Usage example: Euler characteristic curves
==========================================

This notebook provides usage examples for the ``euchar.curve`` module.

-  Euler characteristic curves of 2D and 3D images with values sampled
   from uniform distributions.
-  Euler characteristic curves of finite point sets in
   :math:`\mathbb{R}^2` and :math:`\mathbb{R}^3`, obtained using the
   Alpha filtrations on the finite point sets

--------------

.. code:: ipython3

    import numpy as np

.. code:: ipython3

    import euchar.utils
    from euchar.curve import image_2D, image_3D, filtration
    from euchar.filtrations import alpha_filtration_2D, alpha_filtration_3D, inverse_density_filtration
    from euchar.display import piecewise_constant_curve

.. code:: ipython3

    import matplotlib.pyplot as plt
    plt.style.use("seaborn-whitegrid")
    plt.rcParams.update({"font.size": 16})
    from seaborn import distplot

--------------

Synthetic data
--------------

Images

.. code:: ipython3

    np.random.seed(0)
    m = 32
    max_intensity = 256
    
    img_2D = np.random.randint(0, max_intensity, size=(m, m))
    img_3D = np.random.randint(0, max_intensity, size=(m, m, m))

Finite point sets

.. code:: ipython3

    np.random.seed(0)
    N = 100
    points_2D = np.random.rand(N, 2)
    points_3D = np.random.rand(N, 3)

--------------

Euler characteristic curves of 2D and 3D image
----------------------------------------------

For the following computation, the ``vector_2D_changes`` is
automatically computed by ``image_2D()``.

.. code:: ipython3

    ecc_2D = image_2D(img_2D)

To avoid recomputing it every time, it can be passed as a parameter to
``image_2D()``.

Precompute it with

::

   vector_2D_changes = euchar.utils.vector_all_euler_changes_in_2D_images()

For the following computation, the vector of all possible Euler changes
in the case of 3D images needs to be precomputed and saved to a file.

For example one could do this by running

::

   vector_3D_changes = euchar.utils.vector_all_euler_changes_in_3D_images()
   np.save("vector_3D_changes.npy", vector_3D_changes)

.. code:: ipython3

    vector_3D_changes = np.load("vector_3D_changes.npy")
    ecc_3D = image_3D(img_3D, vector_3D_changes)

We can then plot the Euler characteristic curves as piecewise constant
curves.

.. code:: ipython3

    fig, ax = plt.subplots(1, 2, figsize=(14,4))
    plt.subplots_adjust(wspace=0.3)
    
    domain = np.arange(256)
    ax[0].plot(domain, ecc_2D, color="royalblue")
    ax[0].set(title="Euler char curve - 2D image", xlim=[-20, 280], ylim=[-150, 150])
    ax[1].plot(domain, ecc_3D, color="royalblue")
    ax[1].set(title="Euler char curve - 3D image", xlim=[-20, 280], ylim=[-3_800, 3_800]);



.. image:: output_17_0.png


--------------

Euler characteristic curves of finite point sets
------------------------------------------------

We obtain the Alpha filtrations in the form of arrays of indices of
``points_2D`` and ``points_3D``.

.. code:: ipython3

    simplices_2D, alpha_2D = alpha_filtration_2D(points_2D)
    simplices_3D, alpha_3D = alpha_filtration_3D(points_3D)

.. code:: ipython3

    fig, ax = plt.subplots(1, 2, figsize=(14,4))
    plt.subplots_adjust(wspace=0.3)
    _ = distplot(alpha_2D, ax=ax[0])
    ax[0].set(title="Distribution 2D miniball radiuses")
    _ = distplot(alpha_3D, ax=ax[1])
    ax[1].set(title="Distribution 3D miniball radiuses");



.. image:: output_21_0.png


We produce arrays ``bins_2D`` and ``bins_3D``, used to discretize the
domains of the distributions of 2D and 3D Alpha parametrizations.

.. code:: ipython3

    bins_2D = np.linspace(0.0, 0.8,  num=200)
    filt_2D = filtration(simplices_2D, alpha_2D, bins_2D)
    
    bins_3D = np.linspace(0.0, 1, num=200)
    filt_3D = filtration(simplices_3D, alpha_3D, bins_3D)

We plot the resulting Euler characteristic curves as piecewise constant
curves.

.. code:: ipython3

    fig, ax = plt.subplots(1, 2, figsize=(14,4))
    plt.subplots_adjust(wspace=0.3)
    
    ax[0].plot(bins_2D, filt_2D, color="royalblue")
    ax[0].set(title="Euler char curve - 2D points", xlim=[-0.02, 0.82], ylim=[-50, 150])
    ax[1].plot(bins_3D, filt_3D, color="royalblue")
    ax[1].set(title="Euler char curve - 3D points", xlim=[-0.02, 1.02], ylim=[-50, 150]);



.. image:: output_25_0.png