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

    Click :ref:`here <sphx_glr_download_auto_examples_edges_plot_edge_filter.py>` to download the full example code
.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_edges_plot_edge_filter.py:


==============
Edge operators
==============

Edge operators are used in image processing within edge detection algorithms.
They are discrete differentiation operators, computing an approximation of the
gradient of the image intensity function.




.. code-block:: python

    ==============
    Edge operators
    ==============

    Edge operators are used in image processing within edge detection algorithms.
    They are discrete differentiation operators, computing an approximation of the
    gradient of the image intensity function.

    """
    import numpy as np
    import matplotlib.pyplot as plt

    from skimage.data import camera
    from skimage.filters import roberts, sobel, sobel_h, sobel_v, scharr, \
        scharr_h, scharr_v, prewitt, prewitt_v, prewitt_h, farid_v, farid_h

    image = camera()
    edge_roberts = roberts(image)
    edge_sobel = sobel(image)

    fig, ax = plt.subplots(ncols=2, sharex=True, sharey=True,
                           figsize=(8, 4))

    ax[0].imshow(edge_roberts, cmap=plt.cm.gray)
    ax[0].set_title('Roberts Edge Detection')

    ax[1].imshow(edge_sobel, cmap=plt.cm.gray)
    ax[1].set_title('Sobel Edge Detection')

    for a in ax:
        a.axis('off')

    plt.tight_layout()
    plt.show()




.. code-block:: pytb

    Traceback (most recent call last):
      File "/build/skimage-Lp2Zl4/skimage-0.16.2/doc/examples/edges/plot_edge_filter.py", line 1
        ==============
        ^
    SyntaxError: invalid syntax




Different operators compute different finite-difference approximations of
the gradient. For example, the Scharr filter results in a less rotational
variance than the Sobel filter that is in turn better than the Prewitt
filter [1]_ [2]_ [3]_. The difference between the Prewitt and Sobel filters
and the Scharr filter is illustrated below with an image that is the
discretization of a rotation- invariant continuous function. The
discrepancy between the Prewitt and Sobel filters, and the Scharr filter is
stronger for regions of the image where the direction of the gradient is
close to diagonal, and for regions with high spatial frequencies. For the
example image the differences between the filter results are very small and
the filter results are visually almost indistinguishable.

.. [1] https://en.wikipedia.org/wiki/Sobel_operator#Alternative_operators

.. [2] B. Jaehne, H. Scharr, and S. Koerkel. Principles of filter design.
       In Handbook of Computer Vision and Applications. Academic Press,
       1999.

.. [3] https://en.wikipedia.org/wiki/Prewitt_operator



.. code-block:: python


    x, y = np.ogrid[:100, :100]
    # Rotation-invariant image with different spatial frequencies
    img = np.exp(1j * np.hypot(x, y) ** 1.3 / 20.).real

    edge_sobel = sobel(img)
    edge_scharr = scharr(img)
    edge_prewitt = prewitt(img)

    diff_scharr_prewitt = edge_scharr - edge_prewitt
    diff_scharr_sobel = edge_scharr - edge_sobel
    max_diff = np.max(np.maximum(diff_scharr_prewitt, diff_scharr_sobel))

    fig, axes = plt.subplots(nrows=2, ncols=2, sharex=True, sharey=True,
                             figsize=(8, 8))
    ax = axes.ravel()

    ax[0].imshow(img, cmap=plt.cm.gray)
    ax[0].set_title('Original image')

    ax[1].imshow(edge_scharr, cmap=plt.cm.gray)
    ax[1].set_title('Scharr Edge Detection')

    ax[2].imshow(diff_scharr_prewitt, cmap=plt.cm.gray, vmax=max_diff)
    ax[2].set_title('Scharr - Prewitt')

    ax[3].imshow(diff_scharr_sobel, cmap=plt.cm.gray, vmax=max_diff)
    ax[3].set_title('Scharr - Sobel')

    for a in ax:
        a.axis('off')

    plt.tight_layout()
    plt.show()


As in the previous example, here we illustrate the rotational invariance of
the filters. The top row shows a rotationally invariant image along with the
angle of its analytical gradient. The other two rows contain the difference
between the different gradient approximations (Sobel, Prewitt, Scharr &
Farid) and analytical gradient.

The Farid & Simoncelli derivative filters [4]_, [5]_  are the most
rotationally invariant, but require a 5x5 kernel, which is computationally
more intensive than a 3x3 kernel.

.. [4] Farid, H. and Simoncelli, E. P., "Differentiation of discrete
       multidimensional signals", IEEE Transactions on Image Processing 13(4):
       496-508, 2004. :DOI:`10.1109/TIP.2004.823819`

.. [5] Wikipedia, "Farid and Simoncelli Derivatives." Available at:
       <https://en.wikipedia.org/wiki/Image_derivatives#Farid_and_Simoncelli_Derivatives>



.. code-block:: python



    x, y = np.mgrid[-10:10:255j, -10:10:255j]
    img = np.sin(x ** 2 + y ** 2)

    imgx = 2 * x * np.cos(x ** 2 + y ** 2)
    imgy = 2 * y * np.cos(x ** 2 + y ** 2)


    def angle(dx, dy):
        return np.mod(np.arctan2(dy, dx), np.pi)


    true_angle = angle(imgx, imgy)

    angle_farid = angle(farid_h(img), farid_v(img))
    angle_sobel = angle(sobel_h(img), sobel_v(img))
    angle_scharr = angle(scharr_h(img), scharr_v(img))
    angle_prewitt = angle(prewitt_h(img), prewitt_v(img))


    def diff_angle(angle_1, angle_2):
        return np.minimum(np.pi - np.abs(angle_1 - angle_2),
                          np.abs(angle_1 - angle_2))


    diff_farid = diff_angle(true_angle, angle_farid)
    diff_sobel = diff_angle(true_angle, angle_sobel)
    diff_scharr = diff_angle(true_angle, angle_scharr)
    diff_prewitt = diff_angle(true_angle, angle_prewitt)

    fig, axes = plt.subplots(nrows=3, ncols=2, sharex=True, sharey=True,
                             figsize=(8, 8))
    ax = axes.ravel()

    ax[0].imshow(img, cmap=plt.cm.gray)
    ax[0].set_title('Original image')

    ax[1].imshow(true_angle, cmap=plt.cm.hsv)
    ax[1].set_title('Analytical gradient angle')

    ax[2].imshow(diff_sobel, cmap=plt.cm.inferno, vmin=0, vmax=0.02)
    ax[2].set_title('Sobel error')

    ax[3].imshow(diff_prewitt, cmap=plt.cm.inferno, vmin=0, vmax=0.02)
    ax[3].set_title('Prewitt error')

    ax[4].imshow(diff_scharr, cmap=plt.cm.inferno, vmin=0, vmax=0.02)
    ax[4].set_title('Scharr error')

    cax = ax[5].imshow(diff_farid, cmap=plt.cm.inferno, vmin=0, vmax=0.02)
    ax[5].set_title('Farid error')

    fig.subplots_adjust(right=0.8)
    cbar_ax = fig.add_axes([0.90, 0.10, 0.02, 0.50])
    fig.colorbar(cax, cax=cbar_ax, ticks=[0, 0.01, 0.02])

    for a in ax:
        a.axis('off')

    plt.show()

**Total running time of the script:** ( 0 minutes  0.000 seconds)


.. _sphx_glr_download_auto_examples_edges_plot_edge_filter.py:


.. only :: html

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



  .. container:: sphx-glr-download

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



  .. container:: sphx-glr-download

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


.. only:: html

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

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