Journal of Mathematical Imaging and Vision

, Volume 57, Issue 3, pp 402–422 | Cite as

Learning the Geometric Structure of Manifolds with Singularities Using the Tensor Voting Graph

  • Shay DeutschEmail author
  • Gérard Medioni


We present a general framework that addresses manifolds with singularities and multiple intersecting manifolds, which is also robust against a large number of outliers. We suggest a hybrid local–global method that leverages the algorithmic capabilities of the tensor voting framework and, unlike tensor voting, is capable of reliably inferring the global structure of complex manifolds by using a unique graph construction, called the tensor voting graph (TVG). Moreover, we propose to explicitly and directly resolve the ambiguities near the intersections with a novel algorithm, which uses the TVG and the positions of the points near the manifold intersections. Experimental results in estimating geodesic distances and clustering demonstrate that our framework outperforms the state of the art, especially on geometric complex settings such as when the tangent spaces at the intersections points are not orthogonal and in the presence of a large amount of outliers.


Tensor voting Manifold learning Unsupervised learning Intersecting manifolds 


  1. 1.
    Tenenbaum, J., de Silva, V., Langford, J.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000)CrossRefGoogle Scholar
  2. 2.
    Roweis, S., Saul, L.: Nonlinear dimensionality reduction by locally linear embedding. Science 290, 2323–2326 (2000)CrossRefGoogle Scholar
  3. 3.
    Belkin, M., Que, Q., Wang, Y., Zhou, X.: Graph laplacians on singular manifolds: toward understanding complex spaces: graph laplacians on manifolds with singularities and boundaries, CoRR, vol. abs/1211.6727 (2012)Google Scholar
  4. 4.
    Mordohai, P., Medioni, G.: Tensor Voting: A Perceptual Organization Approach to Computer Vision and Machine Learning. Morgan & Claypool Publishers, San Rafael (2006)zbMATHGoogle Scholar
  5. 5.
    Wang, Y., Jiang, Y., Wu, Y., Zhou, Z.: Spectral clustering on multiple manifolds. IEEE Trans. Neural Netw. 22(7), 1149–1161 (2011)CrossRefGoogle Scholar
  6. 6.
    Gong, D., Zhao, X., Medioni, G.: Robust multiple manifold structure learning. In: ICML (2012)Google Scholar
  7. 7.
    Goldberg, A.B., Zhu, X., Singh, A., Xu, Z., Nowak, R.: Multi-manifold semi-supervised learning. In: AISTATS, pp. 169–176 (2009)Google Scholar
  8. 8.
    EryArias-Castro, G., Zhang, T.: Spectral clustering based on local pca. In Review (2013)Google Scholar
  9. 9.
    Belkin, M., Niyogi, P.: Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput 15(6), 1373–1396 (2003)CrossRefzbMATHGoogle Scholar
  10. 10.
    Donoho, D.L., Grimes, C.: Hessian eigenmaps: locally linear embedding techniques for high dimensional data. Proc. Natl. Acad. Sci. USA 100, 5591 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Coifman, R.R., Lafon, S., Lee, A.B., Maggioni, M., Warner, F., Zucker, S.: Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps. In: Proceedings of the National Academy of Sciences, pp. 7426–7431 (2005)Google Scholar
  12. 12.
    Zhang, Z., Zha, H.: Principal manifolds and nonlinear dimensionality reduction via tangent space alignment. SIAM J. Sci. Comput. 26(1), 313–338 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Brand, M.: Charting a manifold. Adv. Neural Inf. Process. Syst., pp. 985–992 (2003)Google Scholar
  14. 14.
    Lin, T., Zha, H.: Riemannian manifold learning. IEEE Trans. Pattern Anal. Mach. Intell. 30(5), 796–809 (2008)CrossRefGoogle Scholar
  15. 15.
    Dollár, P., Rabaud, V., Belongie, S.: Non-isometric manifold learning: analysis and an algorithm. In: Proceedings of the 24th International Conference on Machine Learning, pp. 241–248 (2007)Google Scholar
  16. 16.
    Singer, A., Wu, H.: Vector diffusion maps and the connection laplacian. Commun. Pure Appl. Math. 65(8), 1067–1144 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Zelnik-manor, L., Perona, P.: Self-tuning spectral clustering. In: Advances in Neural Information Processing Systems, pp. 1601–1608 (2004)Google Scholar
  18. 18.
    Gionis, A., Hinneburg, A., Papadimitriou, S., Tsaparas, P.: Dimension induced clustering. In: LWA, pp. 109–110 (2005)Google Scholar
  19. 19.
    Vidal, R., Ma, Y., Sastry, S.: Generalized principal component analysis (gpca) (2003)Google Scholar
  20. 20.
    Elhamifar, E., Vidal, R.: Sparse subspace clustering. In: CVPR, pp. 2790–2797 (2009)Google Scholar
  21. 21.
    Chen, G., Lerman, G.: Spectral curvature clustering (scc). Int. J. Comput. Vis. 81(3), 317–330 (2009)CrossRefGoogle Scholar
  22. 22.
    Ng, A., Jordan, M., Weiss, Y.: On spectral clustering: analysis and an algorithm. In: Advances in Neural Information Processing Systems, pp. 849–856 (2001)Google Scholar
  23. 23.
    Deutsch, S., Medioni, G.G.: Unsupervised learning using the tensor voting graph. In: Proceedings of Scale Space and Variational Methods in Computer Vision—5th International Conference, pp. 282–293. SSVM 2015, Lège-Cap Ferret, 31 May–4 June 2015Google Scholar
  24. 24.
    Deutsch, S., Medioni, G.G.: Intersecting manifolds: detection, segmentation, and labeling. In: Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence, pp. 3445–3452. IJCAI 2015, Buenos Aires, 25–31 July 2015Google Scholar
  25. 25.
    Biederman, I.: Recognition-by-components: a theory of human image understanding. Psychol. Rev. 94, 115–147 (1987)CrossRefGoogle Scholar
  26. 26.
    Waltz, D.L.: Generating semantic descriptions from drawings of scenes with shadows. Technical Report, Cambridge, MA (1972)Google Scholar
  27. 27.
    Mordohai, P., Medioni, G.: Dimensionality estimation, manifold learning and function approximation using tensor voting. J. Mach. Learn. Res. 11, 411–450 (2010)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Niyogi, P., Smale, S., Weinberger, S.: Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geom. 39(1), 419–441 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Dijkstra, E.: Communication with an Automatic Computer. Ph.D. thesis, University of Amsterdam (1959)Google Scholar
  30. 30.
    Genovese, C.R., Perone-Pacifico, M., Verdinelli, I., Wasserman, L.: Minimax manifold estimation. J. Mach. Learn. Res. 13, 1263–1291 (2012)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Waltz, D.: Understanding line drawings of scenes with shadows. In: The Psychology of Computer Vision. McGraw-Hill (1975)Google Scholar
  32. 32.
    Mordohai, P., Medioni, G.: Junction inference and classification for figure completion using tensor voting. In: 2012 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, vol. 4, pp. 56 (2004)Google Scholar
  33. 33.
    Tang, C.-K., Medioni, G.G.: Inference of integrated surface, curve, and junction descriptions from sparse 3d data. IEEE Trans. Pattern Anal. Mach. Intell. 20(11), 1206–1223 (1998)CrossRefGoogle Scholar
  34. 34.
    Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17(4), 395–416 (2007)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Li, Z., Guo, J., Cheong, L.F., Zhou, S.Z.: Perspective motion segmentation via collaborative clustering. In: ICCV, pp. 1369–1376 (2013)Google Scholar
  36. 36.
    Arias-Castro, E., Chen, G., Lerman, G.: Spectral clustering based on local linear approximations. Electron. J. Stat. 5, 1537–1587 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Martin, S., Pollock, S.N., Coutsias, E.A., Watson, J.P., Brown, W.M.: Algorithmic dimensionality reduction for molecular structure analysis. J. Chem. Phys. 129(6), 064118 (2008)CrossRefGoogle Scholar
  38. 38.
    Gong, D., Medioni, G.: Dynamic manifold warping for view invariant action recognition. In: IEEE International Conference on Computer Vision, pp. 571–578 (2011)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of Southern CaliforniaLos AngelesUSA

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