Abstract
The necessary and sufficient conditions for being able to estimate scene structure, motion and camera calibration from a sequence of images are very rarely satisfied in practice. What exactly can be estimated in sequences of practical importance, when such conditions are not satisfied? In this paper we give a complete answer to this question. For every camera motion that fails to meet the conditions, we give explicit formulas for the ambiguities in the reconstructed scene, motion and calibration. Such a characterization is crucial both for designing robust estimation algorithms (that do not try to recover parameters that cannot be recovered), and for generating novel views of the scene by controlling the vantage point. To this end, we characterize explicitly all the vantage points that give rise to a valid Euclidean reprojection regardless of the ambiguity in the reconstruction. We also characterize vantage points that generate views that are altogether invariant to the ambiguity. All the results are presented using simple notation that involves no tensors nor complex projective geometry, and should be accessible with basic background in linear algebra.
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References
Avidan, S. and Shashua, A. 1997. Novel view synthesis in tensor space. In Proc. of IEEE Conference on Computer Vision and Pattern Recognition, pp. 1034–1040.
Beardsley, P. and Zisserman, A. 1995. Affine calibration of mobile vehicles. In Europe-China Workshop on Geometric Modeling and Invariants for Computer Vision.
Boufama, B., Mohr, R., and Veillon, F. 1993. Euclidean constraints for uncalibrated reconstruction. In ICCV, Berlin, Germany, pp. 466–470.
Carlsson, S. 1994. Multiple image invariance using the double algebra. In Applications of Invariance in Computer Vision.
Christy, S. and Horaud, R. 1996. Euclidean shape and motion from multiple perspective views via affine iterations. IEEE PAMI, 18(11):1098–1104.
Faugeras, O. 1997. What can be seen in three dimensions with an uncalibrated stereo rig? INRIA Research Report, No. 3225.
Faugeras, O. and Papadopoulo, T. 1995. Grassmann-Cayley algebra for modeling systems of cameras and the algebraic equations of the manifold of trifocal tensors. In Proc. of the IEEE Workshop of Representation of Visual Scenes.
Hartley, R. 1994. Lines and points in three views; and integrated approach. In Proc. of the Image Understanding Workshop.
Heyden, A. 1998. Reduced multilinear constraints—Theory and experiments. International Journal of Computer Vision, 30(2):5–26.
Heyden, A. and Äström, K. 1997. Algebraic properties of multilinear constraints. Mathematical Methods in Applied Sciences, 20(13):1135–1162.
Heyden, A., Sparr, G., and Äström, K. 1997. Perception and action using multilinear forms. In Algebraic Frames for the Perception-Action Cycle, G., Sommer and J., Koenderink (Eds.), Springer-Verlag, pp. 54–65.
Luong, Q.-T. and Faugeras, O. 1997. Self-calibration of a moving camera from point correspondences and fundamental matrices. IJCV, 22(3):261–289.
Luong, Q.-T. and Vieville, T. 1994. Canonical representations for the geometries of multiple projective views. ECCV, 589–599.
Ma, Y., Vidal, R., Košecká, J., and Sastry, S. 1999. Camera selfcalibration: Geometry and algorithms. IEEE Transactions on PAMI, submitted. UC Berkeley Technical Report UCB/ERL No. M99/32.
Maybank, S. 1993. Theory of Reconstruction from Image Motion. Springer-Verlag.
Maybank, S. and Faugeras, O. 1992. A theory of self-calibration of a moving camera. IJCV, 8(2):123–152.
Maybank, S. and Shashua, A. 1988. Ambiguity in reconstruction from images of six points. In ICCV, Bombay, India, pp. 703–708.
Moon, T., Van Gool, L., Van Dients, M., and Pauwels, E. 1993. Affine reconstruction from perspective pairs obtained by a translating camera. In Applications of Invariance in Computer Vision, J.L. Mundy and A. Zisserman (Eds.), pp. 297–316.
Sturm, P. 1997. Critical motion sequences for monocular self-calibration and uncalibrated Euclidean reconstruction. CVPR, pp. 1100–1105.
Triggs, B. to appear. The geometry of projective reconstruction I: Matching constraints and the joint image. International Journal of Computer Vision.
Werman, M. and Shashua, A. 1995. The study of 3D-from-2D using elimination. In Proc. of Europe-China Workshop on Geometrical Modeling and Invariants for Computer Vision, Xi'an, China, pp. 94–101.
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Ma, Y., Soatto, S., Košecká, J. et al. Euclidean Reconstruction and Reprojection Up to Subgroups. International Journal of Computer Vision 38, 219–229 (2000). https://doi.org/10.1023/A:1008143307025
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DOI: https://doi.org/10.1023/A:1008143307025