Skip to main content
Log in

\({L_q}\)-Closest-Point to Affine Subspaces Using the Generalized Weiszfeld Algorithm

  • Published:
International Journal of Computer Vision Aims and scope Submit manuscript

Abstract

This paper presents a method for finding an \(L_q\)-closest-point to a set of affine subspaces, that is a point for which the sum of the q-th power of orthogonal distances to all the subspaces is minimized, where \(1 \le q < 2\). We give a theoretical proof for the convergence of the proposed algorithm to a unique \(L_q\) minimum. The proposed method is motivated by the \(L_q\) Weiszfeld algorithm, an extremely simple and rapid averaging algorithm, that finds the \(L_q\) mean of a set of given points in a Euclidean space. The proposed algorithm is applied to the triangulation problem in computer vision by finding the \(L_q\)-closest-point to a set of lines in 3D. Our experimental results for the triangulation problem confirm that the \(L_q\)-closest-point method, for \(1 \le q < 2\), is more robust to outliers than the \(L_2\)-closest-point method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. Gradient descent with backtracking means head in the descent downhill gradient direction. If this does not result in a decreased cost, then back up by making a smaller step, until the cost decreases.

References

  • Aftab, K., Hartley, R., & Trumpf, J. (submitted). Generalized Weiszfeld algorithms for Lq optimization.

  • Ameri, B., & Fritsch, D. (2000). Automatic 3d building reconstruction using plane-roof structures. Washington, DC: ASPRS.

    Google Scholar 

  • Brimberg, J. (2003). Further notes on convergence of the weiszfeld algorithm. Yugoslav Journal of Operations Research, 13(2), 199–206.

    Article  MathSciNet  MATH  Google Scholar 

  • Brimberg, J., & Chen, R. (1998). A note on convergence in the single facility minisum location problem. Computers & Mathematics with Applications, 35(9), 25–31.

    Article  MathSciNet  MATH  Google Scholar 

  • Brimberg, J., & Love, R. F. (1993). Global convergence of a generalized iterative procedure for the minisum location problem with lp distances. Operations Research, 41(6), 1153–1163.

    Article  MathSciNet  MATH  Google Scholar 

  • Chartrand, R., & Yin, W. (2008). Iteratively reweighted algorithms for compressive sensing. In IEEE International Conference on Acoustics, Speech and Signal Processing.

  • Chi, E. C., & Lange, K. (2014). A look at the generalized heron problem through the lens of majorization-minimization. The American Mathematical Monthly, 121(2), 95–108.

    Article  MathSciNet  MATH  Google Scholar 

  • Daubechies, I., DeVore, R., Fornasier, M., & Gunturk, S. (2008). Iteratively re-weighted least squares minimization: Proof of faster than linear rate for sparse recovery. In 42nd Annual Conference on Information Sciences and Systems.

  • Dick, A.R., Torr, P.H., Ruffle, S.J., & Cipolla, R. (2001). Combining single view recognition and multiple view stereo for architectural scenes. In IEEE International Conference on Computer Vision

  • Eckhardt, U. (1980). Weber’s problem and weiszfeld’s algorithm in general spaces. Mathematical Programming, 18(1), 186–196.

    Article  MathSciNet  MATH  Google Scholar 

  • Eldar, Y., & Mishali, M. (2009). Robust recovery of signals from a structured union of subspaces. IEEE Transactions on Information Theory, 55(11), 5302–5316.

    Article  MathSciNet  Google Scholar 

  • Fletcher, P. T., Venkatasubramanian, S., & Joshi, S. (2009). The geometric median on Riemannian manifolds with application to robust atlas estimation. NeuroImage, 45(1), S143–S152.

  • Furukawa, Y., Curless, B., Seitz, S., & Szeliski, R. (2009). Manhattan-world stereo. In IEEE Conference on Computer Vision and Pattern Recognition.

  • Furukawa, Y., Curless, B., Seitz, S.M., & Szeliski, R. (2009). Reconstructing building interiors from images. In IEEE International Conference on Computer Vision.

  • Hartley, R., Aftab, K., & Trumpf, J. (2011). L1 rotation averaging using the Weiszfeld algorithm. In IEEE Conference on Computer Vision and Pattern Recognition.

  • Hartley, R., Trumpf, J., Dai, Y., & Li, H. (2013). Rotation averaging. International Journal of Computer Vision, 103(3), 267–305.

    Article  MathSciNet  MATH  Google Scholar 

  • Hartley, R. I., & Sturm, P. (1997). Triangulation. Computer Vision and Image Understanding, 68(2), 146–157.

    Article  Google Scholar 

  • Hartley, R. I., & Zisserman, A. (2004). Multiple view geometry in computer vision (2nd ed.). Cambridge: Cambridge University Press.

    Book  MATH  Google Scholar 

  • Henry, P., Krainin, M., Herbst, E., Ren, X., & Fox, D. (2010). Rgb-d mapping: Using depth cameras for dense 3d modeling of indoor environments. In the 12th International Symposium on Experimental Robotics (ISER).

  • Kleiman, S., & Laksov, D. (1972). Schubert calculus. American Mathematical Monthly, 79, 1061–1082.

    Article  MathSciNet  MATH  Google Scholar 

  • Lourakis, M. A., & Argyros, A. (2009). SBA: A Software Package for Generic Sparse Bundle Adjustment. ACM Transactions on Mathematical Software (TOMS), 36(1), 2.

    Article  MathSciNet  Google Scholar 

  • Luenberger, D. G. (2003). Linear and nonlinear programming. New York: Springer.

    MATH  Google Scholar 

  • Ma, R. (2004). Building model reconstruction from lidar data and aerial photographs. Ph.D. thesis, The Ohio State University.

  • Mordukhovich, B., & Nam, N. M. (2011). Applications of variational analysis to a generalized fermat-torricelli problem. Journal of Optimization Theory and Applications, 148(3), 431–454.

    Article  MathSciNet  MATH  Google Scholar 

  • Mordukhovich, B. S., Nam, N. M., & Salinas, J, Jr. (2012). Solving a generalized heron problem by means of convex analysis. The American Mathematical Monthly, 119(2), 87–99.

    Article  MathSciNet  MATH  Google Scholar 

  • Müller, P., Zeng, G., Wonka, P., & Van Gool, L. (2007). Image-based procedural modeling of facades. ACM Transactions on Graphics, 26(3), 85.

    Article  Google Scholar 

  • Pu, S., & Vosselman, G. (2009). Knowledge based reconstruction of building models from terrestrial laser scanning data. ISPRS Journal of Photogrammetry and Remote Sensing, 64(6), 575–584.

    Article  Google Scholar 

  • Remondino, F., & El-Hakim, S. (2006). Image-based 3d modelling: A review. The Photogrammetric Record, 21(115), 269–291.

    Article  Google Scholar 

  • Schindler, K., & Bauer, J. (2003). A model-based method for building reconstruction. In: First IEEE International Workshop on Higher-Level Knowledge in 3D Modeling and Motion Analysis.

  • Semple, J. G., & Kneebone, G. T. (1979). Algebraic Projective Geometry. London: Oxford University Press.

  • Taillandier, F. (2005). Automatic building reconstruction from cadastral maps and aerial images. International Archives of Photogrammetry and Remote Sensing, 36, 105–110.

    Google Scholar 

  • Triggs, B., McLauchlan, P. F., Hartley, R. I., & Fitzgibbon, A. W. (2000). Bundle adjustment—a modern synthesis. In: Vision algorithms: Theory and practice (pp. 298–372). Springer.

  • Vanegas, C.A., Aliaga, D.G., & Benes, B. (2010). Building reconstruction using manhattan-world grammars. In IEEE Conference on Computer Vision and Pattern Recognition.

  • Weiszfeld, E. (1937). Sur le point pour lequel la somme des distances de \(n\) points donnés est minimum. Tohoku Mathematical Journal, 43(355—-386), 2.

  • Werner, T., & Zisserman, A. (2003). New techniques for automated architectural reconstruction from photographs. In European Conference on Computer Vision.

  • Wilczkowiak, M., Trombettoni, G., Jermann, C., Sturm, P., & Boyer, E. (2003). Scene modeling based on constraint system decomposition techniques. In Ninth IEEE International Conference on Computer Vision.

  • Yang, L. (2010). Riemannian median and its estimation. LMS Journal of Computation and Mathematics, 13, 461–479.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This research has been funded by National ICT Australia. National ICT Australia is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the Australian Research Council through the ICT Centre of Excellence program.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Khurrum Aftab.

Additional information

Communicated by M. Hebert.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aftab, K., Hartley, R. & Trumpf, J. \({L_q}\)-Closest-Point to Affine Subspaces Using the Generalized Weiszfeld Algorithm. Int J Comput Vis 114, 1–15 (2015). https://doi.org/10.1007/s11263-014-0791-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11263-014-0791-8

Keywords

Navigation