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

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

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

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

.. _sphx_glr_auto_examples_HeterogeneousFields_plot_mfbf.py:


==============================
Multifractional Brownian field
==============================

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

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

Here, the Hurst function is defined at position
:math:`(x, y)` by the constant function

.. math::
        \beta_{(x,y)}(w) \equiv H_y = \left(\frac{y}{N}\right) h_0
        + \left(\frac{N-y}{N}\right) h_1,

for some fixed values :math:`h_0, h_1`. This leads to a texture whose
roughness linearly varies in the vertical direction. Such a simulation
was used for illustration and experiments in :cite:p:`Vu2020`.

.. note::
  Such simulations are related to the so-called
  **multifractional Brownian field**
  which was defined :cite:p:`Benassi97,Peltier96`. In a harmonizable
  representation, these fields are of the form

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

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

.. GENERATED FROM PYTHON SOURCE LINES 76-118



.. image:: /auto_examples/HeterogeneousFields/images/sphx_glr_plot_mfbf_001.png
    :alt: 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.
    h0 = 0.1  # Hurst index at (x, y) with y=0.
    h1 = 0.9  # Hurst index at (x, y) with y=N-1.

    # Define a fractional Brownian field.
    Z = tbfield()

    # 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 = 'Multifractional Brownian field'

    for y in range(N):
        # Set the Hurst index at local value h.
        t = y / (N - 1)
        h = h0 * (1 - t) + h1 * t
        # Change values of the topothesy and Hurst functions.
        Z.hurst.ChangeParameters(np.array([h]))  # Change the Hurst index.
        # Normalization.
        Z.NormalizeModel()

        # 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(1)
        zy = Z.Simulate(coordy)

        # Keep the simulation at positions  (y, :).
        z.values.reshape(z.M)[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:** ( 9 minutes  1.159 seconds)


.. _sphx_glr_download_auto_examples_HeterogeneousFields_plot_mfbf.py:


.. only :: html

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



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

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



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


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