# Earth movers distance

PyEMD: Fast EMD for Python
June 6, 2017 – 03:45 pm

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.

### Installation

To install the latest release:

Before opening an issue related to installation, please try to install PyEMD in a fresh, empty Python 3 virtual environment and check that the problem persists.

### 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]])

### API

emd(first_histogram, second_histogram, distance_matrix)

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

The arguments to emd_with_flow are the same.

### Limitations and Caveats

• 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 flow matrix does not contain the flows to/from the extra mass bin.
• 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.

### Contributing

To help develop PyEMD, fork the project on GitHub and install the requirements with pip.

The Makefile defines some tasks to help with development:

• default: compile the Cython code into C++ and build the C++ into a Python extension, using the setup.py build command
• build: same as default, but using the cython command
• clean: remove the build directory and the compiled C++ extension
• test: run unit tests with py.test

Tests for different Python environments can be run by installing tox with pip install tox and running the tox command.

### Credit

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

#### Please cite these papers if you use this code:

Ofir Pele and Michael Werman, "A linear time histogram metric for improved SIFT matching, " in 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} }

Ofir Pele and Michael Werman, "Fast and robust earth mover's distances, " in 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} }

Source: github.com
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