PyEMD: Fast EMD for Python

PyEMD is a Python wrapper for Ofir Pele and Michael Werman's implementation of the Earth Mover's Distance that allows it to be used with NumPy. If you use this code, please cite the papers listed at the end of this document.

Usage

>>> from pyemd import emd
>>> import numpy as np
>>> first_histogram = np.array([0.0, 1.0])
>>> second_histogram = np.array([5.0, 3.0])
>>> distance_matrix = np.array([[0.0, 0.5],
...                             [0.5, 0.0]])
>>> emd(first_histogram, second_histogram, distance_matrix)
3.5

You can also get the associated minimum-cost flow:

>>> from pyemd import emd_with_flow
>>> emd_with_flow(first_histogram, second_histogram, distance_matrix)
(3.5, [[0.0, 0.0], [0.0, 1.0]])

You can also calculate the EMD directly from two arrays of observations:

>>> from pyemd import emd_samples
>>> first_array = [1, 2, 3, 4]
>>> second_array = [2, 3, 4, 5]
>>> emd_samples(first_array, second_array, bins=2)
0.5

API Documentation

emd()

emd(first_histogram,
    second_histogram,
    distance_matrix,
    extra_mass_penalty=-1.0)

Arguments:

• first_histogram (np.ndarray): A 1D array of type np.float64 of length N.
• second_histogram (np.ndarray): A 1D array of np.float64 of length N.
• distance_matrix (np.ndarray): A 2D array of np.float64, of size at least N × N. This defines the underlying metric, or ground distance, by giving the pairwise distances between the histogram bins. NOTE: It must represent a metric; there is no warning if it doesn't.

Keyword Arguments:

• extra_mass_penalty (float): The penalty for extra mass. If you want the resulting distance to be a metric, it should be at least half the diameter of the space (maximum possible distance between any two points). If you want partial matching you can set it to zero (but then the resulting distance is not guaranteed to be a metric). The default value is -1.0, which means the maximum value in the distance matrix is used.

Returns: (float) The EMD value.

---

emd_with_flow()

emd_with_flow(first_histogram,
              second_histogram,
              distance_matrix,
              extra_mass_penalty=-1.0)

Arguments are the same as for emd().

Returns: (tuple(float, list(list(float)))) The EMD value and the associated minimum-cost flow.

---

emd_samples()

emd_samples(first_array,
            second_array,
            extra_mass_penalty=-1.0,
            distance='euclidean',
            normalized=True,
            bins='auto',
            range=None)

Arguments:

• first_array (Iterable): An array of samples used to generate a histogram.
• second_array (Iterable): An array of samples used to generate a histogram.

Keyword Arguments:

• extra_mass_penalty (float): Same as for emd().
• distance (string or function): A string or function implementing a metric on a 1D np.ndarray. Defaults to the Euclidean distance. Currently limited to 'euclidean' or your own function, which must take a 1D array and return a square 2D array of pairwise distances.
• normalized (boolean): If true (default), treat histograms as fractions of the dataset. If false, treat histograms as counts. In the latter case the EMD will vary greatly by array length.
• bins (int or string): The number of bins to include in the generated histogram. If a string, must be one of the bin selection algorithms accepted by np.histogram(). Defaults to 'auto', which gives the maximum of the 'sturges' and 'fd' estimators.
• range (tuple(int, int)): The lower and upper range of the bins, passed to numpy.histogram(). Defaults to the range of the union of first_array and second_array. Note: if the given range is not a superset of the default range, no warning will be given.

Returns: (float) The EMD value between the histograms of first_array and second_array.

---

Limitations and Caveats

• emd() and emd_with_flow():
  • The distance_matrix is assumed to represent a metric; there is no check to ensure that this is true. See the documentation in pyemd/lib/emd_hat.hpp for more information.
  • The histograms and distance matrix must be numpy arrays of type np.float64. The original C++ template function can accept any numerical C++ type, but this wrapper only instantiates the template with double (Cython converts np.float64 to double). If there's demand, I can add support for other types.
• emd_with_flow():
  • The flow matrix does not contain the flows to/from the extra mass bin.
• emd_samples():
  • With numpy < 1.15.0, using the default bins='auto' results in an extra call to np.histogram() to determine the bin lengths, since the NumPy bin-selectors are not exposed in the public API. For performance, you may want to set the bins yourself. If numpy >= 1.15 is available, np.histogram_bin_edges() is called instead, which is more efficient.

Credit

• All credit for the actual algorithm and implementation goes to Ofir Pele and Michael Werman. See the relevant paper.
• Thanks to the Cython developers for making this kind of wrapper relatively easy to write.

Please cite these papers if you use this code:

Ofir Pele and Michael Werman. Fast and robust earth mover's distances. Proc. 2009 IEEE 12th Int. Conf. on Computer Vision, Kyoto, Japan, 2009, pp. 460-467.

@INPROCEEDINGS{pele2009,
  title={Fast and robust earth mover's distances},
  author={Pele, Ofir and Werman, Michael},
  booktitle={2009 IEEE 12th International Conference on Computer Vision},
  pages={460--467},
  year={2009},
  month={September},
  organization={IEEE}
}

Ofir Pele and Michael Werman. A linear time histogram metric for improved SIFT matching. Computer Vision - ECCV 2008, Marseille, France, 2008, pp. 495-508.

@INPROCEEDINGS{pele2008,
  title={A linear time histogram metric for improved sift matching},
  author={Pele, Ofir and Werman, Michael},
  booktitle={Computer Vision--ECCV 2008},
  pages={495--508},
  year={2008},
  month={October},
  publisher={Springer}
}