.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/segmentation/plot_euler_number.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code or to run this example in your browser via Binder .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_segmentation_plot_euler_number.py: ========================= Euler number ========================= This example shows an illustration of the computation of the Euler number [1]_ in 2D and 3D objects. For 2D objects, the Euler number is the number of objects minus the number of holes. Notice that if a neighborhood of 8 connected pixels (2-connectivity) is considered for objects, then this amounts to considering a neighborhood of 4 connected pixels (1-connectivity) for the complementary set (holes, background) , and conversely. It is also possible to compute the number of objects using :func:`skimage.measure.label`, and to deduce the number of holes from the difference between the two numbers. For 3D objects, the Euler number is obtained as the number of objects plus the number of holes, minus the number of tunnels, or loops. If one uses 3-connectivity for an object (considering the 26 surrounding voxels as its neighborhood), this corresponds to using 1-connectivity for the complementary set (holes, background), that is considering only 6 neighbors for a given voxel. The voxels are represented here with blue transparent surfaces. Inner porosities are represented in red. .. [1] https://en.wikipedia.org/wiki/Euler_characteristic .. GENERATED FROM PYTHON SOURCE LINES 27-66 .. code-block:: Python from skimage.measure import euler_number, label import matplotlib.pyplot as plt import numpy as np # Sample image. SAMPLE = np.array( [ [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0], [0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0], [1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0], [0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1], [0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], ] ) SAMPLE = np.pad(SAMPLE, 1, mode='constant') fig, ax = plt.subplots() ax.imshow(SAMPLE, cmap=plt.cm.gray) ax.axis('off') e4 = euler_number(SAMPLE, connectivity=1) object_nb_4 = label(SAMPLE, connectivity=1).max() holes_nb_4 = object_nb_4 - e4 e8 = euler_number(SAMPLE, connectivity=2) object_nb_8 = label(SAMPLE, connectivity=2).max() holes_nb_8 = object_nb_8 - e8 ax.set_title( f'Euler number for N4: {e4} ({object_nb_4} objects, {holes_nb_4} ' f'holes), \n for N8: {e8} ({object_nb_8} objects, ' f'{holes_nb_8} holes)' ) plt.show() .. image-sg:: /auto_examples/segmentation/images/sphx_glr_plot_euler_number_001.png :alt: Euler number for N4: 2 (2 objects, 0 holes), for N8: 0 (1 objects, 1 holes) :srcset: /auto_examples/segmentation/images/sphx_glr_plot_euler_number_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 67-74 3-D objects =========== In this example, a 3-D cube is generated, then holes and tunnels are added. Euler number is evaluated with 6 and 26 neighborhood configuration. This code is inpired by https://matplotlib.org/devdocs/gallery/mplot3d/voxels_numpy_logo.html .. GENERATED FROM PYTHON SOURCE LINES 74-156 .. code-block:: Python def make_ax(grid=False): ax = plt.figure().add_subplot(projection='3d') ax.grid(grid) ax.set_axis_off() return ax def explode(data): """visualization to separate voxels Data voxels are separated by 0-valued ones so that they appear separated in the matplotlib figure. """ size = np.array(data.shape) * 2 data_e = np.zeros(size - 1, dtype=data.dtype) data_e[::2, ::2, ::2] = data return data_e # shrink the gaps between voxels def expand_coordinates(indices): """ This collapses together pairs of indices, so that the gaps in the volume array will have a zero width. """ x, y, z = indices x[1::2, :, :] += 1 y[:, 1::2, :] += 1 z[:, :, 1::2] += 1 return x, y, z def display_voxels(volume): """ volume: (N,M,P) array Represents a binary set of pixels: objects are marked with 1, complementary (porosities) with 0. The voxels are actually represented with blue transparent surfaces. Inner porosities are represented in red. """ # define colors red = '#ff0000ff' blue = '#1f77b410' # upscale the above voxel image, leaving gaps filled = explode(np.ones(volume.shape)) fcolors = explode(np.where(volume, blue, red)) # Shrink the gaps x, y, z = expand_coordinates(np.indices(np.array(filled.shape) + 1)) # Define 3D figure and place voxels ax = make_ax() ax.voxels(x, y, z, filled, facecolors=fcolors) # Compute Euler number in 6 and 26 neighborhood configuration, that # correspond to 1 and 3 connectivity, respectively e26 = euler_number(volume, connectivity=3) e6 = euler_number(volume, connectivity=1) plt.title(f'Euler number for N26: {e26}, for N6: {e6}') plt.show() # Define a volume of 7x7x7 voxels n = 7 cube = np.ones((n, n, n), dtype=bool) # Add a tunnel c = int(n / 2) cube[c, :, c] = False # Add a new hole cube[int(3 * n / 4), c - 1, c - 1] = False # Add a hole in neighborhood of previous one cube[int(3 * n / 4), c, c] = False # Add a second tunnel cube[:, c, int(3 * n / 4)] = False display_voxels(cube) .. image-sg:: /auto_examples/segmentation/images/sphx_glr_plot_euler_number_002.png :alt: Euler number for N26: 1, for N6: 0 :srcset: /auto_examples/segmentation/images/sphx_glr_plot_euler_number_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 0.732 seconds) .. _sphx_glr_download_auto_examples_segmentation_plot_euler_number.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: binder-badge .. image:: images/binder_badge_logo.svg :target: https://mybinder.org/v2/gh/scikit-image/scikit-image/v0.23.1?filepath=notebooks/auto_examples/segmentation/plot_euler_number.ipynb :alt: Launch binder :width: 150 px .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_euler_number.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_euler_number.py ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_