.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/special_functions/wavelets/plot_01_haar_wavelet.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_api_gallery_special_functions_wavelets_plot_01_haar_wavelet.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

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

.. _sphx_glr_api_gallery_special_functions_wavelets_plot_01_haar_wavelet.py:


Haar's wavelet: averages and differences at every scale
=======================================================

Haar's 1910 orthonormal system of step functions splits a signal into a
coarse average plus details at each scale: pairwise sums and differences
over sqrt 2, repeated on the sums. A jump shows up as a single large
coefficient per level, exactly where it happens.

.. GENERATED FROM PYTHON SOURCE LINES 12-18

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.special_functions import discrete_wavelet_transform, inverse_discrete_wavelet_transform
    from mathematicskit.special_functions.visualizers.plots import plot_wavelet_decomposition








.. GENERATED FROM PYTHON SOURCE LINES 19-21

One level by hand
-----------------

.. GENERATED FROM PYTHON SOURCE LINES 21-27

.. code-block:: Python


    x = np.array([4.0, 2.0, 5.0, 5.0, 1.0, 3.0, 0.0, 0.0])
    one = discrete_wavelet_transform(x, "haar", level=1)
    print("pairwise sums / sqrt 2       :", np.round(one.approximation, 4))
    print("pairwise differences / sqrt 2:", np.round(one.details[0], 4))





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

 .. code-block:: none

    pairwise sums / sqrt 2       : [4.2426 7.0711 2.8284 0.    ]
    pairwise differences / sqrt 2: [ 1.4142  0.     -1.4142  0.    ]




.. GENERATED FROM PYTHON SOURCE LINES 28-30

A piecewise-constant signal is sparse in the Haar basis
-------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 30-43

.. code-block:: Python


    n = 256
    signal = np.where(np.arange(n) < 100, 1.0, -0.5) + np.where((np.arange(n) >= 170) & (np.arange(n) < 200), 2.0, 0.0)
    result = discrete_wavelet_transform(signal, "haar", level=5)
    coefficients = np.concatenate([result.approximation, *result.details])
    print(f"nonzero Haar coefficients: {np.sum(np.abs(coefficients) > 1e-12)} of {n}")
    print(f"perfect reconstruction error: {np.max(np.abs(inverse_discrete_wavelet_transform(result) - signal)):.1e}")

    fig, (ax0, ax1) = plt.subplots(2, 1, figsize=(8, 7))
    ax0.step(np.arange(n), signal, where="post")
    ax0.set_title("Piecewise-constant signal")
    plot_wavelet_decomposition(result, ax=ax1)
    fig.tight_layout()



.. image-sg:: /api/gallery/special_functions/wavelets/images/sphx_glr_plot_01_haar_wavelet_001.png
   :alt: Piecewise-constant signal, Wavelet decomposition (haar)
   :srcset: /api/gallery/special_functions/wavelets/images/sphx_glr_plot_01_haar_wavelet_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    nonzero Haar coefficients: 16 of 256
    perfect reconstruction error: 8.9e-16





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

   **Total running time of the script:** (0 minutes 0.147 seconds)


.. _sphx_glr_download_api_gallery_special_functions_wavelets_plot_01_haar_wavelet.py:

.. only:: html

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

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/special_functions/wavelets/plot_01_haar_wavelet.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_haar_wavelet.ipynb <plot_01_haar_wavelet.ipynb>`

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

      :download:`Download Python source code: plot_01_haar_wavelet.py <plot_01_haar_wavelet.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_haar_wavelet.zip <plot_01_haar_wavelet.zip>`


.. only:: html

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

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