Geometric Tree Kernels: Classification of COPD from Airway Tree Geometry

  • Aasa Feragen
  • Jens Petersen
  • Dominik Grimm
  • Asger Dirksen
  • Jesper Holst Pedersen
  • Karsten Borgwardt
  • Marleen de Bruijne
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7917)


Methodological contributions: This paper introduces a family of kernels for analyzing (anatomical) trees endowed with vector valued measurements made along the tree. While state-of-the-art graph and tree kernels use combinatorial tree/graph structure with discrete node and edge labels, the kernels presented in this paper can include geometric information such as branch shape, branch radius or other vector valued properties. In addition to being flexible in their ability to model different types of attributes, the presented kernels are computationally efficient and some of them can easily be computed for large datasets (N ~10.000) of trees with 30 − 600 branches. Combining the kernels with standard machine learning tools enables us to analyze the relation between disease and anatomical tree structure and geometry. Experimental results: The kernels are used to compare airway trees segmented from low-dose CT, endowed with branch shape descriptors and airway wall area percentage measurements made along the tree. Using kernelized hypothesis testing we show that the geometric airway trees are significantly differently distributed in patients with Chronic Obstructive Pulmonary Disease (COPD) than in healthy individuals. The geometric tree kernels also give a significant increase in the classification accuracy of COPD from geometric tree structure endowed with airway wall thickness measurements in comparison with state-of-the-art methods, giving further insight into the relationship between airway wall thickness and COPD. Software: Software for computing kernels and statistical tests is available at .


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  1. 1.
    Bach, F.R.: Graph kernels between point clouds. In: ICML, pp. 25–32 (2008)Google Scholar
  2. 2.
    Bishop, C.M.: Pattern Recognition and Machine Learning. Springer (2006)Google Scholar
  3. 3.
    Borgwardt, K.M., Kriegel, H.-P.: Shortest-path kernels on graphs. In: ICDM (2005)Google Scholar
  4. 4.
    Chang, C.-C., Lin, C.-J.: LIBSVM: A library for support vector machines. ACM Trans. Int. Syst. and Tech. 2, 27:1–27:27 (2011), Software available at
  5. 5.
    Cortes, C., Haffner, P., Mohri, M.: Rational kernels: Theory and algorithms. JMLR 5, 1035–1062 (2004)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Feragen, A.: Complexity of computing distances between geometric trees. In: Gimel’farb, G., Hancock, E., Imiya, A., Kuijper, A., Kudo, M., Omachi, S., Windeatt, T., Yamada, K. (eds.) SSPR & SPR 2012. LNCS, vol. 7626, pp. 89–97. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  7. 7.
    Feragen, A., Hauberg, S., Nielsen, M., Lauze, F.: Means in spaces of tree-like shapes. In: ICCV (2011)Google Scholar
  8. 8.
    Feragen, A., Petersen, J., Owen, M., Lo, P., Thomsen, L.H., Wille, M.M.W., Dirksen, A., de Bruijne, M.: A hierarchical scheme for geodesic anatomical labeling of airway trees. In: Ayache, N., Delingette, H., Golland, P., Mori, K. (eds.) MICCAI 2012, Part III. LNCS, vol. 7512, pp. 147–155. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  9. 9.
    Gretton, A., Borgwardt, K.M., Rasch, M.J., Schölkopf, B., Smola, A.J.: A kernel two-sample test. JMLR 13, 723–773 (2012)Google Scholar
  10. 10.
    Hackx, M., Bankier, A.A., Genevois, P.A.: Chronic Obstructive Pulmonary Disease: CT quantification of airways disease. Radiology 265(1), 34–48 (2012)CrossRefGoogle Scholar
  11. 11.
    Hasegawa, M., Nasuhara, Y., Onodera, Y., Makita, H., Nagai, K., Fuke, S.I., Ito, Y., Betsuyaku, T., Nishimura, M.: Airflow Limitation and Airway Dimensions in Chronic Obstructive Pulmonary Disease. Am. J. Respir. Crit. Care Med. 173(12), 1309–1315 (2006)CrossRefGoogle Scholar
  12. 12.
    Haussler, D.: Convolution kernels on discrete structures. Technical report, Department of Computer Science, University of California at Santa Cruz (1999)Google Scholar
  13. 13.
    Leslie, C., Kuang, R.: Fast kernels for inexact string matching. In: Schölkopf, B., Warmuth, M.K. (eds.) COLT/Kernel 2003. LNCS (LNAI), vol. 2777, pp. 114–128. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  14. 14.
    Lo, P., van Ginneken, B., Reinhardt, J.M., de Bruijne, M.: Extraction of Airways from CT (EXACT’09). In: 2. Int. WS. Pulm. Im. Anal., pp. 175–189 (2009)Google Scholar
  15. 15.
    Mahé, P., Ueda, N., Akutsu, T., Perret, J.-L., Vert, J.-P.: Extensions of marginalized graph kernels. In: ICML (2004)Google Scholar
  16. 16.
    Miller, E., Owen, M., Provan, J.S.: Averaging metric phylogenetic trees (2012) (Preprint),
  17. 17.
    Nye, T.M.W.: Principal components analysis in the space of phylogenetic trees. Ann. Statist. 39(5), 2716–2739 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Petersen, J., Nielsen, M., Lo, P., Saghir, Z., Dirksen, A., de Bruijne, M.: Optimal graph based segmentation using flow lines with application to airway wall segmentation. In: Székely, G., Hahn, H.K. (eds.) IPMI 2011. LNCS, vol. 6801, pp. 49–60. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  19. 19.
    Shervashidze, N., Schweitzer, P., van Leeuwen, E.J., Mehlhorn, K., Borgwardt, K.M.: Weisfeiler-Lehman graph kernels. JMLR 12, 2539–2561 (2011)Google Scholar
  20. 20.
    Sørensen, L., Lo, P., Dirksen, A., Petersen, J., de Bruijne, M.: Dissimilarity-based classification of anatomical tree structures. In: Székely, G., Hahn, H.K. (eds.) IPMI 2011. LNCS, vol. 6801, pp. 475–485. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  21. 21.
    Vishwanathan, S.V.N., Schraudolph, N.N., Kondor, R.I., Borgwardt, K.M.: Graph kernels. JMLR 11, 1201–1242 (2010)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Vishwanathan, S.V.N., Smola, A.J.: Fast kernels for string and tree matching. In: NIPS, pp. 569–576 (2002)Google Scholar
  23. 23.
    Washko, G.R., Dransfield, T., Estepar, R.S.J., Diaz, A., Matsuoka, S., Yamashiro, T., Hatabu, H., Silverman, E.K., Bailey, W.C., Reilly, J.J.: Airway wall attenuation: a biomarker of airway disease in subjects with COPD. J. Appl. Phys. 107(1), 185–191 (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Aasa Feragen
    • 1
    • 2
  • Jens Petersen
    • 1
  • Dominik Grimm
    • 2
  • Asger Dirksen
    • 4
  • Jesper Holst Pedersen
    • 5
  • Karsten Borgwardt
    • 2
    • 3
  • Marleen de Bruijne
    • 1
    • 6
  1. 1.Department of Computer ScienceUniversity of CopenhagenDenmark
  2. 2.Max Planck Institute for Intelligent Systems and Max Planck Institute for Developmental BiologyTübingenGermany
  3. 3.Zentrum für BioinformatikEberhard Karls Universität TübingenGermany
  4. 4.Lungemedicinsk Afdeling, Gentofte HospitalDenmark
  5. 5.Department of Cardiothoracic SurgeryRigshospitaletDenmark
  6. 6.Erasmus MC - University Medical Center RotterdamThe Netherlands

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