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 


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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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