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.. "auto_examples\HeterogeneousFields\plot_amfbf1.py"
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.. only:: html

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

        Click :ref:`here <sphx_glr_download_auto_examples_HeterogeneousFields_plot_amfbf1.py>`
        to download the full example code

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

.. _sphx_glr_auto_examples_HeterogeneousFields_plot_amfbf1.py:


============================================
Anisotropic Multifractional Brownian field 1
============================================

.. codeauthor:: Frédéric Richard <frederic.richard_at_univ-amu.fr>

In this example, we show how to simulate heterogeneous textures whose
pattern orientation spatially varies. Such textures can be obtained
by simulating a turning band field with a space-dependent Hurst function
represented as a step function.

Here, the Hurst function :math:`\beta_{(x,y)}` is defined at position
:math:`(x, y)` by translating an initial Hurst function :math:`\beta_0`

.. math::
        \beta_{(x,y)}(s)  = \beta_0\left(s - \frac{y}{N} \phi_1\right),
        \forall s \in \mathbb{R}.

for some fixed orientation :math:`\phi_1`. The orientation of the
generated texture varies in the vertical direction while its roughness and
degree of anisotropy remain the same. Such a simulation
was used for illustration and experiments in :cite:p:`Vu2020`.

.. note::
  Such simulations are related to the so-called
  **anisotropic multifractional Brownian field**
  which was investigated :cite:p:`Vu2020`. In a harmonizable
  representation, these fields are defined as

  .. math::
      Z(x, y) = \int_{\mathbb{R}^2}
      (e^{i\langle (x, y), w \rangle} -1 ) \tau_{(x, y)}(w)
      \vert w \vert^{-\beta_{x, y}(w) -1}
      dW(w),

  where :math:`dW` stands for a complex Brownian measure.

.. GENERATED FROM PYTHON SOURCE LINES 78-127



.. image:: /auto_examples/HeterogeneousFields/images/sphx_glr_plot_amfbf1_001.png
    :alt: Anisotropic Multifractional Brownian field
    :class: sphx-glr-single-img





.. code-block:: default

    import numpy as np
    from afbf import coordinates, sdata, tbfield

    N = 256  # Image size.

    pid = np.pi / 8  # length of a step of the Hurst function.
    dpid = np.pi / 16  # length of the transition between steps.
    phi1 = np.pi / 2  # orientation at y=N-1

    # Define a field.
    Z = tbfield('afbf-smooth')

    # Change the parameters of the Hurst function.
    Z.hurst.ChangeParameters(
        np.array([0.8, 0.2]),
        np.array([- pid - dpid, - pid, pid, pid + dpid])
        )
    Z.hurst.trans = 1  # Indicate where to locate step transition.
    Z.NormalizeModel()  # Normalize the model.

    # Set coordinates.
    coord = coordinates(N)  # field coordinates.
    coordy = coordinates()  # subset of coordinates.
    x = np.reshape(np.arange(1, N+1), (N, 1))

    # Initialize an image.
    z = sdata(coord)
    z.name = 'Anisotropic Multifractional Brownian field'

    for y in range(N):
        # Translate the Hurst and topothesy functions.
        trans = y / (N - 1) * phi1
        Z.hurst.ApplyTransforms(trans)
        Z.topo.ApplyTransforms(trans)

        # Restrict to coordinates (:, y).
        coordy.DefineNonUniformLocations(
            np.concatenate((x, (y+1) * np.ones((N, 1))), axis=1))
        coordy.N = coord.N

        # Simulate the current field keeping a same random seed.
        np.random.seed(3)
        zy = Z.Simulate(coordy)

        # Keep the simulation at positions  (y, :).
        z.values.reshape(N, N)[y, :] = zy.values.reshape(1, N)[0, 0:N]

    # Display the simulation.
    z.Display()


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

   **Total running time of the script:** ( 8 minutes  50.157 seconds)


.. _sphx_glr_download_auto_examples_HeterogeneousFields_plot_amfbf1.py:


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 .. container:: sphx-glr-footer
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