Abstract
The paper has two main contributions: The first is a set of methods for computing structure and motion for m≥ 3 views of 6 points. It is shown that a geometric image error can be minimized over all views by a simple three parameter numerical optimization. Then, that an algebraic image error can be minimized over all views by computing the solution to a cubic in one variable. Finally, a minor point, is that this “quasi-linear” linear solution enables a more concise algorithm, than any given previously, for the reconstruction of 6 points in 3 views.
The second contribution is an m view n ≥ 6 point robust reconstruction algorithm which uses the 6 point method as a search engine. This extends the successful RANSAC based algorithms for 2-views and 3-views to m views. The algorithm can cope with missing data and mismatched data and may be used as an efficient initializer for bundle adjustment.
The new algorithms are evaluated on synthetic and real image sequences, and compared to optimal estimation results (bundle adjustment).
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References
P. Beardsley, P. Torr, and A. Zisserman. 3D model acquisition from extended image sequences. In Proc. ECCV, LNCS 1064/1065, pages 683–695. Springer-Verlag, 1996.
S. Carlsson. Duality of reconstruction and positioning from projective views. In IEEE Workshop on Representation of Visual Scenes, Boston, 1995.
S. Carlsson and D. Weinshall. Dual computation of projective shape and camera positions from multiple images. IJCV, 1998. in Press.
M. A. Fischler and R. C. Bolles. Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography. Comm. ACM, 24(6):381–395, 1981.
A. W. Fitzgibbon and A. Zisserman. Automatic camera recovery for closed or open image sequences. In Proc. ECCV, pages 311–326. Springer-Verlag, Jun 1998.
R. I. Hartley. Euclidean reconstruction from uncalibrated views. In J.L. Mundy, A. Zisserman, and D. Forsyth, editors, Proc. 2nd European-US Workshop on In-variance, Azores, pages 187–202, 1993.
R. I. Hartley. Projective reconstruction and invariants from multiple images. IEEE T-PAMI, 16:1036–1041, October 1994.
R. I. Hartley. Multilinear relationships between coordinates of corresponding image points and lines. In Proceedings of the Sophus Lie Symposium, Nordfjordeid, Norway (not published yet), 1995.
R. I. Hartley. Minimizing algebraic error. Phil. Trans. R. Soc. Lond. A, 356(1740):1175–1192, 1998.
A. Heyden. Projective structure and motion from image sequences using subspace methods. In Scandinavian Conference on Image Analysis, Lappenraanta, 1997.
D. Jacobs. Linear fitting with missing data: Applications to structure from motion and to characterizing intensity images. In Proc. CVPR, pages 206–212, 1997.
S. Mahamud and M. Hebert. Iterative_projective reconstruction from multiple views. In Proc. CVPR, 2000.
S. J. Maybank and A. Shashua. Ambiguity in reconstruction from images of six points. In Proc. ICCV, pages 703–708, 1998.
P. F. McLauchlan and D. W. Murray. A unifying framework for structure from motion recovery from image sequences. In Proc. ICCV, pages 314–320, 1995.
L. Quan. Invariants of 6 points from 3 uncalibrated images. In J. O. Eckland, editor, Proc. ECCV, pages 459–469. Springer-Verlag, 1994.
L. Quan, A. Heyden, and F. Kahl. Minimal projective reconstruction with missing data. In Proc. CVPR, pages 210–216, Jun 1999.
I. D. Reid and D. W. Murray. Active tracking of foveated feature clusters using affine structure. IJCV, 18(1):41–60, 1996.
P. Sturm and W. Triggs. A factorization based algorithm for multi-image projective structure and motion. In Proc. ECCV, pages 709–720, 1996.
C. Tomasi and T. Kanade. Shape and motion from image streams under orthography: A factorization approach. IJCV, 9(2):137–154, Nov 1992.
P. H. S. Torr and D. W. Murray. The development and comparison of robust methods for estimating the fundamental matrix. IJCV, 24(3):271–300, 1997.
P. H. S. Torr and A. Zisserman. Robust parameterization and computation of the trifocal tensor. Image and Vision Computing, 15:591–605, 1997.
P. H. S. Torr and A. Zisserman. Robust computation and parameterization of multiple view relations. In Proc. ICCV, pages 727–732, Jan 1998.
W. Triggs. Factorization methods for projective structure and motion. In Proc. CVPR, pages 845–851, 1996.
X. Yan, X. Dong-hui, P. Jia-xiong, and D. Ming-yue. The unique solution of projective invariants of six points from four uncalibrated images. Pattern Recognition, 30(3):513–517, 1997.
Z. Zhang, R. Deriche, O. D. Faugeras, and Q. Luong. A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry. Artificial Intelligence, 78:87–119, 1995.
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Schaffalitzky, F., Zisserman, A., Hartley, R.I., Torr, P.H.S. (2000). A Six Point Solution for Structure and Motion. In: Computer Vision - ECCV 2000. ECCV 2000. Lecture Notes in Computer Science, vol 1842. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45054-8_41
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DOI: https://doi.org/10.1007/3-540-45054-8_41
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