EMD-L1: An Efficient and Robust Algorithm for Comparing Histogram-Based Descriptors

  • Haibin Ling
  • Kazunori Okada
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3953)


We propose a fast algorithm, EMD-L 1, for computing the Earth Mover’s Distance (EMD) between a pair of histograms. Compared to the original formulation, EMD-L 1 has a largely simplified structure. The number of unknown variables in EMD-L 1 is O(N) that is significantly less than O(N 2) of the original EMD for a histogram with N bins. In addition, the number of constraints is reduced by half and the objective function is also simplified. We prove that the EMD-L 1 is formally equivalent to the original EMD with L 1 ground distance without approximation. Exploiting the L 1 metric structure, an efficient tree-based algorithm is designed to solve the EMD-L 1 computation. An empirical study demonstrates that the new algorithm has the time complexity of O(N 2), which is much faster than previously reported algorithms with super-cubic complexities. The proposed algorithm thus allows the EMD to be applied for comparing histogram-based features, which is practically impossible with previous algorithms. We conducted experiments for shape recognition and interest point matching. EMD-L 1 is applied to compare shape contexts on the widely tested MPEG7 shape dataset and SIFT image descriptors on a set of images with large deformation, illumination change and heavy noise. The results show that our EMD-L 1-based solutions outperform previously reported state-of-the-art features and distance measures in solving the two tasks.


Image Retrieval Interest Point Robust Algorithm Shape Match Shape Context 
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.
    Belongie, S., Malik, J., Puzicha, J.: Shape Matching and Object Recognition Using Shape Context. IEEE Trans. on PAMI 24(24), 509–522 (2002)Google Scholar
  2. 2.
    Cohen, S., Guibas, L.: The Earth Mover’s Distance under Transformation Sets. In: ICCV, vol. II, pp. 1076–1083 (1999)Google Scholar
  3. 3.
    Grauman, K., Darrell, T.: Fast Contour Matching Using Approximate Earth Mover’s Distance. In: CVPR, vol. I, pp. 220–227 (2004)Google Scholar
  4. 4.
    Grauman, K., Darrell, T.: The Pyramid Match Kernel: Discriminative Classification with Sets of Image Features. In: ICCV, vol. II, pp. 1458–1465 (2005)Google Scholar
  5. 5.
    Harris, C., Stephens, M.: A combined corner and edge detector. In: Alvey Vision Conference, pp. 147–151 (1988)Google Scholar
  6. 6.
    Hillier, F.S., Lieberman, G.J.: Introduction to Mathematical Programming. McGraw-Hill, New York (1990)Google Scholar
  7. 7.
    Indyk, P., Thaper, N.: Fast Image Retrieval via Embeddings. In: 3rd Workshop on Statistical and computational Theories of Vision, Nice, France (2003)Google Scholar
  8. 8.
    Ke, Y., Sukthankar, R.: PCA-SIFT: a more distinctive representation for local image descriptors. In: CVPR, vol. II, pp. 506–513 (2004)Google Scholar
  9. 9.
    Lazebnik, S., Schmid, C., Ponce, J.: A sparse texture representation using affine-invariant regions. IEEE Trans. PAMI 27(8), 1265–1278 (2005)Google Scholar
  10. 10.
    Latecki, L.J., Lakamper, R., Eckhardt, U.: Shape Descriptors for Non-rigid Shapes with a Single Closed Contour. In: CVPR, vol. I, pp. 424–429 (2000)Google Scholar
  11. 11.
    Levina, E., Bickel, P.: The Earth Mover’s Distance is the Mallows Distance: Some Insights from Statistics. In: ICCV, pp. 251–256 (2001)Google Scholar
  12. 12.
    Lin, J.: Divergence measures based on the Shannon entropy. IEEE Trans. On Information Theory 37(1), 145–151 (1991)zbMATHCrossRefGoogle Scholar
  13. 13.
    Ling, H., Jacobs, D.W.: Using the Inner-Distance for Classification of Articulated Shapes. In: CVPR, vol. II, pp. 719–726 (2005)Google Scholar
  14. 14.
    Ling, H., Okada, K.: An Efficient Earth Mover’s Distance Algorithm for Robust Histogram Comparison (2006) (in submission)Google Scholar
  15. 15.
    Lowe, D.: Distinctive Image Features from Scale-Invariant Keypoints. IJCV 60(2), 91–110 (2004)CrossRefGoogle Scholar
  16. 16.
    Mikolajczyk, K., Schmid, C.: A Performance Evaluation of Local Descriptors. IEEE Trans. on PAMI 27(10), 1615–1630 (2005)Google Scholar
  17. 17.
    Mokhtarian, F., Abbasi, S., Kittler, J.: Efficient and Robust Retrieval by Shape Content through Curvature Scale Space. In: Image Databases and Multi-Media Search, pp. 51–58. World Scientific, Singapore (1997)Google Scholar
  18. 18.
    Mortensen, E.N., Deng, H., Shapiro, L.: A SIFT Descriptor with Global Context. In: CVPR, vol. I, pp. 184–190 (2005)Google Scholar
  19. 19.
    Peleg, S., Werman, M., Rom, H.: A Unified Approach to the Change of Resolution:Space and Gray-level. IEEE Trans. on PAMI 11, 739–742 (1989)Google Scholar
  20. 20.
    Rubner, Y., Tomasi, C., Guibas, L.J.: The Earth Mover’s Distance as a Metric for Image Retrieval. IJCV 40(2), 99–121 (2000)zbMATHCrossRefGoogle Scholar
  21. 21.
    Sebastian, T.B., Klein, P.N., Kimia, B.B.: On Aligning Curves. IEEE Trans. on PAMI 25(1), 116–125 (2003)Google Scholar
  22. 22.
    Thayananthan, A., Stenger, B., Torr, P.H.S., Cipolla, R.: Shape Context and Chamfer Matching in Cluttered Scenes. In: CVPR, vol. I, pp. 127–133 (2003)Google Scholar
  23. 23.
    Tu, Z., Yuille, A.L.: Shape Matching and Recognition-Using Generative Models and Informative Features. In: Pajdla, T., Matas, J(G.) (eds.) ECCV 2004. LNCS, vol. 3023, pp. 195–209. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  24. 24.
    Wesolowsky, G.: TheWeber Problem: History and Perspectives. Location Science 1(1), 5–23 (1993)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Haibin Ling
    • 1
  • Kazunori Okada
    • 2
  1. 1.Computer Science Dept., Center for Automation ResearchUniversity of MarylandCollege ParkUSA
  2. 2.Imaging and Visualization DepartmentSiemens Corporate Research, Inc.PrincetonUSA

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