Advertisement

Building a Weighted Complex from Data

Chapter
  • 2.8k Downloads

Abstract

In some applications, both the neighborhood structure of the data and the weights of the cells are naturally and directly defined by the problem at hand (e.g., road networks, social networks, communication networks, chemical graph theory or surface simplification). However, in many other applications the appropriate representation of the data to be analyzed is not provided (e.g., machine learning). Therefore, to use the tools of discrete calculus, a practitioner must determine the topology and weights of the graph or complex from the data that is most appropriate for solving the problem. In this chapter, we will discuss different techniques for generating a meaningful weighted complex from an embedding or from the data itself. Our focus will be primarily on generating weighted edges and faces from node and/or edge data, but we additionally demonstrate how these techniques may be applied to weighting higher-order structures.

Keywords

Edge Weight Rotation System Laplacian Matrix Cycle Basis Locality Preserve Projection 
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.

References

  1. 15.
    Awate, S.P., Whitaker, R.T.: Unsupervised, information-theoretic, adaptive image filtering for image restoration. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(3), 364–376 (2006) CrossRefGoogle Scholar
  2. 22.
    Barber, C.B., Dobkin, D.P., Huhdanpaa, H.: The quickhull algorithm for convex hulls. ACM Transactions on Mathematical Software 22(4), 469–483 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 23.
    Belkin, M., Niyogi, P.: Laplacian eigenmaps and spectral techniques for embedding and clustering. In: Advances in Neural Information Processing Systems, vol. 14, pp. 585–591. MIT Press, Cambridge (2001) Google Scholar
  4. 30.
    Berger, F., Gritzmann, P., de Vries, S.: Minimum cycle bases for network graphs. Algorithmica 40(1), 51–62 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 33.
    Bezem, G.J., van Leeuwen, J.: Enumeration in graphs. Technical Report RUU-CS-87-7, Rijksuniversiteit Utrecht (1987) Google Scholar
  6. 38.
    Black, M.J., Sapiro, G., Marimont, D.H., Heeger, D.: Robust anisotropic diffusion. IEEE Transactions on Image Processing 7(3), 421–432 (1998) CrossRefGoogle Scholar
  7. 54.
    Boykov, Y., Kolmogorov, V.: Computing geodesics and minimal surfaces via graph cuts. In: Proceedings of International Conference on Computer Vision, vol. 1 (2003) Google Scholar
  8. 61.
    Bruckstein, A.M., Netravali, A.N., Richardson, T.J.: Epi-convergence of discrete elastica. Applicable Analysis 79(1–2), 137–171 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 63.
    Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Modeling and Simulation 4(2), 490–530 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 66.
    Cassell, A.C., Henderson, J.C., Ramachandran, K.: Cycle bases of minimal measure for the structural analysis of skeletal structures by the flexibility method. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 350, 61–70 (1976) zbMATHCrossRefGoogle Scholar
  11. 78.
    Chua, L.O., Chen, L.: On optimally sparse cycle and coboundary basis for a linear graph. IEEE Transactions on Circuit Theory CT-20, 495–503 (1973) Google Scholar
  12. 87.
    Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms. MIT Press, Cambridge (2001) zbMATHGoogle Scholar
  13. 93.
    Cremers, D., Grady, L.: Statistical priors for efficient combinatorial optimization via graph cuts. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) Computer Vision—ECCV 2006. Lecture Notes in Computer Science, vol. 3, pp. 263–274. Springer, Berlin (2006) CrossRefGoogle Scholar
  14. 100.
    de Pina, J.: Applications of shortest path methods. Ph.D. thesis, University of Amsterdam, Netherlands (1995) Google Scholar
  15. 102.
    Desbrun, M., Hirani, A.N., Leok, M., Marsden, J.E.: Discrete exterior calculus. arXiv:math.DG/0508341 (2005)
  16. 113.
    Doi, E., Lewicki, M.S.: Relations between the statistical regularities of natural images and the response properties of the early visual system. In: Japanese Cognitive Science Society, SIG P&P, pp. 1–8 (2005) Google Scholar
  17. 123.
    Elmoataz, A., Lézoray, O., Bougleux, S.: Nonlocal discrete regularization on weighted graphs: A framework for image and manifold processing. IEEE Transactions on Image Processing 17(7), 1047–1060 (2008) MathSciNetCrossRefGoogle Scholar
  18. 144.
    Geman, D., Reynolds, G.: Constrained restoration and the discovery of discontinuities. IEEE Transactions on Pattern Analysis and Machine Intelligence 14(3), 367–383 (1992) CrossRefGoogle Scholar
  19. 146.
    Geman, S., McClure, D.: Statistical methods for tomographic image reconstruction. In: Proc. 46th Sess. Int. Stat. Inst. Bulletin ISI, vol. 52, pp. 4–21 (1987) Google Scholar
  20. 147.
    Gibbons, A.: Algorithmic Graph Theory. Cambridge University Press, Cambridge (1989) Google Scholar
  21. 155.
    Golynski, A., Horton, J.: A polynomial time algorithm to find the minimum cycle basis of a regular matroid. In: Lecture Notes in Computer Science, pp. 200–209. Springer, Berlin (2002) Google Scholar
  22. 157.
    Gotsman, C., Kaligosi, K., Mehlhorn, K., Michail, D., Pyrga, E.: Cycle bases of graphs and sampled manifolds. Computer Aided Geometric Design 24(8–9), 464–480 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 158.
    Grady, L.: Space-variant computer vision: A graph-theoretic approach. Ph.D. thesis, Boston University, Boston, MA (2004) Google Scholar
  24. 164.
    Grady, L., Jolly, M.P.: Weights and topology: A study of the effects of graph construction on 3D image segmentation. In: Metaxas, D.N. et al. (eds.) Proc. of MICCAI 2008. Lecture Notes in Computer Science, vol. 5241, pp. 153–161. Springer, Berlin (2008) Google Scholar
  25. 165.
    Grady, L., Schiwietz, T., Aharon, S., Westermann, R.: Random walks for interactive alpha-matting. In: Villanueva, J.J. (ed.) Proc. of Fifth IASTED International Conference on Visualization, Imaging and Image Processing, pp. 423–429. Acta Press, Benidorm (2005) Google Scholar
  26. 167.
    Grady, L., Schwartz, E.L.: Faster graph-theoretic image processing via small-world and quadtree topologies. In: Proc. of 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 2, pp. 360–365. IEEE Comput. Soc., Washington (2004) CrossRefGoogle Scholar
  27. 169.
    Grady, L., Schwartz, E.L.: Isoperimetric partitioning: A new algorithm for graph partitioning. SIAM Journal on Scientific Computing 27(6), 1844–1866 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 174.
    Gremban, K.: Combinatorial preconditioners for sparse, symmetric diagonally dominant linear systems. Ph.D. thesis, Carnegie Mellon University, Pittsburgh, PA (1996) Google Scholar
  29. 175.
    Gremban, K., Miller, G., Zagha, M.: Performance evaluation of a new parallel preconditioner. In: Proceedings of 9th International Parallel Processing Symposium, pp. 65–69. IEEE Comput. Soc., Santa Barbara (1995) CrossRefGoogle Scholar
  30. 176.
    Gross, J., Tucker, T.: Topological Graph Theory. Dover, New York (2001) zbMATHGoogle Scholar
  31. 177.
    Gross, J.L., Yellen, J.: Graph Theory and Its Applications. CRC Press, Boca Raton (1998) Google Scholar
  32. 193.
    He, X., Niyogi, P.: Locality preserving projections. In: Advances in Neural Information Processing Systems, vol. 16, pp. 153–160. MIT Press, Cambridge (2003) Google Scholar
  33. 194.
    Heiler, M., Keuchel, J., Schnorr, C.: Semidefinite clustering for image segmentation with a-priori knowledge. In: Proc. of the 27th DAGM Symposium, pp. 309–317. Springer, Berlin (2005) Google Scholar
  34. 206.
    Horton, J.: A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM Journal on Computing 16, 358 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 212.
    Ishikawa, H.: Higher-order clique reduction in binary graph cut. In: Proc. of CVPR (2009) Google Scholar
  36. 215.
    Jaeger, F.: A survey of the cycle double cover conjecture. In: Alspach, B.R., Godsil, C.D. (eds.) Cycles in Graphs. Annals of Discrete Mathematics, vol. 27, pp. 1–12. Elsevier, Amsterdam (1985) CrossRefGoogle Scholar
  37. 226.
    Kavitha, T., Mehlhorn, K., Michail, D., Paluch, K.: An algorithm for minimum cycle basis of graphs. Algorithmica 52(3), 333–349 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  38. 234.
    Klein, D.: Resistance-distance sum rules. Croatica Chemica Acta 75(2), 633–649 (2002) Google Scholar
  39. 235.
    Klein, D.J., Randić, M.: Resistance distance. Journal of Mathematical Chemistry 12(1), 81–95 (1993) MathSciNetCrossRefGoogle Scholar
  40. 239.
    Kohli, P., Kumar, M., Torr, P.: P3 & beyond: Solving energies with higher order cliques. In: Proc. of IEEE Conference on Computer Vision and Pattern Recognition (2007) Google Scholar
  41. 240.
    Kohn, R.V., Vogelius, M.: Relaxation of a variational method for impedance computed tomography. Communications on Pure and Applied Mathematics 40(6), 745–777 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  42. 243.
    Kolmogorov, V., Rother, C.: Minimizing nonsubmodular functions with graph cuts—a review. IEEE Transactions on Pattern Analysis and Machine Intelligence 29(7), 1274–1279 (2007) CrossRefGoogle Scholar
  43. 244.
    Kolmogorov, V., Zabih, R.: What energy functions can be minimized via graph cuts? IEEE Transactions on Pattern Analysis and Machine Intelligence 26(2), 147–159 (2004) CrossRefGoogle Scholar
  44. 245.
    Komodakis, N., Paragios, N.: Higher-order clique reduction in binary graph cut. In: Proc. of CVPR (2009) Google Scholar
  45. 257.
    Levin, A., Lischinski, D., Weiss, Y.: A closed-form solution to natural image matting. IEEE Transactions on Pattern Analysis and Machine Intelligence 30(2), 228–242 (2008) CrossRefGoogle Scholar
  46. 259.
    Liang, Z., Hart, H.: Bayesian image processing of data from constrained source distributions—I. Non-valued, uncorrelated and correlated constraints. Bulletin of Mathematical Biology 49(1), 51–74 (1987) MathSciNetzbMATHGoogle Scholar
  47. 261.
    Liu, H., Wang, J.: A new way to enumerate cycles in graph. In: Proc. of AICT/ICIW. IEEE (2006) Google Scholar
  48. 262.
    Liu, T.: A heuristic algorithm for finding circuits to double cover a bridgeless graph. Technical Report 57-2001, Rutgers (2001) Google Scholar
  49. 268.
    MacLane, S.: A combinatorial condition for planar graphs. Fundamenta Mathematicae 28, 22–32 (1937) zbMATHGoogle Scholar
  50. 273.
    Mateti, P., Deo, N.: On algorithms for enumerating all circuits of a graph. SIAM Journal on Computing 5, 90 (1976) MathSciNetzbMATHCrossRefGoogle Scholar
  51. 278.
    Mehlhorn, K., Michail, D.: Implementing minimum cycle basis algorithms. Journal of Experimental Algorithmics 11 (2007) Google Scholar
  52. 306.
    Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence 12(7), 629–639 (1990) CrossRefGoogle Scholar
  53. 316.
    Rao, V., Murti, V.: Enumeration of all circuits of a graph. Proceedings of the IEEE 57, 700–701 (1969) CrossRefGoogle Scholar
  54. 334.
    Schoenemann, T., Kahl, F., Cremers, D.: Curvature regularity for region-based image segmentation and inpainting: A linear programming relaxation. In: IEEE International Conference on Computer Vision (ICCV), Kyoto, Japan (2009) Google Scholar
  55. 340.
    Seymour, P.: Sums of circuits. In: Bondy, J.A., Murty, U.R.S. (eds.) Graph Theory and Related Topics, pp. 341–355. Academic Press, New York (1979) Google Scholar
  56. 343.
    Shewchuk, J.R.: Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator. In: Lin, M.C., Manocha, D. (eds.) Applied Computational Geometry: Towards Geometric Engineering. Lecture Notes in Computer Science, vol. 1148, pp. 203–222. Springer, Berlin (1996) CrossRefGoogle Scholar
  57. 344.
    Shewchuk, J.: Delaunay refinement algorithms for triangular mesh generation. Computational Geometry: Theory and Applications 22(1–3), 21–74 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  58. 345.
    Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 22(8), 888–905 (2000) CrossRefGoogle Scholar
  59. 355.
    Stahl, S.: Generalized embedding schemes. Journal of Graph Theory 2, 41–52 (1978) MathSciNetzbMATHCrossRefGoogle Scholar
  60. 374.
    Tewari, G., Gotsman, C., Gortler, S.: Meshing genus-1 point clouds using discrete one-forms. Computers & Graphics 30(6), 917–926 (2006) CrossRefGoogle Scholar
  61. 375.
    Tiernan, J.: An efficient search algorithm to find the elementary circuits of a graph. Comm. of the ACM 13(12), 726 (1970) MathSciNetCrossRefGoogle Scholar
  62. 396.
    Watts, D., Strogatz, S.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 440–442 (1998) CrossRefGoogle Scholar
  63. 397.
    Watts, D.J.: Small Worlds: The Dynamics of Networks Between Order and Randomness. Princeton Studies in Complexity. Princeton University Press, Princeton (1999) Google Scholar
  64. 413.
    Yu, S.X., Shi, J.: Segmentation with pairwise attraction and repulsion. In: Proc. of ICCV, vol. 1. IEEE Comput. Soc., Los Alamitos (2001) Google Scholar
  65. 414.
    Yu, S.X., Shi, J.: Understanding popout through repulsion. In: Proc. of CVPR, vol. 2. IEEE Comput. Soc., Los Alamitos (2001) Google Scholar
  66. 419.
    Zhang, F., Hancock, E.R.: Graph spectral image smoothing using the heat kernel. Pattern Recognition 41(11), 3328–3342 (2008) zbMATHCrossRefGoogle Scholar
  67. 420.
    Zhang, F., Li, H., Jiang, A., Chen, J., Luo, P.: Face tracing based geographic routing in nonplanar wireless networks. In: Proc. of the 26th IEEE INFOCOM (2007) Google Scholar
  68. 422.
    Zhang, Z.: Parameter estimation techniques: A tutorial with application to conic fitting. Image and Vision Computing 15(1), 59–76 (1997) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2010

Authors and Affiliations

  1. 1.Siemens Corporate ResearchPrincetonUSA
  2. 2.Athinoula A. Martinos Center for Biomedical Imaging, Department of RadiologyMassachusetts General Hospital, Harvard Medical SchoolCharlestownUSA

Personalised recommendations