Advertisement

An Energy Minimisation Approach to Attributed Graph Regularisation

  • Zhouyu Fu
  • Antonio Robles-Kelly
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4679)

Abstract

In this paper, we propose a novel approach to graph regularisation based on energy minimisation. Our method hinges in the use of a Ginzburg-Landau functional whose extremum is achieved efficiently by a gradient descend optimisation process. As a result of the treatment given in this paper to the regularisation problem, constraints can be enforced in a straightforward manner. This provides a means to solve a number of problems in computer vision and pattern recognition. To illustrate the general nature of our graph regularisation algorithm, we show results on two application vehicles, photometric stereo and image segmentation. Our experimental results demonstrate the efficacy of our method for both applications under study.

Keywords

Computer Vision Image Segmentation Singular Value Decomposition Gaussian Smoothing Graph Regularisation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Nagel, H., Enkelmann, W.: An investigation of smoothness constraints for the estimation of displacement vector fields from image sequences. IEEE Trans. on Pattern Analysis and Machine Intelligence 8, 565–593 (1986)CrossRefGoogle Scholar
  2. 2.
    Terzopoulos, D.: Multilevel computational processes for visual surface reconstruction. Computer Vision, Graphics and Image Understanding 24, 52–96 (1983)CrossRefGoogle Scholar
  3. 3.
    Marr, D., Poggio, T.: A computational theory of human stereo vision. In: Proceedings of the Royal Society of London. Series B, Biological Sciences. vol. 204, pp. 301–328 (1979)Google Scholar
  4. 4.
    Scharstein, D., Szeliski, R.: A taxonomy and evaluation of dense two-frame stereo correspondence algorithms. Int. Journal of Computer Vision 47(13), 7–42 (2002)zbMATHCrossRefGoogle Scholar
  5. 5.
    Boykov, Y., Jolly, M.P.: Interactive graph cuts for optimal boundary & region segmentation of objects in n-d images. In: Intl. Conf. on Computer Vision, pp. 105–112 (2001)Google Scholar
  6. 6.
    Kolmogorov, V., Zabih, R.: Multi-camera scene reconstruction via graph-cuts. In: European Conf. on Comp. Vision, vol. 3, pp. 82–96 (2002)Google Scholar
  7. 7.
    Vogiatzis, G., Torr, P., Cipolla, R.: Multi-view stereo via volumetric graph-cuts. In: IEEE Conf. on Computer Vision and Pattern Recognition, vol. II, pp. 391–398 (2005)Google Scholar
  8. 8.
    Sun, J., Shum, H.Y., Zheng, N.N.: Stereo matching using belief propagation. In: European Conf. on Comp. Vision, pp. 510–524 (2002)Google Scholar
  9. 9.
    Worthington, P.L., Hancock, E.R.: New constraints on data-closeness and needle map consistency for shape-from-shading. IEEE Transactions on Pattern Analysis and Machine Intelligence 21(12), 1250–1267 (1999)CrossRefGoogle Scholar
  10. 10.
    Barron, J.L., Fleet, D.J., Beauchemin, S.S.: Performance of optical flow techniques. Int. Journal of Computer Vision 12(1), 43–77 (1994)CrossRefGoogle Scholar
  11. 11.
    Blum, A., Chawla, S.: Learning from labeled and unlabeld data using graph mincuts. In: Proc. of Intl. Conf. on Machine Learning, pp. 19–26 (2001)Google Scholar
  12. 12.
    Zhu, X., Ghahramani, Z., Lafferty, J.: Semi-supervised learning using gaussian fields and harmonic functions. In: 20th Intl. Conf. on Machine Learning (2003)Google Scholar
  13. 13.
    Zhou, D., Bousquet, O., Lal, T., Weston, J., Schölkopf, B.: Learning with local and global consistency. In: Neural Information Processing Systems (2003)Google Scholar
  14. 14.
    Zhang, F., Hancock, E.R.: Tensor mri regularization via graph diffusion. In: British Machine Vision Conference, vol. II, pp. 589–598 (2006)Google Scholar
  15. 15.
    Chefd’hotel, C., Tschumperle, D., Deriche, R., Faugeras, O.D.: Constrained flows of matrix-valued functions: Application to diffusion tensor regularization. In: European Conf. on Comp. Vision, vol. I, pp. 251–265 (2002)Google Scholar
  16. 16.
    Tschumperle, D., Deriche, R.: Diffusion tensor regularization with constraints preservation. In: IEEE Conf. on Computer Vision and Pattern Recognition, vol. I, pp. 948–953 (2001)Google Scholar
  17. 17.
    Busemann, H.: The geometry of geodesics. Academic Press, London (1955)zbMATHGoogle Scholar
  18. 18.
    Ranicki, A.: Algebraic l-theory and topological manifolds. Cambridge University Press, Cambridge (1955)Google Scholar
  19. 19.
    Hjaltason, G.R., Samet, H.: Properties of embedding methods for similarity searching in metric spaces. IEEE Trans. on Pattern Analysis and Machine Intelligence 25, 530–549 (2003)CrossRefGoogle Scholar
  20. 20.
    Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000)CrossRefGoogle Scholar
  21. 21.
    Roweis, S.T., Saul, L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science 290, 2323–2326 (2000)CrossRefGoogle Scholar
  22. 22.
    Belkin, M., Niyogi, P.: Laplacian eigenmaps and spectral techniques for embedding and clustering. Neural Information Processing Systems 14, 634–640 (2002)Google Scholar
  23. 23.
    Hein, M., Audibert, J., von Luxburg, U.: From graphs to manifolds - weak and strong pointwise consistency of graph laplacians. In: Auer, P., Meir, R. (eds.) COLT 2005. LNCS (LNAI), vol. 3559, pp. 470–485. Springer, Heidelberg (2005)Google Scholar
  24. 24.
    Ginzburg, V., Landau, L.: On the theory of superconductivity. Zh. Eksp. Teor. Fiz. 20, 1064–1082 (1950)Google Scholar
  25. 25.
    Berger, M.: A Panoramic View of Riemannian Geometry. Springer, Heidelberg (2003)zbMATHGoogle Scholar
  26. 26.
    Chavel, I.: Riemannian Geometry: A Modern Introduction. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar
  27. 27.
    Jost, J.: Riemannian Geometry and Geometric Analysis. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  28. 28.
    Chung, F.R.K.: Spectral Graph Theory. American Mathematical Society, Providence (1997)zbMATHGoogle Scholar
  29. 29.
    Nocedal, J., Wright, S.: Numerical Optimization. Springer, Heidelberg (2000)Google Scholar
  30. 30.
    Stone, M.H.: The generalized weierstrass approximation theorem. Mathematics Magazine 21(4), 167–184 (1948)CrossRefGoogle Scholar
  31. 31.
    Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Woodham, R.: Photometric methods for determining surface orientation from multiple images. Optical Engineering 19(1), 139–144 (1980)Google Scholar
  33. 33.
    Yuille, A., Coughlan, J.: Twenty questions, focus of attention, and a*: A theoretical comparison of optimization strategies. In: Energy Minimization Methods in Computer Vision and Pattern Recognition, pp. 197–212 (1999)Google Scholar
  34. 34.
    Hertzmann, A., Seitz, S.: Example-based photometric stereo: Shape reconstruction with general, varying brdfs. IEEE Trans. on Pattern Analysis and Machine Intelligence 27(8), 1254–1264 (2005)CrossRefGoogle Scholar
  35. 35.
    Tomasi, C., Manduchi, R.: Bilateral filtering for gray and color images. In: Intl. Conf. on Computer Vision, pp. 839–846 (1998)Google Scholar
  36. 36.
    Basri, R., Jacobs, D.: Photometric stereo with general, unknown lighting. In: Proc. Computer Vision and Pattern Recognition, pp. 374–381 (2001)Google Scholar
  37. 37.
    Martin, D., Fowlkes, C., Tal, D., Malik, J.: A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics. In: Int. Conf. on Computer Vision. vol. 2, pp. 416–423 (July 2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Zhouyu Fu
    • 1
  • Antonio Robles-Kelly
    • 1
    • 2
  1. 1.Department of Information Engineering, ANU, CanberraAustralia
  2. 2.NICTA RSISE Bldg. 115, Australian National University, ACT 0200Australia

Personalised recommendations