The Community for Technology Leaders
RSS Icon
Subscribe
Issue No.09 - September (2008 vol.30)
pp: 1603-1617
ABSTRACT
This paper presents a new framework for solving geometric structure and motion problems based on {$L_\infty$}-norm. Instead of using the common sum-of-squares cost-function, that is, the {$L_\two$}-norm, the model-fitting errors are measured using the {$L_\infty$}-norm. Unlike traditional methods based on {$L_\two$}, our framework allows for efficient computation of global estimates. We show that a variety of structure and motion problems, for example, triangulation, camera resectioning and homography estimation can be recast as quasi-convex optimization problems within this framework. These problems can be efficiently solved using Second-Order Cone Programming (SOCP) which is a standard technique in convex optimization. The methods have been implemented in Matlab and the resulting toolbox has been made publicly available. The algorithms have been validated on real data in different settings on problems with small and large dimensions and with excellent performance.
INDEX TERMS
Image Processing and Computer Vision, Convex programming, Constrained optimization, Global optimization
CITATION
Fredrik Kahl, Richard Hartley, "Multiple-View Geometry Under the {$L_\infty$}-Norm", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.30, no. 9, pp. 1603-1617, September 2008, doi:10.1109/TPAMI.2007.70824
REFERENCES
[1] S. Agarwal, M.K. Chandraker, F. Kahl, D.J. Kriegman, and S. Belongie, “Practical Global Optimization for Multiview Geometry,” Proc. Ninth European Conf. Computer Vision, pp. 592-605, 2006.
[2] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge Univ. Press, 2004.
[3] A. Fusiello, A. Benedetti, M. Farenzena, and A. Busti, “Globally Convergent Autocalibration Using Interval Analysis,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 26, no. 12, pp. 1633-1638, Dec. 2004.
[4] V.M. Govindu, “Lie-Algebraic Averaging for Globally Consistent Motion Estimation,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, vol. 1, pp. 684-691, 2004.
[5] R. Hartley and F. Schaffalitzky, “${\rm L}_{\infty}$ Minimization in Geometric Reconstruction Problems,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, vol. 1, pp. 504-509, 2004.
[6] R.I. Hartley, “Defense of the Eight-Point Algorithm,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 19, no. 6, pp. 580-593, June 1997.
[7] R.I. Hartley and P. Sturm, “Triangulation,” Computer Vision and Image Understanding, vol. 68, no. 2, pp. 146-157, Nov. 1997.
[8] R.I. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, second ed. Cambridge Univ. Press, 2004.
[9] A. Heyden, “Projective Structure and Motion from Image Sequences Using Subspace Methods,” Proc. 10th Scandinavian Conf. Image Analysis, pp. 963-968, 1997.
[10] F. Kahl, “Multiple View Geometry and the $L_{\infty}\hbox{-}{\rm Norm}$ ,” Proc. 10th Int'l Conf. Computer Vision, pp. 1002-1009, 2005.
[11] F. Kahl and D. Henrion, “Globally Optimal Estimates for Geometric Reconstruction Problems,” Proc. 10th Int'l Conf. Computer Vision, pp. 978-985, 2005.
[12] Q. Ke and T. Kanade, “Quasiconvex Optimization for Robust Geometric Reconstruction,” Proc. 10th Int'l Conf. Computer Vision, pp. 986-993, 2005.
[13] Q. Ke and T. Kanade, “Uncertainty Models in Quasiconvex Optimization for Geometric Reconstruction,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 1199-1205, 2006.
[14] M.S. Lobo, L. Vandenberghe, S.P. Boyd, and H. Lebret, “Applications of Second-Order Cone Programming,” Linear Algebra and Its Applications, vol. 284, pp. 193-228, 1998.
[15] H.C. Longuet-Higgins, “A Computer Algorithm for Reconstructing a Scene from Two Projections,” Nature, vol. 293, pp. 133-135, 1981.
[16] D. Nistér, “Automatic Dense Reconstruction from Uncalibrated Video Sequences,” PhD dissertation, Royal Inst. of Technology KTH, Sweden, 2001.
[17] C. Rother and S. Carlsson, “Linear Multi View Reconstruction and Camera Recovery Using a Reference Plane,” Int'l J. Computer Vision, vol. 49, nos. 2/3, pp. 117-141, 2002.
[18] K. Sim and R. Hartley, “Recovering Camera Motion Using the $L_{\infty}\hbox{-}{\rm Norm}$ ,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 1230-1237, 2006.
[19] K. Sim and R. Hartley, “Removing Outliers Using the $L_{\infty}\hbox{-}{\rm Norm}$ ,” Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 485-492, 2006.
[20] H. Stewénius, F. Schaffalitzky, and D. Nistér, “How Hard Is Three-View Triangulation Really,” Proc. 10th Int'l Conf. Computer Vision, pp. 686-693, 2005.
[21] J.F. Sturm, “Using SeDuMi 1.02, a MATLAB Toolbox for Optimization over Symmetric Cones,” Optimization Methods and Software, vols. 11-12, pp. 625-653, 1999.
[22] P. Sturm and B. Triggs, “A Factorization Based Algorithm for Multi-Image Projective Structure and Motion,” Proc. Fourth European Conf. Computer Vision, pp. 709-720, 1996.
[23] C. Tomasi and T. Kanade, “Shape and Motion from Image Streams under Orthography: A Factorization Method,” Int'l J. Computer Vision, vol. 9, no. 2, pp. 137-154, 1992.
[24] P.H.S. Torr and A.W. Fitzgibbon, “Invariant Fitting of Two View Geometry,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 26, no. 5, pp. 648-650, May 2004.
[25] B. Triggs, P.F. McLauchlan, R.I. Hartley, and A.W. Fitzgibbon, “Bundle Adjustment—A Modern Synthesis,” Proc. ICCV Workshop Vision Algorithms, pp. 298-372, 1999.
[26] M. Uyttendaele, A. Criminisi, S.B. Kang, S. Winder, R. Szeliski, and R. Hartley, “Image-Based Interactive Exploration of Real-World Environments,” IEEE Computer Graphics and Applications, vol. 24, no. 3, pp. 52-63, May/June 2004.
[27] L. Wolf and A. Shashua, “On Projection Matrices $P^{k} \mapsto P^{2}$ , $k = 3, \ldots, 6$ , and Their Applications in Computer Vision,” Proc. Eighth Int'l Conf. Computer Vision, vol. 1, pp. 412-419, 2001.
21 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool