.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/barycenters/plot_free_support_barycenter.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_examples_barycenters_plot_free_support_barycenter.py>`
        to download the full example code

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_barycenters_plot_free_support_barycenter.py:


========================================================
2D free support Wasserstein barycenters of distributions
========================================================

Illustration of 2D Wasserstein and Sinkhorn barycenters if distributions are weighted
sum of Diracs.

.. GENERATED FROM PYTHON SOURCE LINES 11-25

.. code-block:: Python


    # Authors: Vivien Seguy <vivien.seguy@iip.ist.i.kyoto-u.ac.jp>
    #          Rémi Flamary <remi.flamary@polytechnique.edu>
    #          Eduardo Fernandes Montesuma <eduardo.fernandes-montesuma@universite-paris-saclay.fr>
    #
    # License: MIT License

    # sphinx_gallery_thumbnail_number = 2

    import numpy as np
    import matplotlib.pylab as pl
    import ot









.. GENERATED FROM PYTHON SOURCE LINES 26-28

Generate data
-------------

.. GENERATED FROM PYTHON SOURCE LINES 28-51

.. code-block:: Python


    N = 2
    d = 2

    I1 = pl.imread('../../data/redcross.png').astype(np.float64)[::4, ::4, 2]
    I2 = pl.imread('../../data/duck.png').astype(np.float64)[::4, ::4, 2]

    sz = I2.shape[0]
    XX, YY = np.meshgrid(np.arange(sz), np.arange(sz))

    x1 = np.stack((XX[I1 == 0], YY[I1 == 0]), 1) * 1.0
    x2 = np.stack((XX[I2 == 0] + 80, -YY[I2 == 0] + 32), 1) * 1.0
    x3 = np.stack((XX[I2 == 0], -YY[I2 == 0] + 32), 1) * 1.0

    measures_locations = [x1, x2]
    measures_weights = [ot.unif(x1.shape[0]), ot.unif(x2.shape[0])]

    pl.figure(1, (12, 4))
    pl.scatter(x1[:, 0], x1[:, 1], alpha=0.5)
    pl.scatter(x2[:, 0], x2[:, 1], alpha=0.5)
    pl.title('Distributions')





.. image-sg:: /auto_examples/barycenters/images/sphx_glr_plot_free_support_barycenter_001.png
   :alt: Distributions
   :srcset: /auto_examples/barycenters/images/sphx_glr_plot_free_support_barycenter_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    Text(0.5, 1.0, 'Distributions')



.. GENERATED FROM PYTHON SOURCE LINES 52-54

Compute free support Wasserstein barycenter
-------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 54-61

.. code-block:: Python


    k = 200  # number of Diracs of the barycenter
    X_init = np.random.normal(0., 1., (k, d))  # initial Dirac locations
    b = np.ones((k,)) / k  # weights of the barycenter (it will not be optimized, only the locations are optimized)

    X = ot.lp.free_support_barycenter(measures_locations, measures_weights, X_init, b)








.. GENERATED FROM PYTHON SOURCE LINES 62-64

Plot the Wasserstein barycenter
-------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 64-73

.. code-block:: Python


    pl.figure(2, (8, 3))
    pl.scatter(x1[:, 0], x1[:, 1], alpha=0.5)
    pl.scatter(x2[:, 0], x2[:, 1], alpha=0.5)
    pl.scatter(X[:, 0], X[:, 1], s=b * 1000, marker='s', label='2-Wasserstein barycenter')
    pl.title('Data measures and their barycenter')
    pl.legend(loc="lower right")
    pl.show()




.. image-sg:: /auto_examples/barycenters/images/sphx_glr_plot_free_support_barycenter_002.png
   :alt: Data measures and their barycenter
   :srcset: /auto_examples/barycenters/images/sphx_glr_plot_free_support_barycenter_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 74-75

Compute free support Sinkhorn barycenter

.. GENERATED FROM PYTHON SOURCE LINES 75-82

.. code-block:: Python


    k = 200  # number of Diracs of the barycenter
    X_init = np.random.normal(0., 1., (k, d))  # initial Dirac locations
    b = np.ones((k,)) / k  # weights of the barycenter (it will not be optimized, only the locations are optimized)

    X = ot.bregman.free_support_sinkhorn_barycenter(measures_locations, measures_weights, X_init, 20, b, numItermax=15)








.. GENERATED FROM PYTHON SOURCE LINES 83-85

Plot the Wasserstein barycenter
-------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 85-93

.. code-block:: Python


    pl.figure(2, (8, 3))
    pl.scatter(x1[:, 0], x1[:, 1], alpha=0.5)
    pl.scatter(x2[:, 0], x2[:, 1], alpha=0.5)
    pl.scatter(X[:, 0], X[:, 1], s=b * 1000, marker='s', label='2-Wasserstein barycenter')
    pl.title('Data measures and their barycenter')
    pl.legend(loc="lower right")
    pl.show()



.. image-sg:: /auto_examples/barycenters/images/sphx_glr_plot_free_support_barycenter_003.png
   :alt: Data measures and their barycenter
   :srcset: /auto_examples/barycenters/images/sphx_glr_plot_free_support_barycenter_003.png
   :class: sphx-glr-single-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 1.701 seconds)


.. _sphx_glr_download_auto_examples_barycenters_plot_free_support_barycenter.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_free_support_barycenter.ipynb <plot_free_support_barycenter.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_free_support_barycenter.py <plot_free_support_barycenter.py>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_