robust detection of degenerate configurations whilst estimating the fundamental matrix. Technical Report OUEL 2090/96, Oxford University, Dept. of Engineering Science.
Triggs, B. 1995. Matching constraints and the joint image. In Proc. of the 5th Int. Conf. on Computer Vision, IEEE Computer Society Press: Boston, MA, pp. 338-343.
Tsai, R. and Huang, T. 1984. Uniqueness and estimation of three-dimensional motion parameters of rigid objects with curved surface. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6(1):13-26.
Viéville, T., Faugeras, O.D., and Luong, Q.-T. 1996. Motion of points and lines in the uncalibrated case. The International Journal of Computer Vision, 17(1):7-42.
Weinshall, D., Werman, M., and Shashua, A. 1995. Shape tensors for efficient and learnable indexing. IEEEWorkshop on Representation of Visual Scenes, IEEE, pp. 58-65.
Xu, G. and Zhang, Z. 1996. Epipolar Geometry in Stereo, Motion and Object Recognition: A Unified Approach. Kluwer Academic Publishers.
Zeller, C. 1996. Calibration projective affine et euclidienne en vision par ordinateur. Ph.D. Thesis, École Polytechnique.
Zeller, C. and Faugeras, O. 1994. Applications of non-metric vision to some visual guided tasks. In Proc. of the Int. Conf. on Pattern Recognition, Computer Society Press: Jerusalem, Israel, pp. 132-136. A longer version in INRIA Tech. Report RR2308.
Zhang, Z. 1995. Motion and structure of four points from one motion of a stereo rig with unknown extrinsic parameters. IEEE Transactions on Pattern Analysis and Machine Intelligence, 17(12):1222- 1227.
Zhang, Z. 1996a. A new multistage approach to motion and structure estimation: From essential parameters to euclidean motion via fundamental matrix. Research Report 2910, INRIA Sophia-Antipolis, France. Also appeared in Journal of the Optical Society of AmericaA, 14(11):2938-2950, 1997.
Zhang, Z. 1996b. On the epipolar geometry between two images with lens distortion. International Conferences on Pattern Recognition, Vienna, Austria, Vol. I, pp. 407-411.
Zhang, Z. 1996c. Parameter estimation techniques: A tutorial with application to conic fitting. Image and Vision Computing, 15(1):59-76, 1997. Also INRIA Research Report No. 2676, Oct. 1995.
Zhang, Z. and Faugeras, O.D. 1992. 3D Dynamic Scene Analysis: A Stereo Based Approach. Springer: Berlin, Heidelberg.
Zhang, Z., Deriche, R., Luong, Q.-T., and Faugeras, O. 1994. A robust approach to image matching: Recovery of the epipolar geometry. In Proc. International Symposium of Young Investigators on Information\Computer\Control, Beijing, China, pp. 7-28.
Zhang, Z., Deriche, R., Faugeras, O., and Luong, Q.-T. 1995a. A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry. Artificial Intelligence Journal, 78:87-119.
Zhang, Z., Faugeras, O., and Deriche, R. 1995b. Calibrating a binocular stereo through projective reconstruction using both a calibration object and the environment. In Proc. Europe-China Workshop on Geometrical Modelling and Invariants for Computer Vision, R. Mohr and C. Wu (Eds.), Xi'an, China, pp. 253-260. Also appeared in Videre: A Journal of Computer Vision Research, 1(1):58-68, Fall 1997.
Zhang, Z., Luong, Q.-T., and Faugeras, O. 1996. Motion of an uncalibrated stereo rig: Self-calibration and metric reconstruction. IEEE Trans. Robotics and Automation, 12(1):103-113.
Zhuang, X., Wang, T., and Zhang, P. 1992. A highly robust estimator through partially likelihood function modeling and its application in computer vision. IEEE Transactions on Pattern Analysis and Machine Intelligence, 14(1):19-34.
Zisserman, A. 1992. Notes on geometric invariants in vision. BMVC92 Tutorial.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Zhang, Z. Determining the Epipolar Geometry and its Uncertainty: A Review. International Journal of Computer Vision 27, 161–195 (1998). https://doi.org/10.1023/A:1007941100561
Issue Date:
DOI: https://doi.org/10.1023/A:1007941100561