Abstract
We present practical algorithms for stratified autocalibration with theoretical guarantees of global optimality. Given a projective reconstruction, we first upgrade it to affine by estimating the position of the plane at infinity. The plane at infinity is computed by globally minimizing a least squares formulation of the modulus constraints. In the second stage, this affine reconstruction is upgraded to a metric one by globally minimizing the infinite homography relation to compute the dual image of the absolute conic (DIAC). The positive semidefiniteness of the DIAC is explicitly enforced as part of the optimization process, rather than as a post-processing step.
For each stage, we construct and minimize tight convex relaxations of the highly non-convex objective functions in a branch and bound optimization framework. We exploit the inherent problem structure to restrict the search space for the DIAC and the plane at infinity to a small, fixed number of branching dimensions, independent of the number of views. Chirality constraints are incorporated into our convex relaxations to automatically select an initial region which is guaranteed to contain the global minimum.
Experimental evidence of the accuracy, speed and scalability of our algorithm is presented on synthetic and real data.
Article PDF
Similar content being viewed by others
References
Agrawal, M. (2004). On automatic determination of varying focal lengths using semidefinite programming. In International conference on image processing.
Agarwal, S., Chandraker, M., Kahl, F., Belongie, S., & Kriegman, D. (2006). Practical global optimization for multiview geometry. In European conference on computer vision (pp. 592–605).
Agarwal, S., Snavely, N., & Seitz, S. (2008). Fast algorithms for l ∞ problems in multiview geometry. In IEEE conference on computer vision and pattern recognition.
Al-Khayyal, F., & Falk, J. (1983). Jointly constrained biconvex programming. Mathematics of Operations Research, 8, 273–286.
Boyd, S., & Vandenberghe, L. (2004). Convex optimization. Cambridge: Cambridge University Press.
Breuel, T. (2002). A comparison of search strategies for geometric branch and bound algorithms. In European conference on computer vision (pp. 837–850).
Chandraker, M., Agarwal, S., Kahl, F., Nistér, D., & Kriegman, D. (2007a). Autocalibration via rank-constrained estimation of the absolute quadric. In IEEE conference on computer vision and pattern recognition.
Chandraker, M., Agarwal, S., Kriegman, D., & Belongie, S. (2007b). Globally optimal affine and metric upgrades in stratified autocalibration. In International conference on computer vision.
Faugeras, O. D. (1992). What can be seen in three dimensions with an uncalibrated stereo rig. In European conference on computer vision (pp. 563–578). Berlin: Springer.
Faugeras, O., Luong, Q. T., & Maybank, S. (1992). Camera self-calibration: theory and experiments. In European conference on computer vision (pp. 321–334).
Freedman, D. (2003). Effective tracking through tree-search. IEEE Transactions on Pattern Analysis and Machine Intelligence, 25(5), 604–615.
Fusiello, A., Benedetti, A., Farenzena, M., & Busti, A. (2004). Globally convergent autocalibration using interval analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(12), 1633–1638.
Gat, Y. (2003). A branch-and-bound technique for nano-structure image segmentation. In Computer vision and pattern recognition workshop.
Hartley, R. I. (1998). Chirality. International Journal of Computer Vision, 26(1), 41–61.
Hartley, R., & Kahl, F. (2007). Optimal algorithms in multiview geometry. In Asian conference on computer vision (pp. 13–34).
Hartley, R. I., & Zisserman, A. (2004). Multiple view geometry in computer vision. Cambridge: Cambridge University Press.
Hartley, R., Gupta, R., & Chang, T. (1992). Stereo from uncalibrated cameras. In IEEE conference on computer vision and pattern recognition (pp. 761–764), Champaign, USA.
Hartley, R. I., Hayman, E., de Agapito, L., & Reid, I. (1999). Camera calibration and the search for infinity. In International conference on computer vision (pp. 510–517).
Henrion, D., & Lasserre, J. B. (2003). GloptiPoly: global optimization over polynomials with Matlab and SeDuMi. ACM Transactions on Mathematical Software, 29(2), 165–194.
Heyden, A., & Åström, K. (1996). Euclidean reconstruction from constant intrinsic parameters. In International conference on pattern recognition (pp. 339–343).
Horst, R., & Tuy, H. (2006). Global optimization: deterministic approaches. Berlin: Springer.
Kahl, F. (2005). Multiview geometry and the l ∞-norm. In International conference on computer vision.
Kahl, F., & Henrion, D. (2005). Globally optimal estimates for geometric reconstruction problems. In International conference on computer vision (pp. 978–985).
Lampert, C., Blaschko, M., & Hofmann, T. (2008). Beyond sliding windows: object localization by efficient subwindow search. In IEEE conference on computer vision and pattern recognition.
Land, A. H., & Doig, A. G. (1960). An automatic method of solving discrete programming problems. Econometrica, 28(3), 497–520.
Lasserre, J. B. (2001). Global optimization with polynomials and the problem of moments. SIAM Journal on Optimization, 11, 796–817.
Levenberg, K. (1944). A method for the solution of certain non-linear problems in least squares. The Quarterly of Applied Mathematics, 2, 164–168.
Manning, R., & Dyer, C. (2001). Metric self calibration from screw-transform manifolds. In IEEE conference on computer vision and pattern recognition (pp. 590–597).
Marquardt, D. (1963). An algorithm for least-squares estimation of nonlinear parameters. SIAM Journal on Applied Mathematics, 11, 431–441.
McCormick, G. (1976). Computability of global solutions to factorable nonconvex programs. Mathematical Programming, 10, 147–175.
Moore, R. E. (1966). Interval analysis. New York: Prentice-Hall.
Nistér, D. (2004). Untwisting a projective reconstruction. International Journal of Computer Vision, 60(2), 165–183.
Olsson, C., Kahl, F., & Oskarsson, M. (2009). Branch-and-bound methods for euclidean registration problems. IEEE Transactions on Pattern Analysis and Machine Intelligence, 31(5), 783–794.
Pollefeys, M., & Gool, L. V. (1999). Stratified self-calibration with the modulus constraint. IEEE Transactions on Pattern Analysis and Machine Intelligence, 21(8), 707–724.
Pollefeys, M., Koch, R., & Van Gool, L. (1998). Self-calibration and metric reconstruction in spite of varying and unknown internal camera parameters. In International conference on computer vision (pp. 90–95).
Pollefeys, M., Verbiest, F., & Gool, L. J. V. (2002). Surviving dominant planes in uncalibrated structure and motion recovery. In European conference on computer vision (pp. 837–851).
Prajna, S., Papachristodoulou, A., & Parrilo, P. (2002). Introducing SOSTOOLS: a general purpose sum of squares programming solver. In IEEE conference on decision and control.
Schaffalitzky, F. (2000). Direct solution of modulus constraints. In Indian conference on computer vision, graphics and image processing (pp. 314–321).
Sim, K., & Hartley, R. (2006). Recovering camera motion using l ∞ minimization. In IEEE conference on computer vision and pattern recognition (pp. 1230–1237).
Stewénius, H., Schaffalitzky, F., & Nistér, D. (2005). How hard is three-view triangulation really? In International conference on computer vision (pp. 686–693).
Sturm, J. (1999). Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones. Optimization Methods and Software, 11–12, 625–653.
Sturm, P. (2000). A case against Kruppa’s equations for camera self-calibration. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(10), 1199–1204.
Sturm, P., & Triggs, B. (1996). A factorization based algorithm for multi-image projective structure and motion. In European conference on computer vision (pp. 709–720).
Tawarmalani, M., & Sahinidis, N. (2001). Semidefinite relaxations of fractional programs via novel convexification techniques. Journal of Global Optimization, 20, 137–158.
Tawarmalani, M., & Sahinidis, N. (2002). Convex extensions and envelopes of lower semi-continuous functions. Mathematical Programming, 93(2), 247–263.
Tomasi, C., & Kanade, T. (1992). Shape and motion from image streams under orthography: a factorization method. International Journal of Computer Vision, 9(2), 137–154.
Triggs, B. (1997). Autocalibration and the absolute quadric. In IEEE conference on computer vision and pattern recognition (pp. 609–614).
Zongker, D., & Jain, A. (1996). Algorithms for feature selection: an evaluation. In International conference on pattern recognition (pp. 18–22).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Chandraker, M., Agarwal, S., Kriegman, D. et al. Globally Optimal Algorithms for Stratified Autocalibration. Int J Comput Vis 90, 236–254 (2010). https://doi.org/10.1007/s11263-009-0305-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11263-009-0305-2