Skeleton-Based Recognition of Shapes in Images via Longest Path Matching

Conference paper
Part of the Association for Women in Mathematics Series book series (AWMS, volume 1)


We present a novel image recognition method based on the Blum medial axis that identifies shape information present in unsegmented input images. Inspired by prior work matching from a library using only the longest path in the medial axis, we extract medial axes from shapes with clean contours and seek to recognize these shapes within “no isy” images. Recognition consists of matching longest paths from the segmented images into complicated geometric graphs, which are computed via edge detection on the (unsegmented) input images to obtain Voronoi diagrams associated to the edges. We present two approaches: one based on map-matching techniques using the weak Fréchet distance, and one based on a multiscale curve metric after reducing the Voronoi graphs to their minimum spanning trees. This paper serves as a proof of concept for this approach, using images from three shape databases with known segmentability (whale flukes, strawberries, and dancers). Our preliminary results on these images show promise, with both approaches correctly identifying two out of three shapes.


Longest Path Whale Flukes Medial Axis Geometric Graphs Voronoi Trees 
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.



The authors would like to thank the Institute for Pure and Applied Mathematics, the Association for Women in Mathematics, Microsoft Research, the National Science Foundation, and the National Geospatial Agency for support, financial and otherwise, of this collaboration. Kathryn Leonard thanks Matt Feiszli for providing the initial Matlab code for the H 1∕2 metric for closed curves which was modified for this project.


  1. 1.
    Alt, H., Efrat, A., Rote, G., Wenk, C.: Matching planar maps. J. Algorithms 49(2), 262–283 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Aslan, C., Tari, S.: An axis-based representation for recognition. In: ICCV, Beijing, pp. 1339–1346. IEEE Computer Society (2005)Google Scholar
  3. 3.
    Bai, X., Yang, X., Yu, D., Latecki, L.J.: Skeleton-based shape classification using path similarity. Int. J. Pattern Recognit. Artif. Intell. (IJPRAI) 22(4), 733–746 (2008)Google Scholar
  4. 4.
    Belongie, S., Malik, J., Puzicha, J.: Shape matching and object recognition using shape contexts. IEEE Trans. Pattern Anal. Mach. Intell. 24(4), 509–522 (2002)CrossRefGoogle Scholar
  5. 5.
    Belongie, S., Mori, G., Malik, J.: Matching with shape contexts. In: Analysis and Statistics of Shapes, pp. 81–105. Birkhäuser (2005)Google Scholar
  6. 6.
    Blum, H.: A transformation for extracting new descriptors of shape. In: Wathen-Dunn, W. (ed.) Models for the Perception of Speech and Visual Form, pp. 362–380. MIT, Cambridge (1967).
  7. 7.
    Brakatsoulas, S., Pfoser, D., Salas, R., Wenk, C.: On map-matching vehicle tracking data. In: Proceedings of the 31st International Conference on Very Large Data Bases (VLDB’05), Trondheim, pp. 853–864. VLDB Endowment (2005)Google Scholar
  8. 8.
    Chen, D., Driemel, A., Guibas, L.J., Nguyen, A., Wenk, C.: Approximate map matching with respect to the Fréchet distance. In: Müller-Hannemann, M., Werneck, R.F.F. (eds.) ALENEX, San Francisco, pp. 75–83. SIAM (2011)Google Scholar
  9. 9.
    Fei-Fei, L., Fergus, R., Perona, P.: One-shot learning of object categories. IEEE Trans. Pattern Anal. Mach. Intell. 28(4), 594–611 (2006)CrossRefGoogle Scholar
  10. 10.
    Gold, S., Rangarajan, A.: A graduated assignment algorithm for graph matching. IEEE Trans. Pattern Anal. Mach. Intell. 18(4), 377–388 (1996)CrossRefGoogle Scholar
  11. 11.
    Grompone von Gioi, R., Jakubowicz, J., Morel, J.-M., Randall, G.: LSD: a Line Segment Detector. Image Processing On Line (2012)Google Scholar
  12. 12.
    Gudmundsson, J., Smid, M.: Fréchet queries in geometric trees. In: Bodlaender, H.L., Italiano, G.F. (eds.) Algorithms – ESA 2013, Sophia Antipolis. Lecture Notes in Computer Science, vol. 8125, pp. 565–576. Springer, Berlin/Heidelberg (2013)Google Scholar
  13. 13.
    Huttenlocher, D.P., Klanderman, G.A., Rucklidge, W.J.: Comparing images using the Hausdorff distance. IEEE Trans. Pattern Anal. Mach. Intell. 15(9), 850–863 (1993)CrossRefGoogle Scholar
  14. 14.
    Kushnarev, S.: Teichons: solitonlike geodesics on universal Teichmüller space. Exp. Math. 18, 325–336 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Leonard, K., Strawbridge, R., Lindsay, D., Barata, R., Dawson, M., Averion, L.: Minimal geometric representation and strawberry stem detection. In: 2013 13th International Conference on Computational Science and Its Applications (ICCSA), Ho Chi Minh City, pp. 144–149 (2013)Google Scholar
  16. 16.
    Leonard, K., Feiszli, M., Kushnarev, S.: Metric spaces of shapes and applications: compression, curve matching and low-dimensional representation. Geom. Imaging Comput. 1(2), 173–221 (2014)CrossRefGoogle Scholar
  17. 17.
    Leymarie, F.F., Kimia, B.B.: From the infinitely large to the infinitely small: applications of medial symmetry representations of shape. In: Siddiqi, K., Pizer, S.M. (eds.) Medial Representations: Mathematics, Algorithms and Applications. Computational Imaging and Vision, pp. 327–351. Kluwer Academic, Dordrecht (2008). ISBN:978-1-402-08657-1.
  18. 18.
    Lieutier, A.: Any open bounded subset of \(\mathbb{R}^{n}\) has the same homotopy type as its medial axis. Comput.-Aided Des. 36(11), 1029–1046 (2004)CrossRefGoogle Scholar
  19. 19.
    Lin, C.C., Chellappa, R.: Classification of partial 2-D shapes using Fourier descriptors. IEEE Trans. Pattern Anal. Mach. Intell. 5, 686–690 (1987)CrossRefGoogle Scholar
  20. 20.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)CrossRefzbMATHGoogle Scholar
  21. 21.
    Sebastian, T.B., Kimia, B.B.: Curves vs. skeletons in object recognition. Signal Process. 85(2), 247–263 (2005)Google Scholar
  22. 22.
    Sebastian, T.B., Klein, P.N., Kimia, B.B.: Shock-based indexing into large shape databases. In: Proceedings of the 7th European Conference on Computer Vision-Part III (ECCV’02), Copenhagen, pp. 731–746. Springer, London (2002)Google Scholar
  23. 23.
    Sebastian, T.B., Klein, P.N., Kimia, B.B.: Recognition of shapes by editing their shock graphs. IEEE Trans. Pattern Anal. Mach. Intell. 26(5), 550–571 (2004)CrossRefGoogle Scholar
  24. 24.
    Sharon, E., Mumford, D.: 2D-shape analysis using conformal mapping. Int. J. Comput. Vis. 70, 55–75 (2006)CrossRefGoogle Scholar
  25. 25.
    Trinh, N.H., Kimia, B.B.: Skeleton search: category-specific object recognition and segmentation using a skeletal shape model. Int. J. Comput. Vis. 94(2), 215–240 (2011)CrossRefGoogle Scholar
  26. 26.
    Trouve, A., Miller, M.I., Younes, L.: On metrics and Euler-Lagrange equations of computational anatomy. Ann. Rev. Biomed. Eng. 4, 375–405 (2002)CrossRefGoogle Scholar
  27. 27.
    Wenk, C., Salas, R., Pfoser, D.: Addressing the need for map-matching speed: localizing global curve-matching algorithms. In: Proceedings of the 18th International Conference on Scientific and Statistical Database Management (SSDBM’06), Vienna, pp. 379–388. IEEE Computer Society, Washington, DC (2006)Google Scholar

Copyright information

© Springer International Publishing Switzerland & The Association for Women in Mathematics 2015

Authors and Affiliations

  1. 1.Department of Computer EngineeringMiddle East Technical UniversityAnkaraTurkey
  2. 2.Department of Computer ScienceTechnical University of MunichMunichGermany
  3. 3.Department of Mathematics and Computer ScienceSaint Louis UniversitySaint LouisUSA
  4. 4.Research supported in part by NSF grants CCF-1054779 and IIS-1319573Saint LouisUSA
  5. 5.Department of MathematicsDuke UniversityDurhamUSA
  6. 6.Department of MathematicsZhejiang UniversityZhejiangChina
  7. 7.Department of MathematicsCalifornia State University Channel IslandsCamarilloUSA
  8. 8.Research supported in part by NSF grant IIS-0954256CamarilloUSA
  9. 9.Department of Electrical EngineeringNortheastern UniversityBostonUSA
  10. 10.Department of Computer ScienceTulane UniversityNew OrleansUSA
  11. 11.Research supported in part by NSF grant CCF-0643597New OrleansUSA

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