Clustering and Segmentation



Clustering algorithms are used to find communities of nodes that all belong to the same group. This grouping process is also known as image segmentation in image processing. The clustering problem is also deeply connected to machine learning because a solution to the clustering problem may be used to propagate labels from observed data to unobserved data. In general network analysis, the identification of a grouping allows for the analysis of the nodes within each group as separate entities. In this chapter, we use the tools of discrete calculus to examine both the targeted clustering problem (i.e., finding a specific group) and the untargeted clustering problem (i.e., discovering all groups). We additionally show how to apply these clustering models to the clustering of higher-order cells, e.g., to cluster edges.


Cluster Algorithm Image Segmentation Spectral Cluster Cluster Problem Dual Graph 
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.


  1. 1.
    Agarwal, S., Branson, K., Belongie, S.: Higher order learning with graphs. In: Proc. of the 23rd Int. Conf. on Mach. Learn., vol. 148, pp. 17–24 (2006) Google Scholar
  2. 4.
    Allène, C., Audibert, J.Y., Couprie, M., Cousty, J., Keriven, R.: Some links between min cuts, optimal spanning forests and watersheds. In: Proc. of ISMM’07, vol. 2, pp. 253–264 (2007) Google Scholar
  3. 5.
    Aloise, D., Deshpande, A., Hansen, P., Popat, P.: NP-hardness of Euclidean sum-of-squares clustering. Machine Learning 75(2), 245–248 (2009) CrossRefGoogle Scholar
  4. 8.
    Alvino, C.V., Unal, G.B., Slabaugh, G., Peny, B., Fang, T.: Efficient segmentation based on Eikonal and diffusion equations. International Journal of Computer Mathematics 84(9), 1309–1324 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 11.
    Appleton, B., Talbot, H.: Globally optimal surfaces by continuous maximal flows. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(1), 106–118 (2006) CrossRefGoogle Scholar
  6. 13.
    Aronsson, G., Crandall, M.G., Juutinen, P.: A tour of the theory of absolutely minimizing functions. Bulletin of the American Mathematical Society 41(4), 439–505 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 14.
    Arora, S., Rao, S., Vazirani, U.: Geometry, flows, and graph-partitioning algorithms. Communications of the ACM 51(10), 96–105 (2008) CrossRefGoogle Scholar
  8. 17.
    Bae, E., Tai, X.C.: Graph cut optimization for the piecewise constant level set method applied to multiphase image segmentation. In: Proc. of the International Conference of Scale Space and Variational Methods in Computer Vision, pp. 1–13 (2009) Google Scholar
  9. 18.
    Bai, X., Sapiro, G.: A geodesic framework for fast interactive image and video segmentation and matting. In: ICCV (2007) Google Scholar
  10. 28.
    Bengio, Y., Delalleau, O., Roux, N.L., Paiement, J.F., Vincent, P., Ouimet, M.: Learning eigenfunctions links spectral embedding and kernel PCA. Neural Computation 16(10), 2197–2219 (2004) zbMATHCrossRefGoogle Scholar
  11. 29.
    Bengio, Y., Paiement, J., Vincent, P., Delalleau, O., Le Roux, N., Ouimet, M.: Out-of-sample extensions for LLE, Isomap, MDS, eigenmaps, and spectral clustering. In: Proc. of NIPS, pp. 177–184 (2004) Google Scholar
  12. 41.
    Bohland, J., Bokil, H., Pathak, S., Lee, C., Ng, L., Lau, C., Kuan, C., Hawrylycz, M., Mitra, P.: Clustering of spatial gene expression patterns in the mouse brain and comparison with classical neuroanatomy. Methods 50(2), 105–112 (2010) CrossRefGoogle Scholar
  13. 53.
    Boykov, Y., Jolly, M.P.: Interactive graph cuts for optimal boundary and region segmentation of objects in N-D images. In: Proc. of ICCV 2001, pp. 105–112 (2001) Google Scholar
  14. 55.
    Boykov, Y., Kolmogorov, V.: An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision. IEEE Transactions on Pattern Analysis and Machine Intelligence 26(9), 1124–1137 (2004) CrossRefGoogle Scholar
  15. 56.
    Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. IEEE Transactions on Pattern Analysis and Machine Intelligence 23(11), 1222–1239 (2001) CrossRefGoogle Scholar
  16. 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
  17. 68.
    Chambolle, A.: An algorithm for total variation minimization and applications. Journal of Mathematical Imaging and Vision 20(1–2), 89–97 (2004) MathSciNetGoogle Scholar
  18. 70.
    Chan, T., Vese, L.: Active contours without edges. IEEE Transactions on Image Processing 10(2), 266–277 (2001) zbMATHCrossRefGoogle Scholar
  19. 73.
    Chapelle, O., Schölkopf, B., Zien, A. (eds.): Semi-Supervised Learning. MIT Press, Cambridge (2006) Google Scholar
  20. 76.
    Chen, Y., Dong, M., Rege, M.: Gene expression clustering: A novel graph partitioning approach. In: Proceedings of International Joint Conference on Neural Networks (2007) Google Scholar
  21. 80.
    Chung, F.R.K.: The Laplacian of a hypergraph. In: Proc. of a DIMACS Workshop, Discrete Math. Theoret. Comput. Sci., vol. 10, pp. 21–36. Am. Math. Soc., Providence (1993) Google Scholar
  22. 81.
    Chung, F.R.K.: Spectral Graph Theory. Regional Conference Series in Mathematics, vol. 92. Am. Math. Soc., Providence (1997) zbMATHGoogle Scholar
  23. 83.
    Cohen, L.D.: On active contour models and balloons. CVGIP: Image Understanding 53(2), 211–218 (1991) zbMATHCrossRefGoogle Scholar
  24. 84.
    Cohen, L., Cohen, I.: Finite-element methods for active contour models and balloons for 2-D and 3-D images. IEEE Transactions on Pattern Analysis and Machine Intelligence 15(11), 1131–1147 (1993) CrossRefGoogle Scholar
  25. 85.
    Coifman, R., Lafon, S., Lee, A., Maggioni, M., Nadler, B., Warner, F., Zucker, S.: Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps. Proceedings of the National Academy of Sciences of the United States of America 102(21), 7426–7431 (2005) CrossRefGoogle Scholar
  26. 90.
    Couprie, C., Grady, L., Najman, L., Talbot, H.: Power watersheds: A new image segmentation framework extending graph cuts, random walker and optimal spanning forest. In: Proc. of ICCV, pp. 731–738 (2009) Google Scholar
  27. 94.
    Criminisi, A., Sharp, T., Blake, A.: GeoS: Geodesic image segmentation. In: Proc. of ECCV, pp. 99–112 (2008) Google Scholar
  28. 97.
    Darbon, J.: A note on the discrete binary Mumford–Shah model. In: Proc. of the 3rd Int. Conf. on Computer Vision/Computer Graphics Collaboration Techniques. Lecture Notes in Computer Science, pp. 283–294. Springer, Berlin (2007) CrossRefGoogle Scholar
  29. 98.
    Darbon, J., Sigelle, M.: Image restoration with discrete constrained total variation part I: Fast and exact optimization. Journal of Mathematical Imaging and Vision 26(3), 261–276 (2006) MathSciNetCrossRefGoogle Scholar
  30. 106.
    Dheeraj Singaraju, L.G., Vidal, R.: P-brush: Continuous valued MRFs with normed pairwise distributions for image segmentation. In: Proc. of CVPR 2009. IEEE Comput. Soc., Los Alamitos (2009) Google Scholar
  31. 114.
    Donath, W., Hoffman, A.: Algorithms for partitioning of graphs and computer logic based on eigenvectors of connection matrices. IBM Technical Disclosure Bulletin 15, 938–944 (1972) Google Scholar
  32. 121.
    El-Zehiry, N., Xu, S., Sahoo, P., Elmaghraby, A.: Graph cut optimization for the Mumford–Shah model. In: Proc. of VIIP (2007) Google Scholar
  33. 122.
    El-Zehiry, N.Y., Elmaghraby, A.: Brain MRI tissue classification using graph cut optimization of the Mumford–Shah functional. In: Proceedings of Image and Vision Computing, New Zealand, pp. 321–326 (2007) Google Scholar
  34. 128.
    Falcão, A.X., Lotufo, R.A., Araujo, G.: The image foresting transformation. IEEE Transactions on Pattern Analysis and Machine Intelligence 26(1), 19–29 (2004) CrossRefGoogle Scholar
  35. 129.
    Falcão, A.X., Udupa, J.K., Samarasekera, S., Sharma, S., Elliot, B.H., de Lotufo, A.R.: User-steered image segmentation paradigms: Live wire and live lane. Graphical Models and Image Processing 60(4), 233–260 (1998) CrossRefGoogle Scholar
  36. 133.
    Fiedler, M.: Algebraic connectivity of graphs. Czechoslovak Mathematical Journal 23(98), 298–305 (1973) MathSciNetGoogle Scholar
  37. 134.
    Fiedler, M.: Eigenvectors of acyclic matrices. Czechoslovak Mathematical Journal 25(100), 607–618 (1975) MathSciNetGoogle Scholar
  38. 135.
    Fiedler, M.: A property of eigenvectors of nonnegative symmetric matrices and its applications to graph theory. Czechoslovak Mathematical Journal 25(100), 619–633 (1975) MathSciNetGoogle Scholar
  39. 138.
    Fisher, R.A.: The use of multiple measurements in taxonomic problems. Annals of Eugenics 7, 179–188 (1936) CrossRefGoogle Scholar
  40. 148.
    Girvan, M., Newman, M.: Community structure in social and biological networks. Proceedings of the National Academy of Sciences of the United States of America 99(12), 7821–7826 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  41. 159.
    Grady, L.: Multilabel random walker image segmentation using prior models. In: Proc. of 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR, vol. 1, pp. 763–770. IEEE Press, San Diego (2005) Google Scholar
  42. 161.
    Grady, L.: Random walks for image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(11), 1768–1783 (2006) CrossRefGoogle Scholar
  43. 162.
    Grady, L.: Minimal surfaces extend shortest path segmentation methods to 3D. IEEE Transactions on Pattern Analysis and Machine Intelligence 32(2), 321–334 (2010) CrossRefGoogle Scholar
  44. 163.
    Grady, L., Alvino, C.: The piecewise smooth Mumford-Shah functional on an arbitrary graph. IEEE Transactions on Image Processing 18(11), 2547–2561 (2009) MathSciNetCrossRefGoogle Scholar
  45. 168.
    Grady, L., Schwartz, E.L.: Isoperimetric graph partitioning for image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(3), 469–475 (2006) CrossRefGoogle Scholar
  46. 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
  47. 173.
    Greig, D., Porteous, B., Seheult, A.: Exact maximum a posteriori estimation for binary images. Journal of the Royal Statistical Society. Series B 51(2), 271–279 (1989) Google Scholar
  48. 179.
    Guattery, S., Miller, G.: On the quality of spectral separators. SIAM Journal on Matrix Analysis and Applications 19(3), 701–719 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  49. 184.
    Hall, K.M.: An r-dimensional quadratic placement algorithm. Management Science 17(3), 219–229 (1970) zbMATHCrossRefGoogle Scholar
  50. 190.
    Harrison, L.M., Penny, W., Flandin, G., Ruff, C.C., Weiskopf, N., Friston, K.J.: Graph-partitioned spatial priors for functional magnetic resonance images. NeuroImage 43(4), 694–707 (2008) CrossRefGoogle Scholar
  51. 198.
    Higham, D., Kalna, G., Kibble, M.: Spectral clustering and its use in bioinformatics. Journal of Computational and Applied Mathematics 204(1), 25–37 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  52. 216.
    Jain, A., Murty, M., Flynn, P.: Data clustering: A review. ACM Computing Surveys 31(3), 264–323 (1999) CrossRefGoogle Scholar
  53. 222.
    Kass, M., Witkin, A., Terzopoulos, D.: Snakes: Active contour models. International Journal of Computer Vision 1(4), 321–331 (1988) CrossRefGoogle Scholar
  54. 225.
    Kavitha, S., Roomi, S., Ramaraj, N.: Lossy compression through segmentation on low depth-of-field images. Digital Signal Processing 19(1), 59–65 (2009) CrossRefGoogle Scholar
  55. 229.
    Khaira, M.S., Miller, G.L., Sheffler, T.J.: Nested dissection: A survey and comparison of various nested dissection algorithms. Technical Report CMU-CS-92-106R, Computer Science Department, Carnegie Mellon University (1992) Google Scholar
  56. 237.
    Kodres, U.R.: Geometrical positioning of circuit elements in a computer. In: Proceedings of the 1959 AIEE Fall General Meeting. AIEE, New York (1959) No. CP59-1172 Google Scholar
  57. 242.
    Kolmogorov, V., Boykov, Y., Rother, C.: Applications of parametric maxflow in computer vision. In: Proc. of ICCV (2007) Google Scholar
  58. 246.
    Konstantinos, T.: Maximum flow techniques for network clustering. Ph.D. thesis, Princeton University (2002) Google Scholar
  59. 255.
    Lein, E., Hawrylycz, M., Ao, N., Ayres, M., Bensinger, A., Bernard, A., Boe, A., Boguski, M., Brockway, K., Byrnes, E., et al.: Genome-wide atlas of gene expression in the adult mouse brain. Nature 445(7124), 168–176 (2006) CrossRefGoogle Scholar
  60. 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
  61. 263.
    Lloyd, S.P.: Least square quantization in PCM. Technical Report, Bell Telephone Laboratories Paper (1957) Google Scholar
  62. 264.
    Lloyd, S.: Least squares quantization in PCM. IEEE Transactions on Information Theory 28(2), 129–137 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  63. 269.
    Mahajan, M., Nimbhorkar, P., Varadarajan, K.: The planar k-means problem is NP-hard. In: Proceedings of the 3rd International Workshop on Algorithms and Computation, pp. 274–285. Springer, Berlin (2009) Google Scholar
  64. 281.
    Michel, J., Pellegrini, F., Roman, J.: Unstructured graph partitioning for sparse linear system solving. In: Proc. of the 4th International Symposium, IRREGULAR’97, pp. 273–286 (1997) Google Scholar
  65. 287.
    Mortensen, E., Barrett, W.: Interactive segmentation with intelligent scissors. Graphical Models in Image Processing 60(5), 349–384 (1998) zbMATHCrossRefGoogle Scholar
  66. 288.
    Muhammad, A., Egerstedt, M.: Control using higher order Laplacians in network topologies. In: Proc. of the 17th Int. Symp. on Math. Theory of Networks and Systems, pp. 1024–1038 (2006) Google Scholar
  67. 289.
    Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics 42, 577–685 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  68. 292.
    Newman, M.: Modularity and community structure in networks. Proceedings of the National Academy of Sciences of the United States of America 103(23), 8577–8582 (2006) CrossRefGoogle Scholar
  69. 293.
    Nicholls, F., Torr, P.H.S.: Discrete minimum ratio curves and surfaces. In: Proc. of CVPR (2010) Google Scholar
  70. 303.
    Pal, N., Pal, S.: A review on image segmentation techniques. Pattern Recognition 26(9), 1277–1294 (1993) CrossRefGoogle Scholar
  71. 310.
    Pothen, A., Simon, H., Liou, K.P.: Partitioning sparse matrices with eigenvectors of graphs. SIAM Journal on Matrix Analysis and Applications 11(3), 430–452 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  72. 315.
    Rand, W.M.: Objective criteria for the evaluation of clustering methods. Journal of the American Statistical Association 66, 846–850 (1971) CrossRefGoogle Scholar
  73. 320.
    Roerdink, J., Meijster, A.: The watershed transform: definitions, algorithms, and parallelization strategies. Fundamenta Informaticae 41, 187–228 (2000) MathSciNetzbMATHGoogle Scholar
  74. 330.
    Schaeffer, S.: Graph clustering. Computer Science Review 1(1), 27–64 (2007) MathSciNetCrossRefGoogle Scholar
  75. 332.
    Schmalz, M.S., Ritter, G.X.: Region segmentation techniques for object-based image compression: A review. In: Schmalz, M.S. (ed.) Mathematics of Data/Image Coding, Compression, and Encryption VII, with Applications, vol. 5561, pp. 62–75. SPIE, Bellingham (2004) CrossRefGoogle Scholar
  76. 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
  77. 337.
    Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, London (1982) zbMATHGoogle Scholar
  78. 339.
    Sethian, J.: Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge (1999) zbMATHGoogle Scholar
  79. 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
  80. 347.
    Simon, H.D., Teng, S.H.: How good is recursive bisection? SIAM Journal of Scientific Computing 18(5), 1436–1445 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  81. 348.
    Singaraju, D., Grady, L., Sinop, A.K., Vidal, R.: P-brush: A continuous valued MRF for image segmentation. In: Blake, A., Kohli, P., Rother, C. (eds.) Advances in Markov Random Fields for Vision and Image Processing. MIT Press, Cambridge (2010) Google Scholar
  82. 349.
    Singaraju, D., Grady, L., Vidal, R.: Interactive image segmentation of quadratic energies on directed graphs. In: Proc. of CVPR 2008. IEEE Comput. Soc., Los Alamitos (2008) Google Scholar
  83. 350.
    Sinop, A.K., Grady, L.: A seeded image segmentation framework unifying graph cuts and random walker which yields a new algorithm. In: Proc. of ICCV 2007. IEEE Comput. Soc., Los Alamitos (2007) Google Scholar
  84. 351.
    Smith, B.F., Bjørstad, P.E., Gropp, W.: Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge (1996) zbMATHGoogle Scholar
  85. 352.
    Spielman, D.A., Teng, S.H.: Spectral partitioning works: Planar graphs and finite element meshes. Technical Report UCB CSD-96-898, University of California, Berkeley (1996) Google Scholar
  86. 358.
    Strang, G.: Maximum flows through a domain. Mathematical Programming 26, 123–143 (1983) MathSciNetzbMATHCrossRefGoogle Scholar
  87. 363.
    Stuwe, M.: Plateau’s Problem and the Calculus of Variations. Princeton University Press, Princeton (1989) Google Scholar
  88. 364.
    Sullivan, J.M.: A crystalline approximation theorem for hypersurfaces. Ph.D. thesis, Princeton University, Princeton, NJ (1990) Google Scholar
  89. 368.
    Szallasi, Z., Somogyi, R.: Genetic network analysis—The millennium opening version. In: Proc. Pacific Symposium of Biocomputing Tutorial (2001) Google Scholar
  90. 381.
    Toselli, A., Widlund, O.: Domain Decomposition Methods—Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34. Springer, Berlin (2004) Google Scholar
  91. 382.
    Trichili, H., Bouhlel, M.S., Kammoun, F.: Review and evaluation of medical image segmentation using methods of optimal filtering. Journal of Testing and Evaluation 31(5), 398–404 (2003) Google Scholar
  92. 385.
    Unger, M., Pock, T., Bischof, H.: Interactive globally optimal image segmentation. Technical Report 08/02, Inst. for Computer Graphics and Vision, Graz University of Technology (2008) Google Scholar
  93. 386.
    Unger, M., Pock, T., Trobin, W., Cremers, D., Bischof, H.: TVSeg—Interactive total variation based image segmentation. In: Proc. of British Machine Vision Conference (2008) Google Scholar
  94. 392.
    Walshaw, C., Cross, M., Everett, M.: Mesh partitioning and load-balancing for distributed memory parallel systems. In: Topping, B. (ed.) Proc. Parallel & Distributed Computing for Computational Mechanics (1997) Google Scholar
  95. 407.
    Wu, Z., Leahy, R.: An optimal graph theoretic approach to data clustering: Theory and its application to image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 15(11), 1101–1113 (1993) CrossRefGoogle Scholar
  96. 408.
    Xing, E., Karp, R.: CLIFF: Clustering of high-dimensional microarray data via iterative feature filtering using normalized cuts. Bioinformatics 17, 306–315 (2001) CrossRefGoogle Scholar
  97. 409.
    Xu, C., Prince, J.: Snakes, shapes, and gradient vector flow. IEEE Transactions on Image Processing 7(3), 359–369 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  98. 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
  99. 414.
    Yu, S.X., Shi, J.: Understanding popout through repulsion. In: Proc. of CVPR, vol. 2. IEEE Comput. Soc., Los Alamitos (2001) Google Scholar
  100. 415.
    Zachary, W.W.: An information flow model for conflict and fission in small groups. Journal of Anthropological Research 33, 452–473 (1977) Google Scholar
  101. 416.
    Zahn, C.: Graph theoretical methods for detecting and describing Gestalt clusters. IEEE Transactions on Computers 20, 68–86 (1971) zbMATHCrossRefGoogle Scholar
  102. 418.
    Zeng, X., Chen, W., Peng, Q.: Efficiently solving the piecewise constant Mumford–Shah model using graph cuts. Technical Report, Zhejiang University (2006) Google Scholar
  103. 424.
    Zhu, X., Ghahramani, Z., Lafferty, J.: Semi-supervised learning using Gaussian fields and harmonic functions. In: Machine Learning: Proceedings of the Twentieth International Conference on Machine Learning, pp. 912–919 (2003) Google Scholar

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© 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

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